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Simulation of Particle Coagulation and Breakage in a Planar Jet via the DEMM Method
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 102 (2015) 1650 – 1655
The 7th World Congress on Particle Technology (WCPT7)
Simulation of Particle Coagulation and Breakage in a Planar Jet via the DEMM method Fangyang Yuan*, Zhongli Chen , Xiaojun Pan School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China
Abstract Simultaneous numerical simulation of particle coagulation and breakage in a planar jet is performed. Large eddy simulation (LES) is adopted to calculate the flow field and the direct expansion method of moments (DEMM) is employed to describe the evolution of particles undergoing coagulation and breakage. The Aggregation-Breakage Balance Equation is converted to a set of moment equations which contains the n/2th (n=0~4) moments through the DEMM method. The newly proposed DEMM model is confirmed to be valid to describe the evolution of particle size distribution (PSD). The coherent structures dominate the evolution of particle concentration, geometric mean diameter and geometric standard deviation. The particle number concentration decreases and the span of particle diameter gets wider along the centerline as the jet develops. The geometric standard deviation is larger in the shear layer than that in the jet core. © 2015 The Authors. Published by Elsevier This isLtd. an open access article under the CC BY-NC-ND license © 2014 The Authors. Published by Ltd. Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of Chinese Society of Particuology, Institute of Process Engineering, Chinese Selection under responsibility of Chinese Society of Particuology, Institute of Process Engineering, Chinese Academy Academyand ofpeer-review Sciences (CAS). of Sciences (CAS)
Keywords: Direct expansion method of moments; Coagulation; Breakage; Planar jet
1. Introduction Turbulent flows containing particle coagulation and breakage are of great interest in many major industries processes like manufacture of fumed silica, as well as for natural phenomena like evolution of combustion smoke [14]. Simultaneous evolution of particles undergoing several dynamic processes along with the transport processes of diffusion, convection and thermophoresis is unfeasible to be described well by limited experimental methods in the fluid field [5]. Hence, methodologies which can capture the multi-process particle transport are needed. Numerical simulations have been proved effective on the study of multiphase flow [6-9]. * Corresponding author. Tel.: +86 571 86836009; fax: +86 571 86914401. E-mail address: [email protected]
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of Chinese Society of Particuology, Institute of Process Engineering, Chinese Academy of Sciences (CAS)
doi:10.1016/j.proeng.2015.01.301
1651
Fangyang Yuan et al. / Procedia Engineering 102 (2015) 1650 – 1655
The rate of change of the PSD by coagulation and breakage can be predicted by the Aggregation-Breakage Balance Equation (PABBE) [10]. For years, the method of moments (MOM)[6, 11] and the Sectional Method (SM)[5] are proposed and developed to deal with the equation with nonlinear and partial integro-differential characteristics[12]. 2. Fluid flow The flow field consists of a particle-laden flow ejecting from a planar jet into a still environment, see Fig.1. The LES equations governing the incompressible flow are obtained by filtering the N-S equation:
wui wxi
0 ,
(1)
wui wu uj i wt wx j
wW ij w 2ui 1 wp X U wxi wx j wx j wx j
,
(2)
where ui and p are the filtered velocity and pressure, respectively. W ij is the subgrid-scale(SGS) stress, which is expressed by the Smagorinsky- Lily model.
Fig. 1. Configuration of planar jet flow
3. Particle transport 3.1. PABBE and its moment transformation The transport of particles dispersed throughout the fluid is governed by the PABBE, which comes from Particle General Dynamic Equation (PGDE) proposed by Schmoluchowski and developed by Muller. The breakage process which is brought into the equation by Barthelmes et al.[10] comes about a steady state of PSD by the competition between coagulation and breakage, i.e.,
wn(v, t ) U n(v, t ) ( Dt n(v, t )) wt
1v ³ E (v1 , v v1 )n(v1 , t )n(v v1 , t )dv1 20
f
f
v
v
n(v, t ) ³ E (v1 , v)n(v1 , t )dv1 ³ a(v1 )b(v | v1 )n(v1, t )dv1 a(v)n(v, t )
ˈ
(3)
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Fangyang Yuan et al. / Procedia Engineering 102 (2015) 1650 – 1655
where n(v,t) is the particle size distribution, v and v1 are particle volumes, β(v,v1) is the volume-based aggregation kernel which describes the colloid frequency of particles with volume v and v1. In our work, the Dahneke’s expression of coagulation kernel in the entire size regime is been adopted for calculating due to the low Knudson number (Kn < 1)[13, 14]. α(v) is the volume-based breakage kernel that gives the frequency of breakage of particles of volume v , and b(v|v1) is the fragment distribution function. Here, we adopt the same breakage model as the previous work in Gan et al.[15]. 3.2. Direct expansion method of moments (DEMM) The DEMM is proposed based on the TEMOM[16] in which the closure of the moment evolution equations is achieved by means of the Taylor expansion method[13]. Following the derivation of [14], Eq.(3) can be obtained as:
wmk U mk ( Dt mk ) wt
p
p
1 k ¦ ¦ [ijk m1 / m0 mim m jm 2 1 9 m
i 0j 0
where the moment mk is defined by derivatives of the coagulation kernel
mk
p
³
f
0
p
k
1 3
§ ¨k ©
0,
· m 2m , ,..., m ¸ , p p ¹
(4)
v n v, t dv , ζ = kb((μG)/τ*)q and ξijk involves variety of partial k
E , which will be numerically determined. mk+1/3 which relates to the breakage
process can be obtained following the treatment in [11]. In this paper, we set m=2 and p=4, then the PABBE will be composed of five ordinary differential quations (k=0, 1/2, 1, 3/2, 2) and each involves five moments, namely m0, m1/2, m1, m3/2 and m2. This set of equations will be used in the present study to predict the time evolution of the PSD. 4. Results and discussion 4.1. Numerical specifications We introduce the following dimensionless parameters to simplify the calculation and analysis:
x*j
xj D,
t *j
t D U0
u *j ,
uj U0
p* ,
p UU 02
Mk ,
mk mk 0
,
(5)
where the width of the nozzle D=0.0127m which is the characteristic length, U0=(U1-U2)=(9.55-1.91)=7.64m/s is the characteristic velocity, and the Reynolds number is Re = (U1 U2)D/νair =8300. The initial values of five moments are determined according to the following formula:
mk 0
Φ0 v0k 1 k
0,1 / 2,1,3 / 2, 2 ,
(6)
in which the volume fraction at the nozzle exit Φ0 = 2.0×10-5, and the initial particle volume v0 S d03 6 , in which the initial particle diameter is d0 = 1μm. Computations are performed on a domain of L × H = 60D × 41D, which is discreted into structured grids. The model constant of Smagorinsky–Lilly model Cs is set as 0.1. 4.2. Velocity field The self-preservation profiles of the mean stream-wise velocity compared with the experimental data are shown in Fig.2. As can be seen in the figure, the mean axial velocity satisfies the self-similarity distribution at X/Dı10 and can be expressed as a Gaussian curve. The fully developed vorticity contour of the planar jet is shown in Fig.3.
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Fangyang Yuan et al. / Procedia Engineering 102 (2015) 1650 – 1655
Fig. 2. Self-preservation profiles of the mean stream-wise velocity
Fig. 3. Contour of the fully developed vorticity
4.3. Evolution of PSD 4.3.1. Particle number concentration The time-averaged moments develop along with the evolution of fluid field and arrive steady state after t*=300. Fig.4. shows the instantaneous contour of M0 at t*=373 which represents the particle number concentration. The spatial distribution of M0 resembles that of vorticity in Fig.3., which indicates that distribution of number concentration is dominated by the structure of coherent eddies. It can also be seen that the particle number concentration decreases along the centerline as the jet develops which demonstrates the effect of coagulation. The prediction of PSD proves the validity of the DEMM model.
Fig. 4. Contour of zeroth moments M0 at t*=373
4.3.2. Geometric mean diameter Coagulation and breakage of particles in the flow field bring about the change of particle size. The particle geometric diameter can be obtained by d *p
6v
* p
/ S , in which the particle volume 1/3
v*p
M1 / M 0 . The spatial-
temporal evolution of particle mean diameter is presented in Fig.5. At early stage of the development of fluid field, the particles are dominated by coagulation and size of particles grows fast along the centreline of the jet (Fig.5(a)). As time progresses, the maximal particle diameter grows from 2.04 at t*=8 to 2.79 at t*=318. The span of particle diameters gets wider when the distribution of particle diameters turns irregular as the jet flows downstream (Fig.5(d)).
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Fangyang Yuan et al. / Procedia Engineering 102 (2015) 1650 – 1655
(a)t*=8
(b)t*=64
(c) t*=200 (d) t*=318 Fig. 5. Instantaneous contours of the particle diameter dp* at different times
4.3.3. Geometric standard deviation (GSD) Cross-stream profiles of the geometric standard deviation at t*=373 at different streamwise positions is given in Fig.6. The maximal values of the GSD increase from 1.095 at X/D=6 to 1.175 at X/D=26 and the span of geometric standard deviation becomes wider as the jet develops to downstream. The maximal GSD appears near the interface between the jet and the ambient due to a relatively large concentration and velocity gradient.
Fig. 6. Cross-stream profiles of the geometric standard deviation
Vg
at t*=373
5. Conclusions Simultaneous numerical simulation of particle coagulation and breakage in a planar jet is performed. The flow field is calculated by LES and verified by experimental data. The direct expansion method of moments with n/2th (n=0~4) moments is employed to predict the coagulation of particles. The particle dynamics equation is converted to a set of closed ordinary differential equations and solved numerically. The results confirm the validity of the newly proposed DEMM model on describing the evolution of particle size distribution. The coherent structures dominate the evolution of particle concentration, geometric mean diameter and geometric standard deviation. The particle number concentration decreases and the span of particle
Fangyang Yuan et al. / Procedia Engineering 102 (2015) 1650 – 1655
1655
diameter gets wider along the centerline as the jet develops. The geometric standard deviation is larger in the shear layer than that in the jet core.
References [1] S. E. Pratsinis. Simultaneous nucleation, condensation, and coagulation in aerosol reactors. Journal of Colloid and Interface Science, 1988,124:416-427. [2] Y. XiongˈS. E. Pratsinis. Gas phase production of particles in reactive turbulent flows. Journal of Aerosol Science, 1991,22:637-655. [3] F. O. ErnstˈS. E. Pratsinis. Self-preservation and gelation during turbulence-induced coagulation. Journal of aerosol science, 2006,37:123142. [4] M. Z. YuˈJ.Z. LinˈT.L. Chan. Numerical simulation for nucleated vehicle exhaust particulate matters via the TEMOM/LES method. International Journal of Modern Physics C, 2009,20:399-421. [5] S. E. MillerˈS. C. Garrick. Nanoparticle coagulation in a planar jet. Aerosol science and technology, 2004,38:79-89. [6] N. SettumbaˈS. Garrick. Direct numerical simulation of nanoparticle coagulation in a temporal mixing layer via a moment method. Journal of aerosol science, 2003,34:149-167. [7] M. Z. YuˈJ.Z. Lin, T.L. Chan. Numerical simulation of nanoparticle synthesis in diffusion flame reactor. Powder Technology, 2008,181:920. [8] T.L. ChanˈJ.Z LinˈK. Zhou, et al. Simultaneous numerical simulation of nano and fine particle coagulation and dispersion in a round jet. Journal of aerosol science, 2006,37:1545-1561. [9] J.Z. Lin, P.F.Lin, H.J.Chen. Research on the transport and deposition of nanoparticles in a rotating curved pipe. Physics of Fluids, 2009, 21(122001): 1-11. [10] G. BarthelmesˈS. PratsinisˈH. Buggisch. Particle size distributions and viscosity of suspensions undergoing shear-induced coagulation and fragmentation. Chemical Engineering Science, 2003,58:2893-2902. [110] F. YuanˈF. Gan. Evolution of Aerosol Particles in the Rainfall Process via Method of Moments. Abstract and Applied Analysis, 2013,2013: [12] M. Z. YuˈJ.Z. Lin. Nanoparticle-laden flows via moment method: A review. International Journal of Multiphase Flow, 2010,36:144-151. [13] Z.L. ChenˈJ.Z. LinˈM.Z. Yu. Direct expansion method of moments for nanoparticle Brownian coagulation in the entire size regime. Journal of Aerosol Science, 2014,67:28-37. [14] E. OttoˈH. FissanˈS. Park, et al. The log-normal size distribution theory of Brownian aerosol coagulation for the entire particle size range: part II—analytical solution using Dahneke’s coagulation kernel. Journal of aerosol science, 1999,30:17-34. [15] F.J. GanˈJ.Z. LinˈM.Z. Yu. Particle Size Distribution in a Planar Jet Flow Under-Going Shear-Induced Coagulation and Breakage. Journal of Hydrodynamics, Ser. B, 2010,22:445-455. [16]M. Z. YuˈJ.Z. Lin, T.L. Chan. A new moment method for solving the coagulation equation for particles in Brownian motion. Aerosol Science and Technology, 2008,42:705-713.
ScienceDirect Procedia Engineering 102 (2015) 1650 – 1655
The 7th World Congress on Particle Technology (WCPT7)
Simulation of Particle Coagulation and Breakage in a Planar Jet via the DEMM method Fangyang Yuan*, Zhongli Chen , Xiaojun Pan School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China
Abstract Simultaneous numerical simulation of particle coagulation and breakage in a planar jet is performed. Large eddy simulation (LES) is adopted to calculate the flow field and the direct expansion method of moments (DEMM) is employed to describe the evolution of particles undergoing coagulation and breakage. The Aggregation-Breakage Balance Equation is converted to a set of moment equations which contains the n/2th (n=0~4) moments through the DEMM method. The newly proposed DEMM model is confirmed to be valid to describe the evolution of particle size distribution (PSD). The coherent structures dominate the evolution of particle concentration, geometric mean diameter and geometric standard deviation. The particle number concentration decreases and the span of particle diameter gets wider along the centerline as the jet develops. The geometric standard deviation is larger in the shear layer than that in the jet core. © 2015 The Authors. Published by Elsevier This isLtd. an open access article under the CC BY-NC-ND license © 2014 The Authors. Published by Ltd. Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of Chinese Society of Particuology, Institute of Process Engineering, Chinese Selection under responsibility of Chinese Society of Particuology, Institute of Process Engineering, Chinese Academy Academyand ofpeer-review Sciences (CAS). of Sciences (CAS)
Keywords: Direct expansion method of moments; Coagulation; Breakage; Planar jet
1. Introduction Turbulent flows containing particle coagulation and breakage are of great interest in many major industries processes like manufacture of fumed silica, as well as for natural phenomena like evolution of combustion smoke [14]. Simultaneous evolution of particles undergoing several dynamic processes along with the transport processes of diffusion, convection and thermophoresis is unfeasible to be described well by limited experimental methods in the fluid field [5]. Hence, methodologies which can capture the multi-process particle transport are needed. Numerical simulations have been proved effective on the study of multiphase flow [6-9]. * Corresponding author. Tel.: +86 571 86836009; fax: +86 571 86914401. E-mail address: [email protected]
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of Chinese Society of Particuology, Institute of Process Engineering, Chinese Academy of Sciences (CAS)
doi:10.1016/j.proeng.2015.01.301
1651
Fangyang Yuan et al. / Procedia Engineering 102 (2015) 1650 – 1655
The rate of change of the PSD by coagulation and breakage can be predicted by the Aggregation-Breakage Balance Equation (PABBE) [10]. For years, the method of moments (MOM)[6, 11] and the Sectional Method (SM)[5] are proposed and developed to deal with the equation with nonlinear and partial integro-differential characteristics[12]. 2. Fluid flow The flow field consists of a particle-laden flow ejecting from a planar jet into a still environment, see Fig.1. The LES equations governing the incompressible flow are obtained by filtering the N-S equation:
wui wxi
0 ,
(1)
wui wu uj i wt wx j
wW ij w 2ui 1 wp X U wxi wx j wx j wx j
,
(2)
where ui and p are the filtered velocity and pressure, respectively. W ij is the subgrid-scale(SGS) stress, which is expressed by the Smagorinsky- Lily model.
Fig. 1. Configuration of planar jet flow
3. Particle transport 3.1. PABBE and its moment transformation The transport of particles dispersed throughout the fluid is governed by the PABBE, which comes from Particle General Dynamic Equation (PGDE) proposed by Schmoluchowski and developed by Muller. The breakage process which is brought into the equation by Barthelmes et al.[10] comes about a steady state of PSD by the competition between coagulation and breakage, i.e.,
wn(v, t ) U n(v, t ) ( Dt n(v, t )) wt
1v ³ E (v1 , v v1 )n(v1 , t )n(v v1 , t )dv1 20
f
f
v
v
n(v, t ) ³ E (v1 , v)n(v1 , t )dv1 ³ a(v1 )b(v | v1 )n(v1, t )dv1 a(v)n(v, t )
ˈ
(3)
1652
Fangyang Yuan et al. / Procedia Engineering 102 (2015) 1650 – 1655
where n(v,t) is the particle size distribution, v and v1 are particle volumes, β(v,v1) is the volume-based aggregation kernel which describes the colloid frequency of particles with volume v and v1. In our work, the Dahneke’s expression of coagulation kernel in the entire size regime is been adopted for calculating due to the low Knudson number (Kn < 1)[13, 14]. α(v) is the volume-based breakage kernel that gives the frequency of breakage of particles of volume v , and b(v|v1) is the fragment distribution function. Here, we adopt the same breakage model as the previous work in Gan et al.[15]. 3.2. Direct expansion method of moments (DEMM) The DEMM is proposed based on the TEMOM[16] in which the closure of the moment evolution equations is achieved by means of the Taylor expansion method[13]. Following the derivation of [14], Eq.(3) can be obtained as:
wmk U mk ( Dt mk ) wt
p
p
1 k ¦ ¦ [ijk m1 / m0 mim m jm 2 1 9 m
i 0j 0
where the moment mk is defined by derivatives of the coagulation kernel
mk
p
³
f
0
p
k
1 3
§ ¨k ©
0,
· m 2m , ,..., m ¸ , p p ¹
(4)
v n v, t dv , ζ = kb((μG)/τ*)q and ξijk involves variety of partial k
E , which will be numerically determined. mk+1/3 which relates to the breakage
process can be obtained following the treatment in [11]. In this paper, we set m=2 and p=4, then the PABBE will be composed of five ordinary differential quations (k=0, 1/2, 1, 3/2, 2) and each involves five moments, namely m0, m1/2, m1, m3/2 and m2. This set of equations will be used in the present study to predict the time evolution of the PSD. 4. Results and discussion 4.1. Numerical specifications We introduce the following dimensionless parameters to simplify the calculation and analysis:
x*j
xj D,
t *j
t D U0
u *j ,
uj U0
p* ,
p UU 02
Mk ,
mk mk 0
,
(5)
where the width of the nozzle D=0.0127m which is the characteristic length, U0=(U1-U2)=(9.55-1.91)=7.64m/s is the characteristic velocity, and the Reynolds number is Re = (U1 U2)D/νair =8300. The initial values of five moments are determined according to the following formula:
mk 0
Φ0 v0k 1 k
0,1 / 2,1,3 / 2, 2 ,
(6)
in which the volume fraction at the nozzle exit Φ0 = 2.0×10-5, and the initial particle volume v0 S d03 6 , in which the initial particle diameter is d0 = 1μm. Computations are performed on a domain of L × H = 60D × 41D, which is discreted into structured grids. The model constant of Smagorinsky–Lilly model Cs is set as 0.1. 4.2. Velocity field The self-preservation profiles of the mean stream-wise velocity compared with the experimental data are shown in Fig.2. As can be seen in the figure, the mean axial velocity satisfies the self-similarity distribution at X/Dı10 and can be expressed as a Gaussian curve. The fully developed vorticity contour of the planar jet is shown in Fig.3.
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Fangyang Yuan et al. / Procedia Engineering 102 (2015) 1650 – 1655
Fig. 2. Self-preservation profiles of the mean stream-wise velocity
Fig. 3. Contour of the fully developed vorticity
4.3. Evolution of PSD 4.3.1. Particle number concentration The time-averaged moments develop along with the evolution of fluid field and arrive steady state after t*=300. Fig.4. shows the instantaneous contour of M0 at t*=373 which represents the particle number concentration. The spatial distribution of M0 resembles that of vorticity in Fig.3., which indicates that distribution of number concentration is dominated by the structure of coherent eddies. It can also be seen that the particle number concentration decreases along the centerline as the jet develops which demonstrates the effect of coagulation. The prediction of PSD proves the validity of the DEMM model.
Fig. 4. Contour of zeroth moments M0 at t*=373
4.3.2. Geometric mean diameter Coagulation and breakage of particles in the flow field bring about the change of particle size. The particle geometric diameter can be obtained by d *p
6v
* p
/ S , in which the particle volume 1/3
v*p
M1 / M 0 . The spatial-
temporal evolution of particle mean diameter is presented in Fig.5. At early stage of the development of fluid field, the particles are dominated by coagulation and size of particles grows fast along the centreline of the jet (Fig.5(a)). As time progresses, the maximal particle diameter grows from 2.04 at t*=8 to 2.79 at t*=318. The span of particle diameters gets wider when the distribution of particle diameters turns irregular as the jet flows downstream (Fig.5(d)).
1654
Fangyang Yuan et al. / Procedia Engineering 102 (2015) 1650 – 1655
(a)t*=8
(b)t*=64
(c) t*=200 (d) t*=318 Fig. 5. Instantaneous contours of the particle diameter dp* at different times
4.3.3. Geometric standard deviation (GSD) Cross-stream profiles of the geometric standard deviation at t*=373 at different streamwise positions is given in Fig.6. The maximal values of the GSD increase from 1.095 at X/D=6 to 1.175 at X/D=26 and the span of geometric standard deviation becomes wider as the jet develops to downstream. The maximal GSD appears near the interface between the jet and the ambient due to a relatively large concentration and velocity gradient.
Fig. 6. Cross-stream profiles of the geometric standard deviation
Vg
at t*=373
5. Conclusions Simultaneous numerical simulation of particle coagulation and breakage in a planar jet is performed. The flow field is calculated by LES and verified by experimental data. The direct expansion method of moments with n/2th (n=0~4) moments is employed to predict the coagulation of particles. The particle dynamics equation is converted to a set of closed ordinary differential equations and solved numerically. The results confirm the validity of the newly proposed DEMM model on describing the evolution of particle size distribution. The coherent structures dominate the evolution of particle concentration, geometric mean diameter and geometric standard deviation. The particle number concentration decreases and the span of particle
Fangyang Yuan et al. / Procedia Engineering 102 (2015) 1650 – 1655
1655
diameter gets wider along the centerline as the jet develops. The geometric standard deviation is larger in the shear layer than that in the jet core.
References [1] S. E. Pratsinis. Simultaneous nucleation, condensation, and coagulation in aerosol reactors. Journal of Colloid and Interface Science, 1988,124:416-427. [2] Y. XiongˈS. E. Pratsinis. Gas phase production of particles in reactive turbulent flows. Journal of Aerosol Science, 1991,22:637-655. [3] F. O. ErnstˈS. E. Pratsinis. Self-preservation and gelation during turbulence-induced coagulation. Journal of aerosol science, 2006,37:123142. [4] M. Z. YuˈJ.Z. LinˈT.L. Chan. Numerical simulation for nucleated vehicle exhaust particulate matters via the TEMOM/LES method. International Journal of Modern Physics C, 2009,20:399-421. [5] S. E. MillerˈS. C. Garrick. Nanoparticle coagulation in a planar jet. Aerosol science and technology, 2004,38:79-89. [6] N. SettumbaˈS. Garrick. Direct numerical simulation of nanoparticle coagulation in a temporal mixing layer via a moment method. Journal of aerosol science, 2003,34:149-167. [7] M. Z. YuˈJ.Z. Lin, T.L. Chan. Numerical simulation of nanoparticle synthesis in diffusion flame reactor. Powder Technology, 2008,181:920. [8] T.L. ChanˈJ.Z LinˈK. Zhou, et al. Simultaneous numerical simulation of nano and fine particle coagulation and dispersion in a round jet. Journal of aerosol science, 2006,37:1545-1561. [9] J.Z. Lin, P.F.Lin, H.J.Chen. Research on the transport and deposition of nanoparticles in a rotating curved pipe. Physics of Fluids, 2009, 21(122001): 1-11. [10] G. BarthelmesˈS. PratsinisˈH. Buggisch. Particle size distributions and viscosity of suspensions undergoing shear-induced coagulation and fragmentation. Chemical Engineering Science, 2003,58:2893-2902. [110] F. YuanˈF. Gan. Evolution of Aerosol Particles in the Rainfall Process via Method of Moments. Abstract and Applied Analysis, 2013,2013: [12] M. Z. YuˈJ.Z. Lin. Nanoparticle-laden flows via moment method: A review. International Journal of Multiphase Flow, 2010,36:144-151. [13] Z.L. ChenˈJ.Z. LinˈM.Z. Yu. Direct expansion method of moments for nanoparticle Brownian coagulation in the entire size regime. Journal of Aerosol Science, 2014,67:28-37. [14] E. OttoˈH. FissanˈS. Park, et al. The log-normal size distribution theory of Brownian aerosol coagulation for the entire particle size range: part II—analytical solution using Dahneke’s coagulation kernel. Journal of aerosol science, 1999,30:17-34. [15] F.J. GanˈJ.Z. LinˈM.Z. Yu. Particle Size Distribution in a Planar Jet Flow Under-Going Shear-Induced Coagulation and Breakage. Journal of Hydrodynamics, Ser. B, 2010,22:445-455. [16]M. Z. YuˈJ.Z. Lin, T.L. Chan. A new moment method for solving the coagulation equation for particles in Brownian motion. Aerosol Science and Technology, 2008,42:705-713.