- Email: [email protected]
Quadrature Method of Moments for Nanoparticle Coagulation and Diffusion in the Planar Impinging Jet Flow*
Chin. J . Chem. Eng.,15(6) 828-836
(2007)
Quadrature Method of Moments for Nanoparticle Coagulation and Diffusion in the Planar Impinging Jet Flow* YU Mingzhou(7 fl fl! )a, LIN Jianzhong(#&,$i.)a9bv**, and XIONG Hongbing( #6 ?I%)" a
The State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 3 10027, China China Jiliang University, Hangzhou 310018, China
Abstract A computational model combining large eddy simulation with quadrature moment method was employed to study nanoparticle evolution in a confined impinging jet. The investigated particle size is limited in the transient regime, and the particle collision kernel was obtained by using the theory of flux matching. The simulation was validated by comparing it with the experimental results. The numerical results show coherent structure acts to dominate particle number intensity, size and polydispersity distributions, and it also induce particle-laden jet to be diluted by the ambient. The evolution of particle dynamics in the impinging jet flow are strongly related to the Reynolds number and nozzle-to-plate distance, and their relationships were analyzed. Keywords nanoparticle, coagulation, impinging jet, large eddy simulation, quadrature moment method
1 INTRODUCTION Impingement of nanoparticles (e.g. colloid, bioparticles, macromolecules) on a solid surface occurs in many industrial processes, for example, the contamination control of microelectronic manufacture and cooling, pharmaceutical processing, ventilation during mining and tunneling activities, paint spraying and vehicle exhaust plumes (Fig.1). The impinging jet is also a well-established technique to study the deposition of colloid particles onto a solid surface under well-defined hydrodynamic and physicochemical conditions[11. In these applications, the particle diameters are usually smaller than 1pm and particle coagulation due to Brownian motion dominates over all the jet developing, and thus universal models such as discrete phase model (DPM)[2] for describing particle dynamics are no longer available for the ultrafine particles in these cases. Additionally, the turbulent impingement jet has the pronounced complicated coherent structures in the stagnation region and wall flow region, which usually leads to non-determined particle distribution. Under such circumstances, the studies for ultrafine particle dynamics in impingement jet are greatly limited. However, it is essential to understand the physics of nanoparticles in the above enumerative phenomena.
Figure 1 Schematic diagrams of a planar impinging flow
It should be pointed out that few studies have been reported for particle-laden jet impingement in the literature. These works are mostly limited to experimental research for capturing the evolution of coher-
ent structures[3] and examining the particle deposition[4], without considering the instantaneous influence of coherent structures on the ultrafine particle dynamics. One the other hand, Voss and Finlay[S] and Wang et a1.[6] have investigated the influence of turbulence induced by impingement plate on the particle deagglomeration, but they did not consider the interaction of flow structures and the nanoparticle evolution. Particles in the range between lnm and 1pm stick together as a result of van der Waals forces when once collided, the collision rate of particles can be treated as the coagulation rate and this terminology is used in most of literature[7]. In this region the particle size is smaller than Kolmogorov scale and thus the influence of turbulence on particle deagglomeration is negligible. Under this circumstance, particle dynamics is mainly dominated by the particle coagulation due to Brownian motion. Based on the particle diameter, the particles coagulation can be divided into three different regimes: free-molecule regime (for particles with diameters smaller than lOnm), transition regime (for particles with diameter from lOnm to 100nm) and continuum regime (for particles with diameters larger than lOOnm but smaller than lpm). In different regimes, the particle collision kernels exhibit different coagulation mechanisms[8]. Previous studies for particle coagulation are mostly limited in the free-molecule regime or continuum regime[9], but few in the transition regime[8], because the particle collision is not easily described by mathematical expression in this regime. In order to solve the coagulation equation one needs to decide which coagulation coefficient to employ. In general, the coagulation coefficient in the transition regime is fixed by harmonic mean, but its relative error is proved to be more than other detailed coagulation coefficient[101. The transport of the nanoparticles dispersed throughout the fluid is governed by the aerosol general
Received 2006- 10-13, accepted 2007-09-04.
* Supported by the Ministry of Science and Technology of China (No.2005CCA06900).
** To whom correspondence should be addressed. E-mail: [email protected]
Quadrature Method of Moments for Nanoparticle Coagulationand Diffusion in the Planar Impinging Jet Flow
dynamics equation (GDE)[ 111. The GDE describes particle dynamics under the influence of various physical and chemical phenomena: convection, diffusion, coagulation, surface growth, nucleation, and the other internal or external forces[ 121. Deeply understanding nanoparticle dynamics requires the incorporation the GDE into the computational fluid equations, especially for the processes with complicated flow structures. Up to now, there are several approaches to the modeling and simulation of aerosols based on GDE equation. The moment method (MM) was first introduced by Hulbert and Katz[ 131 and considered to be a powerful method in the field of aerosols because it can provide the representative properties of particles while requiring only the computation of the first three moments[l4]. However, this method may not be the most accurate because it requires the prior assumption for particles size distribution (PSD). Recently, the moment method has been conducted conjunction with direct number simulation (DNS) to investigate the temporal mixing layers[ 151. The sectional method (SM) is another universal technology for resolving the aerosols problems and has been further developed[ 161 after it is proposed by Gelbard and Seinfeld[l7]. In this approach, the continuous size or length range of the internal coordinate is partitioned into a finite series of contiguous sections or bins. This method has the ability to capture the details of early stages of particle clustering where the prior assumption of particle size distribution is not available. A disadvantage of the sectional method is that each section or bin corresponding to a transport equation must be separately solved, which can be prohibitively expensive computationally. An attractive alternative is the quadrature method of moments (QMOM)[IS]. This method was first introduced by McGraw[ 191 for modeling aerosol evolution and has recently been extended to aggregation and breakage problems using both a monovariate and a bivariate PSD[ IS]. The QMOM doesn’t define or produce an explicit size distribution, but the moments could be used with an assumed functional form to obtain a closed set of equations for all moments[20]. This approach is viewed as a competing method that presents the main advantage of being extremely accurate and amenable for coupling with CFD codes[lS]. In this work, the large eddy simulation (LES) combined with the QMOM is performed to investigate the nanoparticle dynamics in the impingement flow systems, which is not reported before. The jet suspended with nanoparticles perpendicularity impinges on a flat plate with an initialization exit velocity, and the particle deposition on the surface of impingement plate due to absorption is not considered. This methodology facilitates the capture of the underlying physics in a time-accurate manner and can describe accurately the nanoparticle evolution under the influence of the physical phenomena: convection, diffusion, and Brownian coagulation, especially in the cases where the flow structures are significantly influenced by the impingement plate. The particles in this study are selected in the transient regime, since the nanoparticle diameters in most of the practical situations range
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from lOnm to 100nm.
2 MATHEMATICAL FORMULATION 2.1 Fluid flow The schematic of a planar impinging jet is shown in Fig.1, in which x and y are the radial and axial directions, respectively. The width of the nozzle D is 25mm and the Reynolds number, defined as Re= DUdv with v being the kinematic viscosity of the fluid, is 10000, 15000, 20000 and 30000. The nozzle-to-plate distances H are 1,2, 3 and 4 nozzle width. The equations governing the incompressible flow, obtained by filtering the Navier-Stokes equation, are:
aq -
,-0
dx;
where q is the filtered velocity, is the filtered pressure, index i, j is taken as 1,2 and refers to the x and y directions, respectively, Tii is the sub-grid scale (SGS) stress tensor. The first SGS stress model, proposed by Smagorinsky[21], is adopted here. His model assumes that the SGS stress follows a gradient-diffusion process, similar to molecular motion. Consequently, Tc is given by the eddy viscosity assumption: qj = 2v,s,- +1-T,,qj (3) 3 where
-
s, is the filtered rate of
the strain tensor,
S.. IJ =(alii/axj+alijlaxi)12; v, =(C,A)*ISI, here C, is the Smagorinsky constant of 0.1, A is the filter width A = (Ax, . Ax,)”2 with Axi(i = 1,2) being i direction mesh size,
Is/= (2susi,)1/2. In the present
study, v, is calculated using the dynamic subgrid model proposed by German0 el al.[22] and developed by Lilly[23]. The width of the test filter in this model was defined as:
(4) The values of the symbols in this equation can be seen in the studies of Lilly[23]. In order to alleviate the dissipation near the impinging plate, the Wall-Adapting Local Eddy-Viscosity (WALE) model is used[24].
2.2 Particle field In this study the volume fraction of particles is and thus the transport properties very small (< of the gas can be approximated by constants. In addition, compared to the Kolmogorov scale, the particle is so small (<100nm) that the influence of solid phase on the gas phase is negligible and the particle fragChin. J. Ch. E. lS(6) 828 (2007)
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mentation and agglomeration due to turbulent shear is not considered. At these circumstances, the dynamics behavior of particles suspended in gas undergoes Brownian diffusion, the transport due to gas convection and Brownian coagulation in the transient regime. Let n(v,t)dv be the particle number in a volume range between v and v+dv per unit mass of gas at time t. Then the coagulation equation for the number concentration function is [25]: an(v9t) +v~un(v,t)=V.r(v,t)Vn(v,t)+ at
1 2
-
l P (vl, v
- vl ) n ( vI ,t ) n ( v - v1,t)dv, - n(v, t)
Here u is the local flow velocity, T(v, t ) is the particle Brownian diffusion constant for volume v, p is the collision kernel in the transient regime. The first term on the left-hand suggest the rate of change of particle number concentration while another term describes the loss or gain of particles due to convection. The fist term on the right-hand denotes the influence of Brownian diffusion on the nanoparticles and the last two terms dominate monomer gain or loss due to Brownian coagulation. In order to express the number density function using particle length L as the internal coordinate, the followed transformation in terms of spherical shape is chosen: v=- ?cL3 6 The relationship between the two expressions of the number density function is:
Then Eq.(5) can be written as aN(L't) + V . uN(L,t) = V . T(L,t)VN(L,t) + at
In order to employ the QMOM method [191, the followed assumption is used:
where NQis the order of the quadrature formulation. L; is the particle diameter and w;is the number intensity. Substituting Eq.( 11) into Eq.( 10) results in the closed coagulation equations based on the particle diameters for kfk: aMk(t)
at
where Pj = P ( L j , L j ) . In order to simplify the treatment and analysis of the interaction between the hydrodynamic and particulate fields, Eq.( 12) is non-dimensionalized using the following relations: *
X'
x . =J , ' D
* t=-
t
DIU,
,
*
u.=',
u.
' uo
L* . = JL.
' r,'
w *.=L w ., M ,* = - M k J
WO
MkO
The characteristic length scale is D, the characteristic velocity is Uo, the reference particle length is b,the reference particle number intensity is wo and the reference value of the kth moment is Mm. Substituting the relations given in Eq.(13) into Eq.( 12) results in the non-dimensionalized .equation for the kth moment, Mk: at*
N(z, t)dz - N(L, t )
+ v .uM,(t) =
ax;
P(L, z)N(z,t)dz
Applying the moment transformation[181: Mk(t)= r N ( L , t ) L k d L the general coagulation equation is resulted: aMk(t) + ~ . u M , ( t ) = at
V.T(L,t)VM,(t)+l r N ( z , t ) X 2 December, 2007
(14) the particle Schmidt number, SC, = v / r ,defined as the ratio of the fluid kinematic viscosity to the particie mass diffusivity[26]. Weights ( wi ) and abscissas ( 4 ) are determined through the product-difference (PD) algorithm from the lower-order moment[l9]. In the transition regime the coagulation rate is
Quadrature Method of Moments for Nanoparticle Coagulation and Diffusion in the Planar Impinging Jet Flow
described neither by the continuum theory nor by simple kinetic theory. Based on the theory of flux matching[27], the coagulation function is mainly expressed as an enhancement of the collision function for the near-continuum regime[8]:
P( ' i
9
' j
) = P c o ("i
'Vj
) f ( Kn)
In Eqs.(ll) and (13) P c 0 ( v i , v j )is the coagulation function in the continuum plus near-continuum regime and is represented by the following equation:
where Kco = 2kBT/(3,u) is the coagulation constant for the continuum plus near-continuum regime, kB is the Boltzmann constant and C ( v )= 1+ Kn [Al + A2 exp(-A, / Kn)] is the slip correction factor. A , = 1.142, A2 = 0.558, A3 = 0.999 are constants determined by Allen and Raabe[28]. Equation (18) can be further represented based on non-dimensionalizedparticle length:
+
(L; L ; ) [ C (L: ) / L: + C ( LJ )I LJ
]
(19) +A2exp(-A3C&/22)].
where C(L;)=l+,-[Al 22
Li & is the coagulation function in the free-molecule regime and given by[25]:
Pfm(v,V )
where C is the mean thermal velocity, and
=(&)
116
Kfm
Since the particle size in this study is smaller than Kolmogorov micro-scale, the turbulent diffusivity for nanoparticles is negligible and thus only Brownian diffusivity in the transition regime is considered. The particle diffusivity coefficient r is[ 121:
(15)
Followed by Dahneke's theory[ 101, the original enhancement functionfiKn) can be obtained[9]: 1+ Kn f (Kn)= 1+ 2Kn + 2Kn2 Otto et a1.[8] reveals that the particle Knudsen number Kn can be considered as the ratio of the coagulation coefficient of the limiting regimes and thus Kn is expressed as follows:
P,, (L:,L; ) = K,,
831
(6k~T/&)112
is the coagulation constant for the free-molecule regime. Equation (20) is represented based on nondimensionalizedparticle length:
3 RESULTS AND DISCUSSION 3.1 Computation conditions In the computation, the filer size for large eddy simulation is taken to be equal to the grid size, and the width of test filter is larger than that of grid filter. The implicit difference scheme in time and the second-order Crank-Niscolson scheme are employed to solve the equations. The QUICK method is adopted for the convective terms in Egs.(6) and (13). The nondimensionlized time step At is a constant, 0.012. The fluid in this study is air at a tempeture of T=300K. The computational grid is comprised of 400 X 800, 600 X 800, 800 X 800 for nozzle-to-plate distance of 2, 3, 4 nozzle width and the computational size in the x direction is 10 nozzle width. The present work concentrates on nanoparticles with diameter below lOOnm and particle density 1000kg.m-3 in the gas flows. In this size regime, particle coagulation mainly attributes to van der Waals forces when once collided. Since some results[ 18,291 show that a quadrature approximation with three nodes (NQ=3) is sufficient to describe the moment evolution with good accuracy, the QMOM calculation in this work was conducted with three nodes (tracking the first six moments, QMOM). For this case of coagulation alone, the initialized particle lengths of three classes at the nozzle exit are
cn=1.0, cf,=2.0 and Go=[c,,+Go]
113
with reference particle length & = lOnm . The corresponding particle number intensity for these three classes are w;, =0.5 , win =0.5 and w;" = O with reference number intensity wo = 1.0 X 1017 particles per m3. In this numerical simulation, the particle length is assumed to be particle diameter and then the above initialization determines the initial particle diameter dPoshould be 12.5nm at nozzle exit using the volume averaged diameter d , = (4RM3/ 3M0)113 / 2 . Knowledge of the initial particle number intensity and diameter is sufficient for determining the initial conditions of the six moments ( M o o= MIn= 1.5 X lo9 , M20 =25 , M30 =4.5X10-7 , M a =8.5X10-'5 , M50 = 1.65 X
). In the initialization computation,
the variables for all moments are initialized by zero except in the nozzle where all moments are given 1. For every time step, locations where particle number Chin. J. Ch. E. 15(6) 828 (2007)
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Chin. J. Ch. E. (Vol. 15, No. 6 )
concentration (Mi) is less than 0.005 are considered to be free of particles.
3.2 Validity test The validity of the present model is tested against results from a simpler model in which only Brownian coagulation is considered. In this case, the influence of gas flow on particle diffusion and transport is neglected and thus Eq.(12) can be reduced to the following form:
NQ
NQ
cL:wiCfijwj i
The contours of vorticity by using the large eidy simulation method at non-dimensionlized time t = 1200 are shown in Fig.3(a). When the jet develops enough, the vortices generate near the nozzle flow, move downstream and finally induce secondary vortices emergence in the wall jet flow regions, which is similar with the visualization of Fig.3(b) made by Heinz et al.[32]. This process repeats periodically. Fig.4 illustrates the measured and computed axial time-averaged velocity profiles along the jet centerline for Re=30000 and HID= 1, 2. It is seen from the figure, the experimental[33] and numerical data are in good agreement.
( k =0;..,5)
(23)
1
In this work, the results obtained from Eq.(23), i.e. the present QMOM method, are compared with the sectional model and analytical solution[8]. Similar to the solutions of Otto et al.[8], the initial parameters for the comparison calculations are & = 94nm, 0,= 1.8, NO= T = 300K , p = 1000kg.mP3 In Fig.2, the change of the number concentration with dimensionless time K,,NOt predicted by the QMOM method, the analytical solution and sectional model are presented. No significant difference among three curves can be found from Fig.2, which suggests the present QMOM method has an ability to solve the problem involving nanoparticle coagulation. As the QMOM method is incorporated into the computational fluid dynamics (CFD), it is necessary to predict the nanoparticle lunetics in the evolving flows. In fact, the approach combining the QMOM method with the CFD code has been used in many complicated particulate systems such as nanoparticle synthesis in a diffusion flame reactor[30] and particle aggregation-breakage processes in turbulent Taylor-Couette flow[? 11.
(a) By large eddy simul?tion at nondimensionlized time r = 1200
Figure 3
(b) PIV by Heinz et a1.[32] Contours of vorticity in the evolution of vortex 1.2
0.8
0
0.2
0.4
0.6
0.8
1.0
Y'( v / D )
Figure 4 The axial velocity profiles along the jet centerline for Re=30000 _ _ ~ HID=l _ Num; HlD=2 Num; 0 HID= 1 Exp; HID=2 Exp ~
K2"t
Figure 2 The comparison for number concentration decay as a function of dimensionless time o sectional method; - - - analytical method; QMOM method (this study)
3.4 Effect of coherent structures on the nanoparticle dispersion and coagulation Figure 5 depicts the instantaneous contours of the non-dimensionalized zero moment Mi , the non-dimensionalized volume-averaged diameters d i
~
~
3.3 Evolution of coherent structures and comparison with experiment In order to understand the relationship between coherent structures and nanoparticle dynamics, the evolution of coherent structures is described firstly. Coherent structures can be illustrated by vorticity field. December, 2007
and polydispersion W when the jet impingement develops enough at nondimensionalized time t* = 1200.
Mi corresponds to the particle number concentration. In theory, coagulation is a process in which particles collide with one another and adhere to form larger particles. The net result is a continuous decrease in number concentration and an increase in particle size.
Quadrature Method of Moments for Nanoparticle Coagulation and Diffusion in the Planar Impinging Jet Flow
833
3.5.1
Pre-impingingjet The particle number intensity and diameter distributions along the jet axis are important to understand the dynamics of the nanoparticle-fluid flow and to explain the behavior of particle diffusion under the influence of impingement plate and particle coagulation. Figs.6(a) and (b) illustrates the time averaged
-
non-dimensionalized particle number intensity Mi -
Figure 5 Instantaneous contoyrs of the non-dimeyionalized zero moment M, ,the particle diameter $, and the polydispersity Win the impinging jet flow with coherent struttures at nondimensionalized time t =1200
and volume averaged diameter di profiles along the jet centerline at HID=2, 3 and 4 for Re=10000, respectively. It is seen that the variance of particle number intensity exhibits the same trend for the different nozzle-to-plate distance, so does the particle diameter. Near the nozzle, the number intensity decreases significantly due to strong particle collision and then decreases slightly under the corporate influence of large coherent structures and particle coagulation. In the very small distance from the impinging surface, the presence of impingement plate leads to the accelerated loss of the particle number intensity. Increasing HID leads to a decrease in particle number intensity and an increase in particle diameter. The influence of Re on the particle number intensity and diameter at the same nozzle-to-plate distance, HID= 3, is shown in Fig.7(a) and Fig.7(b), respectively. It can be seen from this figure increasing Reynolds numbers results in a decrease in particle number intensity while an increase in particle diameter along the jet centerline. 1 .o
The polydispersity index is a measure of the width of the size distribution function. It is defined as W =[(M,M, -Mf)lM,]”2 [34]. From the figure it
0.8
1%
0.6
can be seen that the distributions of Mi , di and W all
0.4
depend similarly on the configuration of coherent structures. In the process of the jet exchanging with the ambients, these coherent structures carry particles to embed into the surrounding gas and thus the particle-laden jet is diluted. Before impinging the plate surface, the particle number intensity decreases downstream under the influence of particle coagulation and diffusion induced by shear vortices, while the particle diameter and polydispersion increase correspondingly. In the stagnation region and wall flow regions, the number intensity decreases in the radial direction, while the mean particle diameter increases correspondingly.
0.2
3.5 Distribution of time averaged particle quantities The solution by using time averaged method allows both qualitative and quantitative assessments of the spatio-temporal evolution of the particle field. In the present work, time averaged data typically employ 8000 samples, which is sufficient for one sweep of the present investigated flows. In fact, quantities obtained in this way correspond to results generated in Reynolds-Averaged Navier-Stokes (RANS) simulations[7] and are all denoted by an over-bar.
0
0.2
0.4 0.6 Y’ ( Y / W -
0.8
1.0
(b) d i Figure 6 Variation of the time-averaged non-dimensionalizedparticle number concentration Mi -
and diameter d i at the jet centerline for the different HID cases with Re =10000 HID: *2; ~ 3 A;4 Chin. J. Ch. E. 15(6) 828 (2007)
Chin. J. Ch. E. (Vol. 15, No. 6)
834 1.0
I
::::B
I
0.12
0.10
I
I
0.2
0
0.4
I
I
I
0.6
0.8
1.0
-2
0
1
2
0
I
2
-1
Y' (.vlD)
2.9 2.8
L '
2.4 -2
1.4 -
0
I
I
-1
X*
0.2
0.4
0.6
0.8
1.0
Y' ( Y O
(b) Figure 7 Variation of the time-averaged non-dimensionalizedparticle number concentration Mi -
and diameter df at the jet centerline for the different Re cases with HID=3 Re: A 10000; + 15000; 720000
dp'
(b) Figure 8 Radial profiles of the time-averaged non-dimensionalizedparticknumber concentration Mi and particle diameter df nearest the impinging surface at different W D cases HID: A2; 7 3 ; + 4 0.20 - 1
3.5.2 Flow in the stagnation region Figure 8 shows the variations of time averaged non-dimensionalized particle number intensity [Fig.8(a)] and volume averaged diameter [Fig.8(b)] approaching the stagnation point within 2 nozzle diameters nearest the impingement surface. In general, the region up to 2 nozzle diameters from the stagnation point can be regarded as the impingement zone of the jet, which is usually received attentions by practical applications. From the figure it can be seen the particle number is largest in the stagnation point and then decrease in the radial direction, but the variance of particle diameter shows the reverse trend. Additionally, increasing the nozzle-to-surface distance, HID, leads to a decrease in particle number intensity while an increase in particle diameter. Ths is because particles have more residence time to coagulate and thus particle number decrease more significantly as HID is higher. 3.5.3 Flow along the impinging surface Figure 9 depicts the axial variances of time averaged non-dimensionalized particle*number intensity and volume averaged diameter at x =3, for the cases of HID=2, 3,4. As shown in Fig.9(a), increasing HID leads to an increase in particle number intensity. In addition, increasing HID broadens the particle profiles in the axial direction as a result of enhanced diffusion of wall jet flow. From the comparison of Fig.9(b), it can be seen that the more HID, the larger particle diameters. The reason is particles have the more chances December, 2007
Is 0.05 -
0.2
0
0.4
0.6
0.8
3.0 2.5
IS 2.0 I .5 0
I
I
1
0.2
0.4 Y'
0.6
dp'
0.8
(b) Figure 9 Effect of the nozzle-to-platedistance on the axial variance of time averaged - non-dimensionalized M,' and volume averaged particle number intensity diameter df at x*=3 for Re=10000 HID: +2; 7 3 ; A 4
Quadrature Method of Moments for Nanoparticle Coagulation and Diffusion in the Planar Impinging Jet Flow
to collide and adhere to form larger ones as H/D increases. The influence of Reynolds number on nanoparticle dynamics in the wall flow jet region was investigated through comparing three different Re cases for H/D=3. Fig.10 shows the variances of time averaged non-dimensionalized particle number intensity and volume ayeraged diameter in the axial direction at the location x =3. From Fig.lO(a) it can be seen that the particle number intensity increases with increasing Reynolds number throughout the whole wall jet region. This is because particles have less residence time to coagulation in the gas flows with high Reynolds number. Contrary to the variance of particle number intensity, the particle diameter decreases as the Reynolds number increases, which is displayed in Fig. 10(b). In Fig.lO(b), the rapid decrease of particle diameter at about y*=O.7 attributes to the very low particle number intensity there.
ment was obtained between model predictions and experimental data. (2) The results show coherent structure evolution has a pronounced effect on the particle number intensity, diameter and polydispersity distributions, and acts to increase the dilution of particle-laden jet. As the particle-laden jet develops downstream, the particle number intensity decreases while particle diameter increases. Throughout the impinging jet flow, increasing Reynolds number or decreasing nozzle-to-plate distance results in an increase in particle number intensity and a decrease in particle diameter. Perpendicular to impinging plate, the particle profile broadens with increasing nozzle-to-plate distance.
NOMENCLATURE C C, c D dp H Kn
kB L, r
Mk NQ n, N p Re T To t
u0 u u,
v, vI W
w 4Y
/3
f A d 0
0.2
0.4
0.6
p v p pp
0.8
y*
Cunningham correction factor Smagorinsky constant mean thermal velocity, m.sCl width of the nozzle, m volume-averaged particle diameter nozzle-to-plate distance, m particle Knudsen number ( Kn = p,, /2p, ) Boltzrnann constant (kB= 1.38054 X 10-23J.K-1) particle diameter, m kth moment of the particle size distribution, m-3 order of the quadrature formulation particle number concentration , m-3 pressure, Pa Reynolds number ( Re = U o D / p) temperature, K SGS stress tensor time, s injection gas velocity, m.s velocity vector, m.s-’ velocity in the ith direction, m s - ’ particle volume, m3 polydispersity index quadrature weight coordinates, m particle collision kernel particle Brownian diffusion constant, m2.s-’ width of the filter, m mean free path of the gas, m gas viscosity, k g 6 l . s - I kinematic viscosity, m2.s I gas density, kgm-3 particle density, kgm
-’
Superscripts
(b)
Figure 10 Effect of the Reynolds number on the axial variance of time-averaged non-dimensionalizedparticle -
835
-
number intensity Mi and particle diameter d; at x*=3 for HID=3 Re: A 10000; r20000; 4 15000
*
REFERENCES 1 2
4 CONCLUSIONS (1) A mathematical model embedding particle dynamics in a confined impinging jet was developed for the simulation of particle evolution subject to large coherent structures. An efficient quadrature moment method was allowed to approximate the general dynamic equation, and the large eddy simulation was used to estimate impinging jet flow. Excellent agree-
non-dimensional quantity
3
van de Ven, T.GM., Kelemen, S.J., “Characterizing polymers with an impinging jet”, J. Colloid Interface Sci., 181, 118-123(1996). Pan, Y.K., Tanaka, T., Tsuji, Y., “Turbulence modulation by dispersed solid particles in rotating channel flows”, Int. J. Multiphase Flow, 28, 527-552(2002). Hwang, S.D., Lee, C.H., Cho, H.H., “Heat transfer and flow structures in axisymmetric impinging jet controlled by vortex pairing”, Int. J . Heat Fluid Flow, 22, 293300(2001).
4 5
Adamczyk, Z., Musial, E., Siwek, B., “Kinetics of particle deposition in the oblique impinging jet cell”, J . Colloid Interface Sci., 269, 53-61(2004). Voss, A., Finlay, W.H., “Deagglomeration of dry powder
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pharmaceutical aerosols”, Int. J. Pharm., 248, 3950(2002). 6 Wang, Z.L., Lange, C.F., Finlay, W.H., “Use of an impinging jet for dispersion of dry powder inhalation aerosols”, Int. J. Pharm., 275, 123-131(2004). 7 Yu, M.Z., Lin, J.Z., Chen, L.H., Chan, T.L., “Large eddy simulation of a planar jet flow with nanoparticle coagulation”, Acta Mechanica Sinaca, 22(4), 293-300(2006). 8 Otto, E., Fissan, H., Park, S.H., Lee, K.W., “The log-normal size distribution theory of Brownian aerosol coagulation for the entire particle size range, part 11-Analytical solution using dahneke’s coagulation kernel”, J . Aerosol. Sci., 30, 17-34(1999). 9 Pratsinis, S.E., Kim, K.S., “Particle coagulation, diffusion and thermophoresis in laminar tube flows”, J. Aerosol. Sci., 20, 101-lll(1989). 10 Dahneke, B., “Simple kinetic theory of Brownian diffusion in vapors and aerosols”, In: Theory of Dispersed Multiphase Flow, Academic Press, New York (1983). 11 Smoluchowski, V., “Versuch einer mathematischen theorie der Koagulationskinetik kollider losungen”, Z. Phys. Chemie-Int. J . Res. Phys. Chem. Chem. Phys., 92,129168(1917). 12 Miller, S.E., Garrick, S.C., “Nanoparticle coagulation in a planar jet”, Aerosol Sci. Technol., 38,79-89(2004). 13 Hulbert, H.M., Katz, S., “Some problems in particle technology, a statistical mechanical formulation”, Chem. Eng. Sci., 19,555-574(1964). 14 Lin, J.Z., Chan, T.L., Liu, S., Zhou, K., Zhou, Y., Lee, S.C., “Effects of coherent structures on nanoparticle coagulation and dispersion in a round jet”, Inter. J . of Nonlinear Sciences and Numerical Simulation, 8( l), 45-54( 2007). 15 Settumba, N., Ganick, S.C., “A comparison of diffusive transport in a moment method for nanoparticle coagulation”, J. Aerosol. Sci., 35, 93-lOl(2004). 16 Landgrebe, J.D., Pratsinis, S.E., “A discrete-sectional model for particulate production by gas-phase chemical reaction and aerosol coagulation in the free-molecular regime”, J. Colloid Inter&ace Sci., 139, 63-86( 1990). 17 Gelbard, F., Seinfeld, J.H., “Simulation of multicomponent aerosol dynamics”, J . Colloid Interface Sci., 78, 485-501( 1980). I8 Marchisio, D.L., Vigil, R.D., Fox, R.O., “Quadrature method of moments for aggregation-breakage processes”, J . Colloid Interface Sci., 258, 322-334(2003). 19 McGraw, R., “Description of aerosol dynamics by the quadrature method of moments”, Aerosol Sci. Technol., 27,255-265( 1997).
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21 22 23 24 25 26 27 28 29 30 31
32 33
34
Rosner, D.E., McGraw, R., Tandon, P., “Multivariate population balances via moment and Monte Carlo simulation methods, an important sol reaction engineering bivariate example and ‘mixing’ moments for the estimate of deposition, scavenging, and optical properties for populations of nonspherical suspended particles”, lnd. Eng. Chem. Res., 42,2699-2711(2003). Smagorinsky, J., “General circulation experiments with the primitive equation”, Mon. Weather Rev., 91, 99164( 1963). Germano, M., Piomelli, U., Moin, P., Cabot, W.H., “A dynamics subgrid-scale eddy viscosity model”, Phys. Fluids, A3, 176*1765(1991). Lilly, D.K., “A proposed modification of the Germano subgrid scale closeure method”, Phys. Fluids, A, 633635( 1992). Nicoud, F., Ducros, F., “Subgrid-scale stress modelling based on the square of the velocity gradient tensor”, Flow Turbul. Combust., 62, 183-200( 1999). Friedlander, S.K., Smoke, Dust and Haze, Fundamentals of Aerosol Behavior, Wiley, New York (2000). Moody, E.G., Collins, L.R., “Effect of mixing on the nucleation and growth of titania particles”, Aerosol Sci. Technol., 37.403-424( 2003). Fuchs, N.A., “Zur Theorie der Koagulation”, Z. Phys. Chemie, 171A, 199-208 (1934). Allen, M.D., Raabe, O.G., “Slip correction measurements of spherical solid aerosol particles in an improved Millikan apparatus”, Aerosol Sci. Technol., 4, 269-286( 1985). Marchisio, D.L., Pikturna, J.T., Fox, R.O., Vigil, R.D., Barresi, A.A., “Quadrature method of moments for population-balance equations”, AIChE J.,49, 1266-1276(2003). Yu, M.Z., Lin, J.Z., Chan, T.L., “Numerical simulation of nanoparticle synthesis in diffusion flame reactor”, Powder Technol., 180,9-20(2007). Marchisio, D.L., Vigil, R.D., Fox, R.O., “Implementation of the quadrature method of moments in CFD codes for aggregation-breakage problems”, Chem. Eng. Sci., 58, 3337-3351(2003). Heinz, O., Ilyushin, B., Markovich, D., “Application of a PDF method for the statistical processing of experimental data”, Int. J . Heat Fluid Flow, 25, 864-874(2004). Baydar, E., Ozmen, Y., “An experimental and numerical investigation on a confined impinging air jet at high Reynolds numbers”, Appl. Therm. Eng., 25, 409421(2005). McGraw, R., Nemesure, S., Schwatrz, S.E., “Properties and evolution of aerosols with size distributions having identical moments”, J . Aerosol. Sci., 29,761-772( 1998).
(2007)
Quadrature Method of Moments for Nanoparticle Coagulation and Diffusion in the Planar Impinging Jet Flow* YU Mingzhou(7 fl fl! )a, LIN Jianzhong(#&,$i.)a9bv**, and XIONG Hongbing( #6 ?I%)" a
The State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 3 10027, China China Jiliang University, Hangzhou 310018, China
Abstract A computational model combining large eddy simulation with quadrature moment method was employed to study nanoparticle evolution in a confined impinging jet. The investigated particle size is limited in the transient regime, and the particle collision kernel was obtained by using the theory of flux matching. The simulation was validated by comparing it with the experimental results. The numerical results show coherent structure acts to dominate particle number intensity, size and polydispersity distributions, and it also induce particle-laden jet to be diluted by the ambient. The evolution of particle dynamics in the impinging jet flow are strongly related to the Reynolds number and nozzle-to-plate distance, and their relationships were analyzed. Keywords nanoparticle, coagulation, impinging jet, large eddy simulation, quadrature moment method
1 INTRODUCTION Impingement of nanoparticles (e.g. colloid, bioparticles, macromolecules) on a solid surface occurs in many industrial processes, for example, the contamination control of microelectronic manufacture and cooling, pharmaceutical processing, ventilation during mining and tunneling activities, paint spraying and vehicle exhaust plumes (Fig.1). The impinging jet is also a well-established technique to study the deposition of colloid particles onto a solid surface under well-defined hydrodynamic and physicochemical conditions[11. In these applications, the particle diameters are usually smaller than 1pm and particle coagulation due to Brownian motion dominates over all the jet developing, and thus universal models such as discrete phase model (DPM)[2] for describing particle dynamics are no longer available for the ultrafine particles in these cases. Additionally, the turbulent impingement jet has the pronounced complicated coherent structures in the stagnation region and wall flow region, which usually leads to non-determined particle distribution. Under such circumstances, the studies for ultrafine particle dynamics in impingement jet are greatly limited. However, it is essential to understand the physics of nanoparticles in the above enumerative phenomena.
Figure 1 Schematic diagrams of a planar impinging flow
It should be pointed out that few studies have been reported for particle-laden jet impingement in the literature. These works are mostly limited to experimental research for capturing the evolution of coher-
ent structures[3] and examining the particle deposition[4], without considering the instantaneous influence of coherent structures on the ultrafine particle dynamics. One the other hand, Voss and Finlay[S] and Wang et a1.[6] have investigated the influence of turbulence induced by impingement plate on the particle deagglomeration, but they did not consider the interaction of flow structures and the nanoparticle evolution. Particles in the range between lnm and 1pm stick together as a result of van der Waals forces when once collided, the collision rate of particles can be treated as the coagulation rate and this terminology is used in most of literature[7]. In this region the particle size is smaller than Kolmogorov scale and thus the influence of turbulence on particle deagglomeration is negligible. Under this circumstance, particle dynamics is mainly dominated by the particle coagulation due to Brownian motion. Based on the particle diameter, the particles coagulation can be divided into three different regimes: free-molecule regime (for particles with diameters smaller than lOnm), transition regime (for particles with diameter from lOnm to 100nm) and continuum regime (for particles with diameters larger than lOOnm but smaller than lpm). In different regimes, the particle collision kernels exhibit different coagulation mechanisms[8]. Previous studies for particle coagulation are mostly limited in the free-molecule regime or continuum regime[9], but few in the transition regime[8], because the particle collision is not easily described by mathematical expression in this regime. In order to solve the coagulation equation one needs to decide which coagulation coefficient to employ. In general, the coagulation coefficient in the transition regime is fixed by harmonic mean, but its relative error is proved to be more than other detailed coagulation coefficient[101. The transport of the nanoparticles dispersed throughout the fluid is governed by the aerosol general
Received 2006- 10-13, accepted 2007-09-04.
* Supported by the Ministry of Science and Technology of China (No.2005CCA06900).
** To whom correspondence should be addressed. E-mail: [email protected]
Quadrature Method of Moments for Nanoparticle Coagulationand Diffusion in the Planar Impinging Jet Flow
dynamics equation (GDE)[ 111. The GDE describes particle dynamics under the influence of various physical and chemical phenomena: convection, diffusion, coagulation, surface growth, nucleation, and the other internal or external forces[ 121. Deeply understanding nanoparticle dynamics requires the incorporation the GDE into the computational fluid equations, especially for the processes with complicated flow structures. Up to now, there are several approaches to the modeling and simulation of aerosols based on GDE equation. The moment method (MM) was first introduced by Hulbert and Katz[ 131 and considered to be a powerful method in the field of aerosols because it can provide the representative properties of particles while requiring only the computation of the first three moments[l4]. However, this method may not be the most accurate because it requires the prior assumption for particles size distribution (PSD). Recently, the moment method has been conducted conjunction with direct number simulation (DNS) to investigate the temporal mixing layers[ 151. The sectional method (SM) is another universal technology for resolving the aerosols problems and has been further developed[ 161 after it is proposed by Gelbard and Seinfeld[l7]. In this approach, the continuous size or length range of the internal coordinate is partitioned into a finite series of contiguous sections or bins. This method has the ability to capture the details of early stages of particle clustering where the prior assumption of particle size distribution is not available. A disadvantage of the sectional method is that each section or bin corresponding to a transport equation must be separately solved, which can be prohibitively expensive computationally. An attractive alternative is the quadrature method of moments (QMOM)[IS]. This method was first introduced by McGraw[ 191 for modeling aerosol evolution and has recently been extended to aggregation and breakage problems using both a monovariate and a bivariate PSD[ IS]. The QMOM doesn’t define or produce an explicit size distribution, but the moments could be used with an assumed functional form to obtain a closed set of equations for all moments[20]. This approach is viewed as a competing method that presents the main advantage of being extremely accurate and amenable for coupling with CFD codes[lS]. In this work, the large eddy simulation (LES) combined with the QMOM is performed to investigate the nanoparticle dynamics in the impingement flow systems, which is not reported before. The jet suspended with nanoparticles perpendicularity impinges on a flat plate with an initialization exit velocity, and the particle deposition on the surface of impingement plate due to absorption is not considered. This methodology facilitates the capture of the underlying physics in a time-accurate manner and can describe accurately the nanoparticle evolution under the influence of the physical phenomena: convection, diffusion, and Brownian coagulation, especially in the cases where the flow structures are significantly influenced by the impingement plate. The particles in this study are selected in the transient regime, since the nanoparticle diameters in most of the practical situations range
829
from lOnm to 100nm.
2 MATHEMATICAL FORMULATION 2.1 Fluid flow The schematic of a planar impinging jet is shown in Fig.1, in which x and y are the radial and axial directions, respectively. The width of the nozzle D is 25mm and the Reynolds number, defined as Re= DUdv with v being the kinematic viscosity of the fluid, is 10000, 15000, 20000 and 30000. The nozzle-to-plate distances H are 1,2, 3 and 4 nozzle width. The equations governing the incompressible flow, obtained by filtering the Navier-Stokes equation, are:
aq -
,-0
dx;
where q is the filtered velocity, is the filtered pressure, index i, j is taken as 1,2 and refers to the x and y directions, respectively, Tii is the sub-grid scale (SGS) stress tensor. The first SGS stress model, proposed by Smagorinsky[21], is adopted here. His model assumes that the SGS stress follows a gradient-diffusion process, similar to molecular motion. Consequently, Tc is given by the eddy viscosity assumption: qj = 2v,s,- +1-T,,qj (3) 3 where
-
s, is the filtered rate of
the strain tensor,
S.. IJ =(alii/axj+alijlaxi)12; v, =(C,A)*ISI, here C, is the Smagorinsky constant of 0.1, A is the filter width A = (Ax, . Ax,)”2 with Axi(i = 1,2) being i direction mesh size,
Is/= (2susi,)1/2. In the present
study, v, is calculated using the dynamic subgrid model proposed by German0 el al.[22] and developed by Lilly[23]. The width of the test filter in this model was defined as:
(4) The values of the symbols in this equation can be seen in the studies of Lilly[23]. In order to alleviate the dissipation near the impinging plate, the Wall-Adapting Local Eddy-Viscosity (WALE) model is used[24].
2.2 Particle field In this study the volume fraction of particles is and thus the transport properties very small (< of the gas can be approximated by constants. In addition, compared to the Kolmogorov scale, the particle is so small (<100nm) that the influence of solid phase on the gas phase is negligible and the particle fragChin. J. Ch. E. lS(6) 828 (2007)
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mentation and agglomeration due to turbulent shear is not considered. At these circumstances, the dynamics behavior of particles suspended in gas undergoes Brownian diffusion, the transport due to gas convection and Brownian coagulation in the transient regime. Let n(v,t)dv be the particle number in a volume range between v and v+dv per unit mass of gas at time t. Then the coagulation equation for the number concentration function is [25]: an(v9t) +v~un(v,t)=V.r(v,t)Vn(v,t)+ at
1 2
-
l P (vl, v
- vl ) n ( vI ,t ) n ( v - v1,t)dv, - n(v, t)
Here u is the local flow velocity, T(v, t ) is the particle Brownian diffusion constant for volume v, p is the collision kernel in the transient regime. The first term on the left-hand suggest the rate of change of particle number concentration while another term describes the loss or gain of particles due to convection. The fist term on the right-hand denotes the influence of Brownian diffusion on the nanoparticles and the last two terms dominate monomer gain or loss due to Brownian coagulation. In order to express the number density function using particle length L as the internal coordinate, the followed transformation in terms of spherical shape is chosen: v=- ?cL3 6 The relationship between the two expressions of the number density function is:
Then Eq.(5) can be written as aN(L't) + V . uN(L,t) = V . T(L,t)VN(L,t) + at
In order to employ the QMOM method [191, the followed assumption is used:
where NQis the order of the quadrature formulation. L; is the particle diameter and w;is the number intensity. Substituting Eq.( 11) into Eq.( 10) results in the closed coagulation equations based on the particle diameters for kfk: aMk(t)
at
where Pj = P ( L j , L j ) . In order to simplify the treatment and analysis of the interaction between the hydrodynamic and particulate fields, Eq.( 12) is non-dimensionalized using the following relations: *
X'
x . =J , ' D
* t=-
t
DIU,
,
*
u.=',
u.
' uo
L* . = JL.
' r,'
w *.=L w ., M ,* = - M k J
WO
MkO
The characteristic length scale is D, the characteristic velocity is Uo, the reference particle length is b,the reference particle number intensity is wo and the reference value of the kth moment is Mm. Substituting the relations given in Eq.(13) into Eq.( 12) results in the non-dimensionalized .equation for the kth moment, Mk: at*
N(z, t)dz - N(L, t )
+ v .uM,(t) =
ax;
P(L, z)N(z,t)dz
Applying the moment transformation[181: Mk(t)= r N ( L , t ) L k d L the general coagulation equation is resulted: aMk(t) + ~ . u M , ( t ) = at
V.T(L,t)VM,(t)+l r N ( z , t ) X 2 December, 2007
(14) the particle Schmidt number, SC, = v / r ,defined as the ratio of the fluid kinematic viscosity to the particie mass diffusivity[26]. Weights ( wi ) and abscissas ( 4 ) are determined through the product-difference (PD) algorithm from the lower-order moment[l9]. In the transition regime the coagulation rate is
Quadrature Method of Moments for Nanoparticle Coagulation and Diffusion in the Planar Impinging Jet Flow
described neither by the continuum theory nor by simple kinetic theory. Based on the theory of flux matching[27], the coagulation function is mainly expressed as an enhancement of the collision function for the near-continuum regime[8]:
P( ' i
9
' j
) = P c o ("i
'Vj
) f ( Kn)
In Eqs.(ll) and (13) P c 0 ( v i , v j )is the coagulation function in the continuum plus near-continuum regime and is represented by the following equation:
where Kco = 2kBT/(3,u) is the coagulation constant for the continuum plus near-continuum regime, kB is the Boltzmann constant and C ( v )= 1+ Kn [Al + A2 exp(-A, / Kn)] is the slip correction factor. A , = 1.142, A2 = 0.558, A3 = 0.999 are constants determined by Allen and Raabe[28]. Equation (18) can be further represented based on non-dimensionalizedparticle length:
+
(L; L ; ) [ C (L: ) / L: + C ( LJ )I LJ
]
(19) +A2exp(-A3C&/22)].
where C(L;)=l+,-[Al 22
Li & is the coagulation function in the free-molecule regime and given by[25]:
Pfm(v,V )
where C is the mean thermal velocity, and
=(&)
116
Kfm
Since the particle size in this study is smaller than Kolmogorov micro-scale, the turbulent diffusivity for nanoparticles is negligible and thus only Brownian diffusivity in the transition regime is considered. The particle diffusivity coefficient r is[ 121:
(15)
Followed by Dahneke's theory[ 101, the original enhancement functionfiKn) can be obtained[9]: 1+ Kn f (Kn)= 1+ 2Kn + 2Kn2 Otto et a1.[8] reveals that the particle Knudsen number Kn can be considered as the ratio of the coagulation coefficient of the limiting regimes and thus Kn is expressed as follows:
P,, (L:,L; ) = K,,
831
(6k~T/&)112
is the coagulation constant for the free-molecule regime. Equation (20) is represented based on nondimensionalizedparticle length:
3 RESULTS AND DISCUSSION 3.1 Computation conditions In the computation, the filer size for large eddy simulation is taken to be equal to the grid size, and the width of test filter is larger than that of grid filter. The implicit difference scheme in time and the second-order Crank-Niscolson scheme are employed to solve the equations. The QUICK method is adopted for the convective terms in Egs.(6) and (13). The nondimensionlized time step At is a constant, 0.012. The fluid in this study is air at a tempeture of T=300K. The computational grid is comprised of 400 X 800, 600 X 800, 800 X 800 for nozzle-to-plate distance of 2, 3, 4 nozzle width and the computational size in the x direction is 10 nozzle width. The present work concentrates on nanoparticles with diameter below lOOnm and particle density 1000kg.m-3 in the gas flows. In this size regime, particle coagulation mainly attributes to van der Waals forces when once collided. Since some results[ 18,291 show that a quadrature approximation with three nodes (NQ=3) is sufficient to describe the moment evolution with good accuracy, the QMOM calculation in this work was conducted with three nodes (tracking the first six moments, QMOM). For this case of coagulation alone, the initialized particle lengths of three classes at the nozzle exit are
cn=1.0, cf,=2.0 and Go=[c,,+Go]
113
with reference particle length & = lOnm . The corresponding particle number intensity for these three classes are w;, =0.5 , win =0.5 and w;" = O with reference number intensity wo = 1.0 X 1017 particles per m3. In this numerical simulation, the particle length is assumed to be particle diameter and then the above initialization determines the initial particle diameter dPoshould be 12.5nm at nozzle exit using the volume averaged diameter d , = (4RM3/ 3M0)113 / 2 . Knowledge of the initial particle number intensity and diameter is sufficient for determining the initial conditions of the six moments ( M o o= MIn= 1.5 X lo9 , M20 =25 , M30 =4.5X10-7 , M a =8.5X10-'5 , M50 = 1.65 X
). In the initialization computation,
the variables for all moments are initialized by zero except in the nozzle where all moments are given 1. For every time step, locations where particle number Chin. J. Ch. E. 15(6) 828 (2007)
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Chin. J. Ch. E. (Vol. 15, No. 6 )
concentration (Mi) is less than 0.005 are considered to be free of particles.
3.2 Validity test The validity of the present model is tested against results from a simpler model in which only Brownian coagulation is considered. In this case, the influence of gas flow on particle diffusion and transport is neglected and thus Eq.(12) can be reduced to the following form:
NQ
NQ
cL:wiCfijwj i
The contours of vorticity by using the large eidy simulation method at non-dimensionlized time t = 1200 are shown in Fig.3(a). When the jet develops enough, the vortices generate near the nozzle flow, move downstream and finally induce secondary vortices emergence in the wall jet flow regions, which is similar with the visualization of Fig.3(b) made by Heinz et al.[32]. This process repeats periodically. Fig.4 illustrates the measured and computed axial time-averaged velocity profiles along the jet centerline for Re=30000 and HID= 1, 2. It is seen from the figure, the experimental[33] and numerical data are in good agreement.
( k =0;..,5)
(23)
1
In this work, the results obtained from Eq.(23), i.e. the present QMOM method, are compared with the sectional model and analytical solution[8]. Similar to the solutions of Otto et al.[8], the initial parameters for the comparison calculations are & = 94nm, 0,= 1.8, NO= T = 300K , p = 1000kg.mP3 In Fig.2, the change of the number concentration with dimensionless time K,,NOt predicted by the QMOM method, the analytical solution and sectional model are presented. No significant difference among three curves can be found from Fig.2, which suggests the present QMOM method has an ability to solve the problem involving nanoparticle coagulation. As the QMOM method is incorporated into the computational fluid dynamics (CFD), it is necessary to predict the nanoparticle lunetics in the evolving flows. In fact, the approach combining the QMOM method with the CFD code has been used in many complicated particulate systems such as nanoparticle synthesis in a diffusion flame reactor[30] and particle aggregation-breakage processes in turbulent Taylor-Couette flow[? 11.
(a) By large eddy simul?tion at nondimensionlized time r = 1200
Figure 3
(b) PIV by Heinz et a1.[32] Contours of vorticity in the evolution of vortex 1.2
0.8
0
0.2
0.4
0.6
0.8
1.0
Y'( v / D )
Figure 4 The axial velocity profiles along the jet centerline for Re=30000 _ _ ~ HID=l _ Num; HlD=2 Num; 0 HID= 1 Exp; HID=2 Exp ~
K2"t
Figure 2 The comparison for number concentration decay as a function of dimensionless time o sectional method; - - - analytical method; QMOM method (this study)
3.4 Effect of coherent structures on the nanoparticle dispersion and coagulation Figure 5 depicts the instantaneous contours of the non-dimensionalized zero moment Mi , the non-dimensionalized volume-averaged diameters d i
~
~
3.3 Evolution of coherent structures and comparison with experiment In order to understand the relationship between coherent structures and nanoparticle dynamics, the evolution of coherent structures is described firstly. Coherent structures can be illustrated by vorticity field. December, 2007
and polydispersion W when the jet impingement develops enough at nondimensionalized time t* = 1200.
Mi corresponds to the particle number concentration. In theory, coagulation is a process in which particles collide with one another and adhere to form larger particles. The net result is a continuous decrease in number concentration and an increase in particle size.
Quadrature Method of Moments for Nanoparticle Coagulation and Diffusion in the Planar Impinging Jet Flow
833
3.5.1
Pre-impingingjet The particle number intensity and diameter distributions along the jet axis are important to understand the dynamics of the nanoparticle-fluid flow and to explain the behavior of particle diffusion under the influence of impingement plate and particle coagulation. Figs.6(a) and (b) illustrates the time averaged
-
non-dimensionalized particle number intensity Mi -
Figure 5 Instantaneous contoyrs of the non-dimeyionalized zero moment M, ,the particle diameter $, and the polydispersity Win the impinging jet flow with coherent struttures at nondimensionalized time t =1200
and volume averaged diameter di profiles along the jet centerline at HID=2, 3 and 4 for Re=10000, respectively. It is seen that the variance of particle number intensity exhibits the same trend for the different nozzle-to-plate distance, so does the particle diameter. Near the nozzle, the number intensity decreases significantly due to strong particle collision and then decreases slightly under the corporate influence of large coherent structures and particle coagulation. In the very small distance from the impinging surface, the presence of impingement plate leads to the accelerated loss of the particle number intensity. Increasing HID leads to a decrease in particle number intensity and an increase in particle diameter. The influence of Re on the particle number intensity and diameter at the same nozzle-to-plate distance, HID= 3, is shown in Fig.7(a) and Fig.7(b), respectively. It can be seen from this figure increasing Reynolds numbers results in a decrease in particle number intensity while an increase in particle diameter along the jet centerline. 1 .o
The polydispersity index is a measure of the width of the size distribution function. It is defined as W =[(M,M, -Mf)lM,]”2 [34]. From the figure it
0.8
1%
0.6
can be seen that the distributions of Mi , di and W all
0.4
depend similarly on the configuration of coherent structures. In the process of the jet exchanging with the ambients, these coherent structures carry particles to embed into the surrounding gas and thus the particle-laden jet is diluted. Before impinging the plate surface, the particle number intensity decreases downstream under the influence of particle coagulation and diffusion induced by shear vortices, while the particle diameter and polydispersion increase correspondingly. In the stagnation region and wall flow regions, the number intensity decreases in the radial direction, while the mean particle diameter increases correspondingly.
0.2
3.5 Distribution of time averaged particle quantities The solution by using time averaged method allows both qualitative and quantitative assessments of the spatio-temporal evolution of the particle field. In the present work, time averaged data typically employ 8000 samples, which is sufficient for one sweep of the present investigated flows. In fact, quantities obtained in this way correspond to results generated in Reynolds-Averaged Navier-Stokes (RANS) simulations[7] and are all denoted by an over-bar.
0
0.2
0.4 0.6 Y’ ( Y / W -
0.8
1.0
(b) d i Figure 6 Variation of the time-averaged non-dimensionalizedparticle number concentration Mi -
and diameter d i at the jet centerline for the different HID cases with Re =10000 HID: *2; ~ 3 A;4 Chin. J. Ch. E. 15(6) 828 (2007)
Chin. J. Ch. E. (Vol. 15, No. 6)
834 1.0
I
::::B
I
0.12
0.10
I
I
0.2
0
0.4
I
I
I
0.6
0.8
1.0
-2
0
1
2
0
I
2
-1
Y' (.vlD)
2.9 2.8
L '
2.4 -2
1.4 -
0
I
I
-1
X*
0.2
0.4
0.6
0.8
1.0
Y' ( Y O
(b) Figure 7 Variation of the time-averaged non-dimensionalizedparticle number concentration Mi -
and diameter df at the jet centerline for the different Re cases with HID=3 Re: A 10000; + 15000; 720000
dp'
(b) Figure 8 Radial profiles of the time-averaged non-dimensionalizedparticknumber concentration Mi and particle diameter df nearest the impinging surface at different W D cases HID: A2; 7 3 ; + 4 0.20 - 1
3.5.2 Flow in the stagnation region Figure 8 shows the variations of time averaged non-dimensionalized particle number intensity [Fig.8(a)] and volume averaged diameter [Fig.8(b)] approaching the stagnation point within 2 nozzle diameters nearest the impingement surface. In general, the region up to 2 nozzle diameters from the stagnation point can be regarded as the impingement zone of the jet, which is usually received attentions by practical applications. From the figure it can be seen the particle number is largest in the stagnation point and then decrease in the radial direction, but the variance of particle diameter shows the reverse trend. Additionally, increasing the nozzle-to-surface distance, HID, leads to a decrease in particle number intensity while an increase in particle diameter. Ths is because particles have more residence time to coagulate and thus particle number decrease more significantly as HID is higher. 3.5.3 Flow along the impinging surface Figure 9 depicts the axial variances of time averaged non-dimensionalized particle*number intensity and volume averaged diameter at x =3, for the cases of HID=2, 3,4. As shown in Fig.9(a), increasing HID leads to an increase in particle number intensity. In addition, increasing HID broadens the particle profiles in the axial direction as a result of enhanced diffusion of wall jet flow. From the comparison of Fig.9(b), it can be seen that the more HID, the larger particle diameters. The reason is particles have the more chances December, 2007
Is 0.05 -
0.2
0
0.4
0.6
0.8
3.0 2.5
IS 2.0 I .5 0
I
I
1
0.2
0.4 Y'
0.6
dp'
0.8
(b) Figure 9 Effect of the nozzle-to-platedistance on the axial variance of time averaged - non-dimensionalized M,' and volume averaged particle number intensity diameter df at x*=3 for Re=10000 HID: +2; 7 3 ; A 4
Quadrature Method of Moments for Nanoparticle Coagulation and Diffusion in the Planar Impinging Jet Flow
to collide and adhere to form larger ones as H/D increases. The influence of Reynolds number on nanoparticle dynamics in the wall flow jet region was investigated through comparing three different Re cases for H/D=3. Fig.10 shows the variances of time averaged non-dimensionalized particle number intensity and volume ayeraged diameter in the axial direction at the location x =3. From Fig.lO(a) it can be seen that the particle number intensity increases with increasing Reynolds number throughout the whole wall jet region. This is because particles have less residence time to coagulation in the gas flows with high Reynolds number. Contrary to the variance of particle number intensity, the particle diameter decreases as the Reynolds number increases, which is displayed in Fig. 10(b). In Fig.lO(b), the rapid decrease of particle diameter at about y*=O.7 attributes to the very low particle number intensity there.
ment was obtained between model predictions and experimental data. (2) The results show coherent structure evolution has a pronounced effect on the particle number intensity, diameter and polydispersity distributions, and acts to increase the dilution of particle-laden jet. As the particle-laden jet develops downstream, the particle number intensity decreases while particle diameter increases. Throughout the impinging jet flow, increasing Reynolds number or decreasing nozzle-to-plate distance results in an increase in particle number intensity and a decrease in particle diameter. Perpendicular to impinging plate, the particle profile broadens with increasing nozzle-to-plate distance.
NOMENCLATURE C C, c D dp H Kn
kB L, r
Mk NQ n, N p Re T To t
u0 u u,
v, vI W
w 4Y
/3
f A d 0
0.2
0.4
0.6
p v p pp
0.8
y*
Cunningham correction factor Smagorinsky constant mean thermal velocity, m.sCl width of the nozzle, m volume-averaged particle diameter nozzle-to-plate distance, m particle Knudsen number ( Kn = p,, /2p, ) Boltzrnann constant (kB= 1.38054 X 10-23J.K-1) particle diameter, m kth moment of the particle size distribution, m-3 order of the quadrature formulation particle number concentration , m-3 pressure, Pa Reynolds number ( Re = U o D / p) temperature, K SGS stress tensor time, s injection gas velocity, m.s velocity vector, m.s-’ velocity in the ith direction, m s - ’ particle volume, m3 polydispersity index quadrature weight coordinates, m particle collision kernel particle Brownian diffusion constant, m2.s-’ width of the filter, m mean free path of the gas, m gas viscosity, k g 6 l . s - I kinematic viscosity, m2.s I gas density, kgm-3 particle density, kgm
-’
Superscripts
(b)
Figure 10 Effect of the Reynolds number on the axial variance of time-averaged non-dimensionalizedparticle -
835
-
number intensity Mi and particle diameter d; at x*=3 for HID=3 Re: A 10000; r20000; 4 15000
*
REFERENCES 1 2
4 CONCLUSIONS (1) A mathematical model embedding particle dynamics in a confined impinging jet was developed for the simulation of particle evolution subject to large coherent structures. An efficient quadrature moment method was allowed to approximate the general dynamic equation, and the large eddy simulation was used to estimate impinging jet flow. Excellent agree-
non-dimensional quantity
3
van de Ven, T.GM., Kelemen, S.J., “Characterizing polymers with an impinging jet”, J. Colloid Interface Sci., 181, 118-123(1996). Pan, Y.K., Tanaka, T., Tsuji, Y., “Turbulence modulation by dispersed solid particles in rotating channel flows”, Int. J. Multiphase Flow, 28, 527-552(2002). Hwang, S.D., Lee, C.H., Cho, H.H., “Heat transfer and flow structures in axisymmetric impinging jet controlled by vortex pairing”, Int. J . Heat Fluid Flow, 22, 293300(2001).
4 5
Adamczyk, Z., Musial, E., Siwek, B., “Kinetics of particle deposition in the oblique impinging jet cell”, J . Colloid Interface Sci., 269, 53-61(2004). Voss, A., Finlay, W.H., “Deagglomeration of dry powder
Chin. J. Ch. E. 15(6) 828 (2007)
836
Chin. J. Ch. E. (Vol. 15, No. 6)
pharmaceutical aerosols”, Int. J. Pharm., 248, 3950(2002). 6 Wang, Z.L., Lange, C.F., Finlay, W.H., “Use of an impinging jet for dispersion of dry powder inhalation aerosols”, Int. J. Pharm., 275, 123-131(2004). 7 Yu, M.Z., Lin, J.Z., Chen, L.H., Chan, T.L., “Large eddy simulation of a planar jet flow with nanoparticle coagulation”, Acta Mechanica Sinaca, 22(4), 293-300(2006). 8 Otto, E., Fissan, H., Park, S.H., Lee, K.W., “The log-normal size distribution theory of Brownian aerosol coagulation for the entire particle size range, part 11-Analytical solution using dahneke’s coagulation kernel”, J . Aerosol. Sci., 30, 17-34(1999). 9 Pratsinis, S.E., Kim, K.S., “Particle coagulation, diffusion and thermophoresis in laminar tube flows”, J. Aerosol. Sci., 20, 101-lll(1989). 10 Dahneke, B., “Simple kinetic theory of Brownian diffusion in vapors and aerosols”, In: Theory of Dispersed Multiphase Flow, Academic Press, New York (1983). 11 Smoluchowski, V., “Versuch einer mathematischen theorie der Koagulationskinetik kollider losungen”, Z. Phys. Chemie-Int. J . Res. Phys. Chem. Chem. Phys., 92,129168(1917). 12 Miller, S.E., Garrick, S.C., “Nanoparticle coagulation in a planar jet”, Aerosol Sci. Technol., 38,79-89(2004). 13 Hulbert, H.M., Katz, S., “Some problems in particle technology, a statistical mechanical formulation”, Chem. Eng. Sci., 19,555-574(1964). 14 Lin, J.Z., Chan, T.L., Liu, S., Zhou, K., Zhou, Y., Lee, S.C., “Effects of coherent structures on nanoparticle coagulation and dispersion in a round jet”, Inter. J . of Nonlinear Sciences and Numerical Simulation, 8( l), 45-54( 2007). 15 Settumba, N., Ganick, S.C., “A comparison of diffusive transport in a moment method for nanoparticle coagulation”, J. Aerosol. Sci., 35, 93-lOl(2004). 16 Landgrebe, J.D., Pratsinis, S.E., “A discrete-sectional model for particulate production by gas-phase chemical reaction and aerosol coagulation in the free-molecular regime”, J. Colloid Inter&ace Sci., 139, 63-86( 1990). 17 Gelbard, F., Seinfeld, J.H., “Simulation of multicomponent aerosol dynamics”, J . Colloid Interface Sci., 78, 485-501( 1980). I8 Marchisio, D.L., Vigil, R.D., Fox, R.O., “Quadrature method of moments for aggregation-breakage processes”, J . Colloid Interface Sci., 258, 322-334(2003). 19 McGraw, R., “Description of aerosol dynamics by the quadrature method of moments”, Aerosol Sci. Technol., 27,255-265( 1997).
December, 2007
20
21 22 23 24 25 26 27 28 29 30 31
32 33
34
Rosner, D.E., McGraw, R., Tandon, P., “Multivariate population balances via moment and Monte Carlo simulation methods, an important sol reaction engineering bivariate example and ‘mixing’ moments for the estimate of deposition, scavenging, and optical properties for populations of nonspherical suspended particles”, lnd. Eng. Chem. Res., 42,2699-2711(2003). Smagorinsky, J., “General circulation experiments with the primitive equation”, Mon. Weather Rev., 91, 99164( 1963). Germano, M., Piomelli, U., Moin, P., Cabot, W.H., “A dynamics subgrid-scale eddy viscosity model”, Phys. Fluids, A3, 176*1765(1991). Lilly, D.K., “A proposed modification of the Germano subgrid scale closeure method”, Phys. Fluids, A, 633635( 1992). Nicoud, F., Ducros, F., “Subgrid-scale stress modelling based on the square of the velocity gradient tensor”, Flow Turbul. Combust., 62, 183-200( 1999). Friedlander, S.K., Smoke, Dust and Haze, Fundamentals of Aerosol Behavior, Wiley, New York (2000). Moody, E.G., Collins, L.R., “Effect of mixing on the nucleation and growth of titania particles”, Aerosol Sci. Technol., 37.403-424( 2003). Fuchs, N.A., “Zur Theorie der Koagulation”, Z. Phys. Chemie, 171A, 199-208 (1934). Allen, M.D., Raabe, O.G., “Slip correction measurements of spherical solid aerosol particles in an improved Millikan apparatus”, Aerosol Sci. Technol., 4, 269-286( 1985). Marchisio, D.L., Pikturna, J.T., Fox, R.O., Vigil, R.D., Barresi, A.A., “Quadrature method of moments for population-balance equations”, AIChE J.,49, 1266-1276(2003). Yu, M.Z., Lin, J.Z., Chan, T.L., “Numerical simulation of nanoparticle synthesis in diffusion flame reactor”, Powder Technol., 180,9-20(2007). Marchisio, D.L., Vigil, R.D., Fox, R.O., “Implementation of the quadrature method of moments in CFD codes for aggregation-breakage problems”, Chem. Eng. Sci., 58, 3337-3351(2003). Heinz, O., Ilyushin, B., Markovich, D., “Application of a PDF method for the statistical processing of experimental data”, Int. J . Heat Fluid Flow, 25, 864-874(2004). Baydar, E., Ozmen, Y., “An experimental and numerical investigation on a confined impinging air jet at high Reynolds numbers”, Appl. Therm. Eng., 25, 409421(2005). McGraw, R., Nemesure, S., Schwatrz, S.E., “Properties and evolution of aerosols with size distributions having identical moments”, J . Aerosol. Sci., 29,761-772( 1998).