Large eddy simulation of particle aggregation in turbulent jets

Journal of Aerosol Science 111 (2017) 1–17

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Large eddy simulation of particle aggregation in turbulent jets a

b

I. Pesmazoglou , A.M. Kempf , S. Navarro-Martinez

a,⁎

MARK

a

Department of Mechanical Engineering, Imperial College, London SW7 2AZ, United Kingdom Chair of Fluid Dynamics, Institute for Combustion and Gasdynamics, and CENIDE, Center for Nano-Integration, Universität Duisburg-Essen, Duisburg 47048, Germany b

AR TI CLE I NF O

AB S T R A CT

Keywords: Aggregation Coagulation Agglomeration Particles Lagrangian Turbulence

Aggregation is an inter-particle process that involves a multitude of different physical and chemical mechanisms. Aggregation processes often occur within turbulent flows; for example in spray drying, soot formation, or nanoparticle formation. When the concentration of particles is very large, a direct simulation of individual particles is not possible and alternative approaches are needed. The present work follows the stochastic aggregation modelling based on a Lagrangian framework by Pesmazoglou, Kempf, and Navarro-Martinez (2016) and implements it in the Large Eddy Simulation context. The new coupled model is used to investigate particle aggregation in turbulent jets. Two cases are considered: an existent Direct Numerical Simulation of nanoparticle agglomeration in a planar jet and an experimental configuration of SiO2 nanoparticles in a round jet. The results show a good agreement in both cases, demonstrating the advantages of the Lagrangian framework to model agglomeration and it capacity to describe the full particle size distribution.

1. Introduction Aggregation (also termed coagulation and agglomeration) in turbulent flows is present in a range of industrial, commercial and physical processes like spray drying, spray atomization, pharmaceutic inhalers or soot formation and growth. Aggregation can have a strong influence on the Particle Size Distribution (PSD) and therefore controls the particle relaxation times and velocities Gavaises, Theodorakakos, Bergeles, and Brenn (1996). It is also the driving process behind soot formation in combustion Zucca, Marchisio, Baressi, and Fox (2006). Aggregation can be both wanted and unwanted, depending on the application, so there is an increasing effort to combine numerical modelling and experiments to investigate the aggregation process in order to control the PSD. Aggregation is the product of an inelastic collision event between two particles; particles can be in the gaseous, liquid, or solid phase. A particle of volume υ (or mass m) is thereby formed by the aggregation of two particles with volumes υ′ and υ − υ′. The resulting volume is subject to the level of coalescence (also termed ‘sintering’ for solids), i.e. the extent to which the colliding-pair volumes merge. Depending on the phase of the particle (solid, liquid, or gaseous), the local thermodynamic conditions (e.g. temperature), and the collision process (angle of attack, relative velocity, etc.), the particle will attach to the ‘receiver’ particle, forming fractal aggregate structures (such as soot) or spherical aggregates. In the present study, the authors consider particles in the nanoscale, where instantaneous coalescence of the colliding pair is assumed. Turbulence can increase or reduce aggregation depending on the particle size, turbulence intensity and Reynolds numbers. Despite the importance of flows with turbulent-aggregation, there have been relatively few attempts to use high-fidelity turbulent solvers coupled with aggregation. In the present work, the aggregation model presented by Pesmazoglou, Kempf, and Navarro-

⁎

Corresponding author. E-mail address: [email protected] (S. Navarro-Martinez).

http://dx.doi.org/10.1016/j.jaerosci.2017.06.002 Received 28 November 2016; Received in revised form 26 April 2017; Accepted 5 June 2017 Available online 08 June 2017 0021-8502/ © 2017 Elsevier Ltd. All rights reserved.

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Martinez (2016) is coupled to Large Eddy Simulations (LES), using the spray-pdf approach by Williams (1958). The aggregation model was validated against analytic solutions Pesmazoglou et al. (2016), the spray-pdf approach was validated by Pesmazoglou et al. Pesmazoglou, Kempf, and Navarro-Martinez, (2013, 2014). The model focuses on applications with large numbers of particles in the domain, > O (108) , mostly on the nano-scale, where one-to-one collision approaches are intractable. The new model is used to investigate particle aggregation in planar and cylindrical turbulent jets, compared to Direct Numerical Simulations (DNS) and experimental data. 1.1. Aggregation in turbulent flows There is not an all-encompassing effect of turbulence on aggregation and the effect depends on the relative length-scales of the turbulent motion and aggregate size. If the particles are of comparable size to the flow length-scales, turbulence directly influences the collision process. Derksen (2012) performed DNS of homogeneous isotropic turbulence, where particles larger than the Kolmogorov scales were injected. He showed that turbulence promotes collisions, leading to more aggregation events, but also that the fluid deformation may lead to breakage of the formed aggregates. Derksen noted that higher levels of turbulence lead to the formation of smaller aggregates. Similar results were reported by Anne-Archard, d'Olce, Tourbin, and Frances (2013) in their experimental investigation of silica nanoparticle aggregation under turbulent-shear flow conditions. For nanoparticles, the Brownian mechanism dominates the collision process. Turbulence influences the process indirectly by altering the concentration, which in turn determines the aggregation levels. The local concentration may vary significantly (instantaneously) from the local mean values, typically leading to more aggregation as a result of the non-linear dependency of aggregation on number concentration. The eddy diffusion rate of nanoparticles far exceeds their Brownian diffusion rate, so that turbulent eddies will transport the particles in ensembles wherein aggregation will still occur Johannessen, Pratsinis, and Livbjerg (2001). Consequently, turbulence can influence the PSD even when the diameter of the particle is much smaller than the Kolmogorov length scale: dp ⪡ηk . Siera-Pallares et al. (2012) suggested that the turbulent mixing time-scale also influences the aggregation process. In regions where the mixing time-scales is large, the residence time for aggregation is also large, allowing for more collisions which lead to bigger cluster sizes. Arguably, this does not necessarily mean that the number of collisions exceeds this in a typical jet core, as observed by Nijdam, Starner, and Langrish (2004). However, due to the longer residence times, particles may undergo sequential collisions and create bigger cluster sizes compared to equivalent particles in the jet core. In the context of particle-laden turbulent jets, there have been a number of attempts to simulate and measure the effects of flowcoupling on the aggregation process. Miller and Garrick (2004) performed the DNS of nanoparticle coagulation in a planar jet. They used a sectional (discretization) method for the solution of the Population Balance Equation (PBE, see next Section). They found the largest particles at the shear layer surrounding the jet potential core. A DNS of a mixing layer with Brownian aggregation by Garrick, Lehtinen, and Zachariah (2006) showed that large vortical flow motion reduced the particle growth due to the diluting effect of ‘engulfed’ particle-free fluid. Yu, Lin, Chen, and Chan (2006) performed an LES of a turbulent jet with nanoparticle coagulation using the moment PBE. They concluded that coherent turbulent structures influence the particle number concentration and the PSD, while stressing the importance of retrieving instantaneous quantities. The effect of coherent structures was also investigated in an impinging jet by Yu, Lin, and Xiong (2007), using the QMOM approach developed by Marchisio, Vigil, and Fox (2003). The coherent structures diluted the particle-laden fluid, decreasing the aggregation events due to the lower particle concentration. Yan, Luo, Fan, Tsuji, and Cen (2008) performed the DNS of a particle-laden turbulent jet with Re = 3000 , using Lagrangian tracking of the particulate phase and a deterministic hard sphere model to simulate inter-particle collisions. They showed a correlation between the number of collisions and the local particle concentration. Moreover, they also observed the difference between mean and instantaneous particle concentration, stressing the influence of turbulence on particle dispersion. Similarly, Schwarzer, Schwertfirm, Manhart, Schmid, and Peukert (2006) argued the importance of considering spatial and temporal fluctuations when modelling aggregation. Rigopoulos (2010) reviewed the coupling between PBE and a RANS turbulence model and showed that the time-average of the aggregation integrals creates several unknown correlations, see also Friedlander (2000). These correlations may become significant when the relation between the mixing time-scale and the aggregation residence time is considered. Similar correlations appear due to the spatial filtering in the context of LES. Garrick (2011) performed a DNS of a mixing layer and used an Eulerian moment-based method for nanoparticle aggregation. Using an a posteriori filter, they obtained the equivalent LES values for the transported moments of the PBE and the relevant sub-grid aggregation rates. They found that the sub-grid aggregation rates can fluctuate by 100% around their corresponding DNS value, and that the main effect of the sub-grid particle-particle interactions was to decrease the filtered aggregation rates. 2. Modelling aggregation 2.1. The aggregation kernel At the simplest level, the aggregation process involves collisions of particles to form new particles of larger volumes. To model aggregation, both microscopic and phenomenological approaches can be used. In microscopic approaches, each collision event is treated separately, like in the hard sphere collision model of Sundaram and Collins (1997); Yamamoto, Potthoff, Tanaka, Kajishima, and Tsuji (2001). Liao and Lucas (2010) categorise the numerous models for fluid particle coalescence processes and their relation to 2

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findings from binary (microscopic) collision experiments. However, in engineering applications, the number of particles, hereafter Ntot, render microscopic approaches intractable, so that macroscopic approaches are sought. In macroscopic approaches, the number of collisions per time and volume between two size classes i and j, that form a new class k with υk = υi + υj , is represented by Cij. The joint probability of a collision to occur, and an aggregate to form, is quantified by the aggregation kernel (or frequency), βij, and the respective concentrations Nc, i and Nc, j :

Cij = βij Nc, i Nc, j .

(1)

The summation over all possible pairs i, j leading to a size class υk gives rise to the Smoluchowski (1917) equation, also known as the Population Balance Equation (PBE). Considering only aggregation processes, the equation is:

dNc, k 1 = dt 2

∞

∑

βij Nc, i Nc, j − Nc, k ∑ βik Nc, i .

i+j=k

(2)

i=1

The kernel βij incorporates the physico-chemical characteristics of the process. Macroscopic parameters of both the continuous and disperse phases may influence βij depending on the physical mechanism of the aggregation process. In the present work, the free-molecular Brownian aggregation kernel (where the Knudsen number Kn is much greater than 1) and the transition regime kernel for Kn ≈ 1 are used. The Brownian aggregation kernel is

βij =

1 π 2 2 (dp, i + dp, j )2 (uTp , i + uTp, j ) 2 , 4

(3)

where dp, i and dp, j are the diameter of the colliding i and j particles respectively. Here, uTp, i is the mean thermal velocity magnitude of a particle of density ρp and volume υi. The transition regime kernel is:

βij =

2π (dp, i dp, i + dp, j

+ dp, j )(Dp, i + Dp, j )

dp, i + dp, j + 2 δi2 + δ 2j

+

,

8(Dp, i + Dp, j ) 1 2 + u2 )2 (dp, i + dp, j )(uTp ,i Tp, j

(4)

where the diffusion coefficient Dp, i , δi and λp, i are given by:

δi =

(dp, i + λp, i )3 − (dp2, i + λp2, i )3/2 3dp, i λp, i

− d p, i

;

λ p, i =

8Dp, i πuTp, i

;

Dp, i =

kB T Gi. 3πdp, i μf

(5)

In this equation, kB is the Boltzmann constant, T and μf are the uniform phase temperature and viscosity, and Gi is the Cunningham slip-flow correction. Different aggregation kernels can be used with the present model (for details, please refer to work by Pesmazoglou et al. (2016). 2.1.1. Modelling the coupled kernel Solution methods for flow-coupled aggregation follow those for the uncoupled kernel, namely: analytic methods, the method of moments (MOM) with its variations (QMOM, HMOM and DQMOM), sectional methods, Monte Carlo (MC) methods, and combinations thereof. These can be solved either in an Eulerian or in a Lagrangian framework. A complete review of all approaches is outside the scope of this paper, but the reader is pointed to work by Rigopoulos (2010) and references therein. The present paper focuses on stochastic/MC methods for particle populations that are too large to directly resolve one-on-one collisions. Monte Carlo methods simulate the aggregation process within a probabilistic framework, where two clusters merge into a single cluster at a given rate Cij (see work by Wagner, 2003). The evolution of a subset of the real particles is simulated by stochastic parcels. These parcels can represent a set of moments of the PBE, a set of discrete classes of the PBE or a sampled size of the real particle population. Despite the Lagrangian nature of the method, an Eulerian representation of the particle concentration (i.e. number of particles per unit volume) is needed for the aggregation kernels. To estimate the local particle concentration, it is important to define a phenomenologically accurate interaction volume, hereafter Vint. The simplest choice is to consider the aggregation events per cell, Vint = Vcell Zhao and Zheng (2013). However, in such a particle cloud (or parcel) representation, the volume of the parcel depends on the internal dispersion of real particles represented by the parcel. The particle dispersion may increase due to turbulence or decrease due to aggregation events. Litchford and Jeng (1991) suggest that each parcel position is represented by a normal distribution with a mean and standard deviation representative of the parcel radius and expressed in terms of a two-particle velocity correlation function. Alternatively, Johannessen et al. (2001), modelled the effects of the flow field on the parcel by a ‘dilution’ factor. A number of methods exist to model aggregation in Monte Carlo approaches. Koch and Pope (2002) use stochastic parcels to simulate aggregation in homogeneous isotropic turbulence of micron-size particles. The particle depletion within a parcel is modelled using the turbulent shear kernel along with an Interaction by Exchange with the Mean approach Villermaux and Devillon (1972), to represent the depletion of particles due to exchange with other parcels, neglecting inter-parcel aggregation. Sommerfeld (2001) and Ho and Sommerfeld (2002) computed the collision probability of a parcel in a turbulent flow-field by generating a fictitious-particle that determines both the collision probability and the coagulation efficiency. Nijdam, Guo, Fletcher, and Langrish (2004) used the model of Ruger, Hohmann, Sommerfeld, and Kohnen (2000) to calculate the 3 collision frequency between two parcels, given their relative velocity and a relative volume of interaction Vint = b1 x rel , where xrel is the 3

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relative distance between the parcels and the parameter b1 assumes a relation between the interacting volume and that of a sphere (Guo, Fletcher, & Langrish, 2003 used b1 = 5). The two parcels merge if the calculated collision probability is greater than a critical value. Zhao and Zheng (2013) couple the differentially weighted MC method of Zhao, Kruis, and Zheng (2009) and Zheng (2011) with a turbulent flow field. They assume the equality Vint = Vcell and define a ‘full coagulation rule’, whereby all particles of a parcel i collide instantaneously with the most probable partner j, given that i contains less particles than j. The aggregation event is represented by an accept/reject probabilistic coagulation rule, where a random number is generated and compared to an aggregation probability Paggr , i . The aggregation probability is modelled as a Poisson distribution with a mean rate equivalent to the sum of all collision rates within a cell, namely Ci = ∑j Cij , where rates are modified according to the full coagulation rule of Zhao et al. (2009). All collisions are included in a single aggregation event with the most probable partner j of parcel i. The partner is selected either using a cumulative probability method Liffman (1992), or with an acceptance-rejection method by Lin, Lee, and Matsoukas (2002). The time-step Δt needed to integrate Eq. (2) should not influence the aggregation process. To de-couple the aggregation process from the dispersion process, Δt should be small enough that the relative distance between parcels does not change significantly within the time step. To accurately capture aggregation, the collision events should be treated sequentially, since the kernels βij = βij (υi , υj ) change after every collision event. In practice, an aggregation residence time τaggr is used (Zhao & Zheng, 2013), which corresponds to the maximum coagulation rate of a parcel in a cell, τaggr = 1/maxi (Vcell Ci) . If Δt > τaggr , the aggregation is divided into smaller substeps, such that Δtaggr < τaggr , where the rates are recalculated and aggregation events applied. 3. Numerical modelling 3.1. Continuous phase The simulation treats the continuous phase in an Eulerian LES, using the in-house incompressible PsiPhi code: see Kempf, Geurts, and Ofelein (2011) and Pettit, Coriton, Gomez, and Kempf (2011). The Smagorinsky (1963) eddy-viscosity model with the model constant of Cs = 0.173 suggested by Lilly (1967) is used to model the unknown sub grid-scale stresses of the momentum equation. The effect of varying the turbulent viscosity was minimal in the simple flows studied in this work, and as such the value of 0.173 was retained avoiding the cost of using a dynamic modelling approach Cavallo Marincola, Ma, and Kempf (2013). A Central Differencing Scheme, second-order accurate in space, is used for the convective momentum fluxes and a third-order Runge-Kutta Scheme is used for integration in time. The code uses implicit filtering with equally sized cubic cells of length Δ, thus ensuring good numerical accuracy and efficient parallelization. 3.2. The spray-PDF equation and particle dispersion In the context of inertial particle dispersion, Williams (1958) introduced a conservation equation for the particle distribution function. The particle distribution function consists of the joint probability of the realisation of M variables that represent the state of the particulate phase, e.g. position, velocity, temperature or size. For a set of M independent variables, the state vector of the j th particle for a given location in space x and time t is Φ j = {ϕ1j (x, t ), …, ϕMj (x, t )} . The spray-pdf equation describes the evolution of the phase space Ψ = {ψ1, …, ψM } of the random variables ϕi of an ensemble of N particles. The mathematical formulation of the spray-pdf equation is outside the scope of this work; the interested reader may refer to the original work by Klimontovich (1969). The spray-pdf can be reformulated as a trajectory problem to describe the evolution of the random variables ψi = 1 … M , for a set of jindexed stochastic parcels (samples). The final Itô's equivalent set of Stochastic Differential Equations (SDE) is:

dψi j = Ai (Ψ ) dt + Bik (Ψ ) dWt , k + Ji (Ψ ) δNdtj, λ . δNdtj, λ

(6)

is the increment of a stochastic counting process of rate λ. Any continuous where Ai is the drift term, Ji the jump term, and random forcing can be added both to the spray-pdf and the trajectory problem with a Brownian diffusion term quantified by an equivalent diffusion tensor Bik . For a complete description of the process, please refer to work by Pesmazoglou et al., (2013, 2014). The SDE (6) is solved following Jones and Sheen (1999) and implemented in previous work Bini and Jones (2008); Jones, Lyra, and Navarro-Martinez (2011); Pesmazoglou et al. (2013). The parcel motion is governed by the following Lagrangian equations:

j th

d xp = vp dt

d vp =

∼ (Uf − vp) τp

(7)

dt +

Co

ksgs τt

dW

(8)

Here, xp is the position of the pth parcel and vp the parcel velocity. The ensemble average of all parcel paths gives the Eulerian ∼ equivalent of the positions and velocities of the particles. The expression Uf refers to the LES gas velocity; Co = 1 is a dispersion constant Bini and Jones (2008); Pesmazoglou et al. (2013). The unresolved kinetic energy ksgs of the gas phase is calculated assuming equilibrium of the small scales ksgs = 2Δ2 Cs2/3 S͠ ij S͠ ij Bini and Jones (2007), where S͠ ij is the filtered strain tensor. The term dW = ξN0,1 dt is the incremental Wiener term, where ξN0,1 is a random variable sampled from a normal distribution with a mean of 0 and a variance of 1. The particle relaxation time τp is determined from: 4

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τ p−1 =

3 ρf CD ∼ |Uf − vp | 4 ρp dp

(9)

where ρp , ρf are the particle and continuous phase densities, respectively, and dp is the particle diameter. The turbulence interaction time-scale τt is found from: 1/2 2α − 1

⎛ τp ksgs ⎞ τt = τp ⎜ Δ ⎟ ⎝ ⎠

(10)

the constant α = 0.8 is selected to match the tails of the acceleration pdfs following Bini and Jones (2007). ∼ The drag coefficient CD is evaluated using an empirical relation to the particle Reynolds number Rep = (ρp dp |Uf − vp |)/μ , defined by the drag law of Yuen and Chen (1976) assuming spherical particles. For the solution of a given set of particle equations 7–10, the continuous phase properties at the position of a given particle are provided from the Eulerian LES, using a linear interpolation of the cell-centred values of the neighbouring cells. It has to be noted that for particles on the nano-scale, the relaxation time is very small and therefore the stochastic term is also expected to be small so that the particles will act like fluid particles. 3.3. Aggregation model Aggregation is incorporated directly into the spray-pdf by the addition of the particle number concentration and volume to the set of Lagrangian Eq. (6). In aggregation processes, the number of real particles represented by a parcel does not remain constant and changes due to particle interactions. In the present Lagrangian context, the particle population per parcel is a variable in the spray-pdf equation, and the following stochastic trajectory can be formulated:

dnint = −n˙ coll dt ,

(11)

Here, nint is the number of real particles in a given parcel, n˙ coll is the rate of collisions of real particles of the parcel. The number of collisions ncoll that cause a change dnint in a given time dt is modelled along the lines of a jump process with time increments dt and a mean rate τaggr. The parcel concentration is defined as:

np (t ) =

nint (t ) , Vint

(12)

and the Eulerian particle number concentration for a certain sample volume Vsamp centred at a position x is: ∀ i ∈ Vsamp

Nc (x, t ) =

∑ i=1

nint , i , Vsamp

(13)

Δ3

or the In this work, the sampling volume and the volume of interaction are both chosen to be that of a cell, i.e. Vsamp = Vint = Vcell = cube of the filter width. In the present model, the particles of a parcel may collide with a number of different parcels. If every collision event created an additional parcel, then the number of parcels in the simulation would quickly increase to an intractable number. Therefore, it is assumed that the aggregated particles from a colliding parcel-pair are added to one of the two colliding parcels. The parcel that has the larger mean particle diameter accommodates the aggregated particles and is termed ‘receiver’, whereas the other parcel is termed ‘donor’. To simplify the multi-collision process the further assumption is made:

• The receiver parcel always has a greater or equal mean particle diameter compared to that of the donor parcel. • More than one donor particle may collide with a single particle of the receiver parcel but not vice-versa. • Internal parcel collisions are allowed, i.e. particles of the same parcel can collide with each other to form aggregates. The first assumption adds a directional-bias to the aggregation event (small →large). This assumption suggests that if a large population of small particles (represented by a donor parcel i) interacts with a few large particles (receiver j), the large particles may experience more than one collision event. The number of collisions ncoll, ij may be greater than nint , j , however, the total number of collision events is limited to the internal population of the donor parcel. For the same notation of i, j parcels, the above definition boils down to: ∀ j ∈ Vcell

∑ j

⎛ ncoll, ij = min ⎜ ⎝

∀ j ∈ Vcell

∑ j

⎞. n coll, ij , nint , i⎟ ⎠

(14)

The suggested approach prevents collisions from being double counted: although the kernels may be the same βij = βji , the number of collisions ncoll, ij ≠ ncoll, ji is not. The computational advantage of this method is that a second loop to find the most probable partner is not required and array operations can be employed by formulating the tensor ncoll, ij . 5

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Particles in the proposed model are allowed to collide with any set of particles (including particles within the same parcel) within Vcell instead of colliding with a single most probable partner. The size of the newly-formed aggregates is described by the ensemble average size of the receiver parcel. The collision tensor ncoll, ij gives the number of collision events that occur between parcel i and parcel j. A probabilistic approach is used for the simulation of such events, whereby the probability of ncoll, ij collisions to take place between two parcels i and j within a volume Vint and in a time interval t is given by:

Paggr (ncoll, ij , t ) =

1 t , ncoll, ij τaggr

(15)

Here,τaggr is the aggregation residence time: −1 τaggr =

βij nint , i nint , j Vint

.

(16)

A linear increase of probability with t is assumed and the measure Paggr (ncoll, ij , t ) is the cumulative distribution function (CDF) representing the probability of ncoll, ij events to occur at any time in the range [0: t ]. The division by ncoll, ij normalizes the probability to a maximum of Paggr (ncoll, ij , t ) = 1 for any number of collision events. Note that Vint in this study is chosen to be equal to Vcell for any i, j pair. By sampling a random number ξU0,1 from a uniform distribution in the range 0 → 1, the equivalent ij element of the collision tensor for a time-step Δt is constructed as:

ncoll, ij =

⎧ floor ⎨ ⎩ 0,

(

Δt τaggr

)

+ ξU0,1 , ifυi < υj otherwise

(17)

The function ‘floor’ produces the integer part of a real number. Eq. (17) states that the total number of collisions in the direction i → j during an elapsed time Δt for υi < υj is equal to the integer (or definite) number of collisions plus the floor of the sum of the remainder and the random number. If for example Paggr (1, Δt ) = Δt / τaggr = 0.8 , then there is an 80% probability that ξU0,1 ≥ 0.2 and therefore that a collision will take place. (Note that ncoll, ji = 0 if υi > υj .) To optimize the number of operations required and to include the cases where υi = υj without double-counting, the collision array ncoll, ij (17) for all pairs i − j and internal collisions i − i is constructed using a particle sorting algorithm per cell. To ensure that the aggregation events are sufficiently resolved in time, the time-step limit of Zhao and Zheng (2013) is used:

Δtaggr ≤

∀

⎡⎛∀ min ⎢ i ∈ Domain ⎢ ⎜ ⎣⎝

j ∈ Vcell

∑ j

−1

βij nint , i nint , j ⎞ ⎟ Vint , ij ⎠

⎤ ⎥. ⎥ ⎦

(18)

This criterion implies that the time-step is limited by the maximum total aggregation rate of all parcels in the domain. For every parcel i, the effect of the collision events are incorporated to the equivalent trajectories of each partner j. (Note that a constant particle density ρp is assumed for all the following calculations.) The change in velocity due to aggregation is updated using a conservation of linear momentum between the two colliding parcels (with i ≠ j ), viz:

[dvp, j]aggr , ij =

(vp, i − vp, j) 1+

υj nint , j

. (19)

υi ncoll, ij

The equivalent velocity trajectory becomes a function of the variable (ncoll, ij ) sampled from a stochastic jump process and Eq. (8) extends to: ∼ (Uf − vp, j) ksgs dt + Co d W + ∑ [d vp, j]aggr , ij . d vp, j = τp τt (20) ∀ i ∈ Vcell The mean volume of a parcel j is updated using a weighted sum of the colliding particle volumes. It is therefore assumed that each particle of the receiver parcel will increase in volume according to:

[dυj]aggr , ij =

ncoll, ij nint , j

υi

(21)

Therefore, the equivalent trajectory for the average parcel volume of the receiver j due to collisions with all donor parcels i is:

dυj =

∑

[dυj]aggr , ij . (22)

∀ i ∈ Vcell

Finally, the new internal particle number of each parcel is updated according to:

dnint , i = −

∑

ncoll, ij . (23)

∀ j ∈ Vcell

In its generality, the outlined multi-collisional approach for stochastic parcels can be used with different aggregation mechanisms 6

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Fig. 1. Configuration of nanoparticle-laden planar jet.

and kernel modifications, for example to incorporate the effects of a fractal shape. The aggregation model has been validated by Pesmazoglou et al. (2016). In cases where the environment is contaminated, it is possible that a background (noise) particle concentration may be present. Such ‘noise’ can influence the aggregation events and usually there is little knowledge regarding the PSD of the background particles. The one-directional multi-collision approach allows for the calculation of ‘background aggregation’ events for every parcel. In essence, each parcel can be a receiver of background particles. The background particle concentration Nbg is transported in an Eulerian framework as a passive scalar (assuming negligible depletion) and interpolated at the location of a parcel Nbg @ p . The condition (17) can be therefore readily evaluated to give the number of collisions from the background phase ‘i’, with the interpolated concentration nint , i = Nbg @ p , to the parcel ‘j’. 4. Nanoparticle aggregation in a planar jet The aggregation model developed is applied to the nanoparticle-laden planar jet configuration by Miller and Garrick (2004). They used a sectional model for the solution of the PBE evolution (see Section 1.1) and a direct comparison with the present aggregation model can be easily established. 4.1. Configuration and modelling parameters Miller and Garrick (2004) examined a coagulating aerosol in an incompressible, isothermal (T = 300 K) planar jet. Their aim was to investigate the underlying structure of vapour-phase particle growth processes. Fig. 1 shows a sketch of the simulated domain. The same boundary conditions as in the DNS have been used in the x , y directions, with spanwise periodic boundary conditions. The jet diameter is Djet = 1 mm and the velocity distribution is uniform throughout the height of the jet and equal to Ujet = 95 m/s, while the co-flow velocity is Ucf = 0.55Ujet . The resulting Reynolds number based on Ujet and Djet is Re = 4000. To accelerate the development of large scale vortical structures, random perturbations of 3% of Ujet are added in the cross-stream y-direction. The LES domain in the x and y directions extends to 12 × 7Djet and is the same size as in the DNS. The total width of the domain was chosen to be 10Djet. It must be noted that the increased dimensionality of the LES versus the planar DNS may reduce the scales of the vortical structures as energy is transferred to all three directions. However, in the relatively short domain and low Reynolds, the influence of the span-wise direction in the flow field is small and the aggregation models can be compared directly. A grid spacing of Δ = 0.1 mm (Djet / Δ = 10 ) was found sufficient to represent the evolution of the continuous phase. Fig. 2 shows the radial distribution of the continuous phase velocity at four axial locations compared to DNS data. The mean velocity and jet spreading are well represented by the LES and the grid resolution is deemed adequate. Three dimensional effects are not observed in the mean quantities and comparison with the DNS will be restricted to the first three locations. Following Miller and Garrick (2004), mono-disperse particles of dp = 1 nm are seeded with a volume fraction of ϕυ0 = 10−7 at the jet inlet, in a narrow band of width wpin = Djet /2 (see Fig. 1). The particles were restricted to a planar motion by setting their zdirectional velocity to zero. For the given range of particle diameters and air temperature, the Knudsen number is Kn ≫ 1 and the free-molecular Brownian aggregation kernel (3) is used. The sectional method of Miller and Garrick (2004) used 10 bins of variable width to approximate the PBE evolution. Particle inertial effects and consequently particle slip was ignored and spherical droplets and instantaneous coagulation are assumed. The present work uses the same aggregation kernel and major assumptions to facilitate the comparisons between DNS and LES. The initial number of particles per parcel nint (t = 0) is equal to 0.2% of the total population of real particles present in the simulation assuming no aggregation. The LES time-step is Δt ∼ O (10−8) , with a constant CFL = 0.3. An estimate of the aggregation time τaggr ∼ O (10−5) can be obtained using an inlet initial concentration of N0 ∼ O (1020) [#/m3] and a kernel of βij ∼ O (10−15) [m3/#s]. The condition (18) is met with the chosen time step, allowing for expected fluctuations in τaggr due to varying concentration and kernels as the particles grow. 7

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Fig. 4. Normalised PSD at the centreline and the shear layers at three axial locations.

4.2. Results Fig. 3 compares the results for the particle diameters from the LES and DNS at three axial locations, namely x / Djet = 2, 6 and 10. No information regarding the density of the aggregating phase particles ρp is provided in the original work by Miller and Garrick (2004) and the Damköhler number was varied to match the mean diameter observed at x / Djet = 2 (resulting in a Daaggr = 10 ). The mean diameter radial distributions at the remaining axial locations agree well with the results from the DNS. Particles with large diameters concentrate near the shear layer; see Fig. 3. Preferential dispersion effects can be considered negligible, as particles follow the flow field. The particle concentration reduces in the shear layers as the flow is diluted by the particle-free co-flow stream. The particles near the shear layer have therefore larger mean diameters primarily due to longer flow residence times, which implies more sequential collisions leading to larger sizes. These results also agree qualitatively with other studies Cocero, Ferrero, and Miguel (2002), Johannessen et al. (2001); Miller and Garrick (2004), Nijdam, Starner et al. (2004), Yan et al. (2008), and Yu et al. (2006). Fig. 4 compares the PSD at the jet-core ( y ≤ y0.8 ) with and the shear layer ( y > y0.8) at three axial locations x / Djet = 2 , 6 and 10. At the jet centreline, the distribution develops a bi-modal shape with a second mode at dp ≈ 3 nm. This second mode cannot be seen in the mean particle diameters distribution in Fig. 3 and it may hence lead to the incorrect conclusion that all particles have aggregated at least once by the time they reach x / Djet = 2 . In reality, small particles that aggregated once are more likely to aggregate a second time before reaching x / Djet = 2 ; a result which is expected by the dependence of βij with dp. In fact, the population density of dp = 2 nm particles is smaller than that of dp = 3 nm particles. At downstream locations, the population of 1 nm particles further decreases and the PSD broadens, but the observed bi-modality persists. Although in the original PBE method, particles greater than 8 nm could not be captured due to the sectional method, the present results show a maximum diameter of dp = 14 nm at x / Djet = 10 . Fig. 5 depicts the variation of the mean particle diameter < dp > with Daaggr at two axial locations: x / Djet = 2 and 10. For the same flow conditions, particle volume fraction, and initial particle size, the variation of the Damköhler number has a non-linear effect on aggregation. * , the mean diameter decreases again. Increasing Daaggr also increases the mean diameter. However, after a critical value, Daaggr * . This behaviour is observed at both streamwise locations, although there is a difference on the observed critical Daaggr

Fig. 5. Ensemble average particle diameter at two axial locations for the range of Daaggr investigated.

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Fig. 6. Experimental configuration of Zhu et al. (2013) and simulated domain.

This non-linear behaviour in Fig. 5 is explained by the balance between concentration and residence times. Increasing aggregation reduces the concentration upstream, which in turn reduces the number of collisions downstream. The aggregation increases again further downstream due to the longer residence times as the jet decelerates. 5. SiO2 aggregation in a turbulent round jet The next investigated case is the configuration of Zhu, Qi, and Wang (2013), where a Silicon Dioxide (SiO2 )-laden jet diffuses into atmospheric air. Their test case addresses the aggregation and dispersion of the Silica nanoparticles as other particle processes are not present. To the author's knowledge, the present work is the first simulation of this experiment. 5.1. Experimental configuration and modelling parameters The experimental set-up consists of a Nitrogen jet carrying droplets of an aqueous-SiO2 solution issued in an evaporator using an air-blast atomizer. In the evaporator, the droplets were mixed with heated air at 323 K. The solvent was evaporated and the dried nanoparticles were distributed homogeneously within the heated air. The SiO2 -laden air was subsequently passed through a rectifier unit and injected via a converging nozzle with exit diameter of Djet = 20 mm into the environment at atmospheric conditions (see Fig. 6). The jet was turbulent with a bulk velocity of Ujet = 21.4 m/s and a turbulence intensity of u′ jet / Ujet = 1%; which corresponds to a bulk Reynolds number of Re = 23,516 at a nozzle outlet temperature of Tjet = 323 K. Velocity and turbulent intensity radial profiles were taken at several axial locations using a Laser Doppler Anemometry (LDA) system. The PSD and particle concentration were measured using a Scanning Mobility Particle Sizer (SMPS), which consisted of a TSI classifier, a long Differential Mobility Analyzer (DMA) and a Condensation Particle Counter (CPC). The SMPS had a sampling flow rate of 5·10−6m3/s and a sheath flow-rate of 8.3·10−7m3/s . A scanning residence time of 135 s was allowed for every sample. The nanoparticle concentration at the jet outlet was nearly uniform and equal to N0, jet = 340, 000#/cm3 . The diameter-based PSD at the jet exit showed little deviation in the radial direction and had a range of dp = 12 − 700 nm with a Sauter Mean Diameter (SMD) of d32 ≈ 250 nm and a mode at approximately 100 nm. Particle concentration measurements were taken at several radial and axial positions (between 0 and 18Djet). The computational domain extends 12 × 6 × 6Djet with a grid-spacing of Δ = 1 mm, with Djet / Δ = 20 . The mean velocity profile at the inlet of the simulation was directly read from the experimental measurements and the inflow turbulence was generated with a prescribed intensity equal to that of the experimental measurements at x / Djet = 2 with an integral length scale of lI / Δ = 2.2 , where Δ is the grid spacing. A CFL = 0.3 was chosen, which corresponds to Δt = 10−6 . Other than that, the same model parameters as in the previous test case are used. One-way coupling was assumed as the volume fraction at the jet exit was 1.06·10−9. The PSDs at the various measurement points were constructed from the histograms of the parcel diameters. A full simulation including statistical sampling ran for approximately 100,000 CPU hours. The particles were injected in bands at every time-step. Each band had a different seeding rate and diameter distribution in order to match the equivalent experimental concentrations and PSDs. For each seeding band, the CDF for the particle radial location and diameter was generated and stored at the beginning of the simulation. Two random uniform numbers ξU0,1 for every parcel were inversely mapped to the equivalent CDF of the radial position and the particle diameter. A lower- and upper-limit of nint = 10 and 100 was set, respectively, which resulted in a parcel to particle resolution of nint /(Ndomain Vdomain ) = 10−9 . The mean parcel to particle resolution was estimated from the ratio of the steady-state total parcel-to-real particle population present in the domain when no aggregation was used in the simulation. A very high resolution was chosen to enable the detection of small variations in the evolution of the PSD, and to mitigate the errors that would arise if the initial PSD was under-resolved. The Knudsen Kn ∼ 1 and the transition regime aggregation kernel is therefore used (see Eq. (4)). To simplify the numerical treatment of the problem, immediate coalescence and spherical aggregates were assumed. These assumptions have to be considered with care as aggregate morphology, coalescence, and sintering may strongly influence the aggregation process Eggersdorfer and 10

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Fig. 7. Contour plot of temporal average (top) and instantaneous (bottom) filtered axial-velocity of the continuous phase.

Pratsinis (2014). Spherical particles have a smaller surface area compared to aggregates with fractal shapes and the aggregation rate may be hence underestimated. However, (Isabeta & Biscans, 2010) showed that for fumed silica aggregates with a diameter of 200 nm, the mean fractal dimension is relatively low fD ≈ 2.6. Moreover, Jacobson Jacobson (1999) noted that as the population of primary particles in an aggregate increases, its shape becomes more spherical.

5.2. Results 5.2.1. Gas phase Fig. 7 presents the contour plots for the temporal and instantaneous gas velocity fields. The experimental data is well predicted by the LES in the axial direction in both mean and RMS values of velocity (see Fig. 8 ). The computational domain starts at 40 mm ( x / Djet = 2 ) as the incoming turbulence is not known and the experimental values at x / Djet = 2) are used for the incoming boundary conditions. The potential core of the jet persists until x ≈ 100 mm ( x / Djet = 5). Fig. 9 shows the radial profiles of the mean and the RMS uniform phase velocity at the four measurement points of the experiment. The agreement is good and shows that the velocity field is well captured everywhere in the domain. The maximum instantaneous ratio of sub-grid viscosity to laminar is approximately two, which suggest that the LES resolution is sufficient to capture the most energetic eddies. This is in line with results obtained with the same code in Franchetti, Cavallo Marincola, NavarroMartinez, and Kempf (2016).

Fig. 8. Mean and RMS uniform phase velocity variation in the axial direction.

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Fig. 9. Mean and RMS uniform phase velocity radial profiles at the four measurement points x / Djet = 2, 4, 6 and 8.

Fig. 10. Contour plot of temporal average (top) and instantaneous (bottom) particle concentration fields Nc [#/ Δ3 ].

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Fig. 11. Radial particle concentrations Nc at four axial locations.

5.2.2. Disperse phase Fig. 10 shows the instantaneous and time-averaged particle concentration field Nc. There are large spatial fluctuations in concentration with cell-to-cell variations of up to two orders of magnitude. Without any background particle concentration, the LES results significantly underestimate Nc at the measurement positions further downstream (see the particle concentration radial distribution in Fig. 11). A background particle concentration of 80, 000#/cm3 outside the jet was required to observe the measured concentrations near the ‘particle-free’ environment. At the downstream locations from the inlet, the LES underestimates the concentration levels at y ≈ 20 mm (see Fig. 11). These results suggest that the environment, into which the jet issued, could have been contaminated with recirculated nanoparticles from the experiment. Fig. 12 shows the measured and simulated PSD at the inlet of the simulation domain. The background concentration was included in the aggregation process by transporting an additional passive scalar in the Eulerian phase using a TVD scheme. Each parcel interacted with the Eulerian background particle concentration field assuming that all background particles have a dp ≈ 70 nm, equivalent to the mode of the PSD farthest from the centreline at the inflow (see Fig. 12). Fig. 13 compares the measured PSDs to those from LES. The LES results show good agreement with the measurements on the jet centreline at x / Djet = 6 and 8. However, aggregation does not play a big role and there is no big difference with and without aggregation model. The greatest difference is at x = 160mm and r = 3mm , where the simulated PSD does not reproduce the experimental measurements, under-predicting the amount of large particles >300nm . Fig. 14 shows the Brownian aggregation kernel (3) in the free-molecular regime. This kernel peaks at βij ∼ O (10−12m3/#s) , when large particle classes interact with the smallest classes. However, the experimental PSD is not uniform, hence interacting particle sizes tend to be of similar magnitude. The variation of the Brownian aggregation kernel for two interacting particles of the same diameter is shown in Fig. 14 (top right). A conservative choice for the range of particles measured suggest βij ∼ O (10−12m3/#s) and Nc ∼ O (1012#/m3) , a minimum estimate −1 of the aggregation time τaggr = (βij Nc2 Vcell/4) ∼ O (10−2s) , which increases to τaggr ∼ O (1s) if particles of the same size are considered. Using the axial velocity Uf ≈ 15.5 m/s, the flow residence time of a particle in the domain is estimated as τflow ∼ O (10−3s) . The corresponding Damkhöler is Daaggr O (10−1) < 1 and suggests that the number of aggregation events within the domain is not sufficient to significantly alter the simulated PSD. Fig. 15 shows a plot of the cumulative (total) aggregation events that occurred within t = 3.5 s (approximately 3500τflow ). Approximately 200 aggregation events took place in each cell over this time, suggesting a minimum aggregation residence time of

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Fig. 12. Particle size distributions at the simulation inlet for different radial locations. The PSD at r = 18 mm is used for the background aggregation process.

τaggr ∼ O (10−2s ) (within the limits estimated previously), with most aggregation events occurring near the jet centreline. Therefore, although the results from the LES and the predicted levels of aggregation are in good agreement at most locations, the simulation cannot capture the experimental PSD at x = 160mm and r = 3mm . 6. Conclusions The present paper presents an LES-coupled model for particle aggregation and assesses its performance in two different turbulent configurations, a planar and a round jet. The model showed a good agreement in particle statistics with the planar jet DNS of Miller and Garrick (2004), where the long residence time was the main parameter promoting the formation of large particles. The bi-modal shape of the PSD was captured, showing the importance of retrieving the full PSD. The proposed Lagrangian approach has several advantages over the discrete PBE: It allows an unrestricted range of possible particle sizes and does not require a-priori ‘binning’ of the PSD. The computed mean-particle diameter was found to have a non-linear dependency with the aggregation-equivalent Damköhler 14

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Fig. 13. Comparison of measured and simulated PSD at four locations.

number. An increase in aggregation rates does not necessarily mean an increase in overall mean diameter. This behaviour is attributed to the balance between reduced concentration upstream and long residence times downstream. When aggregation kernels are small, few particles are depleted upstream, which in turn produces larger particle concentrations downstream, which combined with longer residence times outweighs the reduced magnitude of the kernel.

Fig. 14. Variation of aggregation kernel for the range of particle-sizes considered (left) and plot of kernel variation for colliding particles with same diameters (top right).

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The model was also applied to the experimental round jet of Zhu et al. (2013). A background concentration of particles was introduced to match the experimental measurements downstream. The LES results agreed well with the measurements of the PSD on the jet centreline. However, an insignificant difference was found when the aggregation model was included. There was a large discrepancy between measured PSD and LES at a single location, where experimental data showed large aggregation. By introducing an order of magnitude analysis, the present paper showed that with the Brownian kernel used, these effects cannot be attributed to aggregation. Overall, the proposed Lagrangian method is able to capture qualitatively and quantitatively PSD and concentration levels in turbulent jet flows, provided that the aggregation kernels are known. References Anne-Archard, D., d'Olce, M., Tourbin, M., & Frances, C. (2013). Aggregation of silica nanoparticles in concentrated suspensions under turbulent, shear and extensional flows. Chemical Engineering Science, 95, 184–193. Bini, M., & Jones, W. P. (2007). Particle acceleration in turbulent flows: A class of nonlinear stochastic models for intermittency. Physics of Fluids, 19 (035104–1). Bini, M., & Jones, W. P. (2008). Large eddy simulation of particle laden turbulent flows. Journal of Fluid Mechanics, 614, 207–252. Cavallo Marincola, F., Ma, T., & Kempf, A. (2013). Large eddy simulations of the Darmstadt turbulent stratified flame series. Proceedings of the Combustion Institute, 34(January (1)), 1307–1315. Cocero, M., Ferrero, S., & Miguel, F. (2002). Crystallization of beta-carotene by continuous gas process: Effect of mixer on crystals formation. In Proceedings of the 4th International symposium on high pressure process technology and chemical engineering. Venive, Italy. Derksen, J. (2012). Direct numerical simulations of aggregation of monosized spherical particles in homogeneous isotropic turbulence. AICHE Journal, 58(8), 2589–2600. Eggersdorfer, M., & Pratsinis, S. (2014). Agglomerates and aggregates of nanoparticles made in the gas phase. Advanced Powder Technology, 25, 71–90. Franchetti, B., Cavallo Marincola, F., Navarro-Martinez, S., & Kempf, A. (2016). Large eddy simulation of a 100 kWth swirling oxy-coal furnace. Fuel, 181, 491–502. Friedlander, S. (2000). Smoke, dust, and haze: fundamentals of aerosol dynamics. Oxford University Press. Garrick, S. (2011). Effects of turbulent fluctuations on nanoparticle coagulation in shear flows. Aerosol Science and Technology, 45, 1272–1285. Garrick, S., Lehtinen, K., & Zachariah, M. (2006). Nanoparticle coagulation via a Navier-Stokes/nodal methodology: Evolution of the particle fields. Journal of Aerosol Science, 37, 555–576. Gavaises, M., Theodorakakos, A., Bergeles, G., & Brenn, G. (1996). Evaluation of the effect of droplet collisions on spray mixing. Proceedings of the Institution of Mechanical Engineers, 210, 465–475. Guo, B., Fletcher, D., & Langrish, T. (2003). Simulation of the agglomeration in a spray using lagrangian particle tracking. Applied Mathematical Modelling, 28, 273–290. Ho, C., & Sommerfeld, M. (2002). Modelling of micro-particle agglomeration in turbulent flows. Chemical Engineering Science, 57, 3073–3084. Isabeta, N., & Biscans, B. (2010). Fractal dimension of fumed silica: Comparison of light scattering and electron microscope methods. Powder Technology, 203(10), 206–210. Jacobson, M. (1999). Fundamentals of atmospheric modeling. Cambridge University Press. Johannessen, T., Pratsinis, S., & Livbjerg, H. (2001). Computational analysis of coagulation and coalescence in the flame synthesis of titania particles. Powder Technology, 118, 242–250. Jones, W., Lyra, S., & Navarro-Martinez, S. (2011). Large eddy simulation of a swirl stabilized spray flame. Proceedings of the Combustion Institute, 33, 2153–2160. Jones, W., & Sheen, D. (1999). A probability density function method for modelling liquid fuel sprays. Flow Turbulence and Combustion, 63, 379–394. Kempf, A., Geurts, B., & Ofelein, J. (2011). Error analysis of large-eddy simulation of the turbulent non-premixed sydney bluff-body flame. Combustion & Flame, 158, 2408–2419. Klimontovich, I. (1969). The statistical theory of non-equilibrium processes in a plasma. Pergamon. Koch, D., & Pope, S. (2002). Coagulation-induced particle-concentration fluctuations in homogeneous, isotropic turbulence. Physics of Fluids, 14(7), 2447–2455. Liao, Y., & Lucas, D. (2010). A literature review on mechanisms and models for the coalescence process of fluid particles. Chemical Engineering Science, 65, 2851–2864. Liffman, K. (1992). A direct simulation Monte-Carlo method for cluster coagulation. Journal of Computational Physics, 100, 116–127. Lilly, D. (1967). The representation of small-scale turbulence in numerical simulation experiments. In Proceedings of the IBM scientific computing symposium on environmental sciences (pp. 195-210), 1. Lin, Y., Lee, K., & Matsoukas, T. (2002). Solution of the population balance equation using constant-number Monte Carlo. Chemical Engineering Science, 57(12), 2241–2252. Litchford, R., & Jeng, S. (1991). Efficient statistical transport model for turbulent particle dispersion in sprays. AIAA Journal, 29, 1443. Marchisio, D., Vigil, R., & Fox, R. (2003). Implementation of the quadrature method of moments in CFD codes for aggregation breakage problems. Chemical Engineering Science, 58(15), 3337–3351. Miller, S., & Garrick, S. (2004). Nanoparticle coagulation in a planar jet. Aerosol Science and Technology, 38, 79–89.

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