A hybrid stochastic/fixed-sectional method for solving the population balance equation

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A hybrid stochastic/fixed-sectional method for solving the population balance equation

Alexandre Bouaniche, Luc Vervisch, Pascale Domingo

To cite this version:

Alexandre Bouaniche, Luc Vervisch, Pascale Domingo. A hybrid stochastic/fixed-sectional method

for solving the population balance equation. Chemical Engineering Science, Elsevier, 2019, 209,

pp.115198. �10.1016/j.ces.2019.115198�. �hal-02313897�

A hybrid Stochastic/Fixed-Sectional method for solving the Population Balance Equation

Alexandre Bouaniche ^a , Luc Vervisch ^a,∗ , Pascale Domingo ^a

a

CORIA - CNRS, Normandie Universit´ e, INSA de Rouen, Technopole du Madrillet, BP 8, 76801 Saint-Etienne-du-Rouvray, France

Accepted for publication in Chemical Engineering Science

Abstract

The dynamics of flowing non-inertial particles undergoing nucleation, surface growth/loss, agglomeration and sometimes breakage, is usually characterised by the particle size distribution function. This distribution evolves accord- ing to a population balance equation. A novel approach combining Monte Carlo and fixed-sectional methods is proposed to minimise the discretisation errors when solving the surface growth/loss term of the population balance equation. The approach relies on a fixed number of stochastic particles and sections, with a numerical algorithm organised to minimise errors even for a moderate number of stochastic particles and sections. Canonical test cases featuring nucleation, agglomeration, and surface growth/loss are simulated.

Results against the analytical solutions confirm the improvement in accu- racy of the novel approach compared with fixed-sectional methods for the same computational effort. The hybrid method is thus of particular interest for simulating problems where surface growth/loss dominates the particles physics.

Keywords: Aerosol modelling; Sectional method; Stochastic method;

Hybrid modeling; Particle Size Distribution; Population Balance Equation, Probability Density Function, Monte Carlo solution

∗

Corresponding author

Email address: [email protected] (Luc Vervisch)

Nomenclature

v particle characteristic size

n(v; x, t) particle number density of size v per unit size N T (x, t) total number particle density

P (v ^∗ ; x, t) probability density function of particle size G(v) particle growth/loss rate

H ˙ o (x, t) nucleation source per unit of flow volume M number of sections

v _i representative size of the i-th section of size N i (x, t) particle number density in the i-th section of size

N _i ^R (x, t) residual particle number density in the i-th section of size A ˙ _i (x, t) agglomeration source in the i-th section of size

A ˙ _T (x, t) total agglomeration source

N _P total number of stochastic particles

n _P

(x, t) number of stochastic particles in the i-th section of size 1. Introduction

The numerical simulation of the dynamics of non-inertial particles in com- plex flows constitutes a real challenge. In many engineering processes, these

5

particles nucleate, their size can grow or be reduced by surface chemical re- actions and, while they are transported by the flow, they can agglomerate, or even break. Aside from the modeling of the complex physical and chemical phenomena at play, the numerics behind the simulation of these flows, which may operate in the turbulent regime, raises numerous issues. Background

10

and comprehensive reviews on these subjects may be found in Ramkrishna (2000); Fox (2003); Marchisio & Fox (2007) and references therein. Along these lines, a large variety of numerical approaches have been discussed in the literature to simulate crystallisation in liquid (Qamar et al., 2007), carbon and soot formation in flames (Leung et al., 1991; Balthasar & Kraft, 2003;

15

Ma et al., 2005; Lindstedt & Louloudi, 2005; Zucca et al., 2006; Patterson &

Kraft, 2007; Eberle et al., 2017; Sewerin & Rigopoulos, 2017; Rodrigues et al., 2018; Aubagnac-Karkar et al., 2018; Schiener & Lindstedt, 2019; Franzelli et al., 2019) and many other chemical engineering applications with non- inertial particles. Works have focused on numerical methods for the direct

20

solving of the particle size distribution after discretisation of the phenom-

ena driving its time evolution (Gelbard & Seinfeld, 1978; Hounslow et al.,

1988; Lister et al., 1995; Kumar & Ramkrishna, 1996a,b, 1997; Rigopoulos

& Jones, 2003; Filbet & Laurenot, 2004; Park & Rogak, 2004; Qamar et al., 2007; Nguyen et al., 2016; Sewerin & Rigopoulos, 2017), while others adress

25

the problems from moments of the distribution (Frenklach, 2002; Mueller et al., 2009; Salenbauch et al., 2019).

The description of these complex multi-phase flows is usually tackled with a statistical formalism. The particle size distribution (PSD) function is in- troduced to collect information on the density of the number of particles, per

30

unit size and per unit flow volume. The time evolution of the PSD is governed by a Population Balance Equation (PBE) (Ramkrishna, 1985, 2000; Solsvik

& Jakobsen, 2015). Aside from the usual unsteady and flow convective terms, the PBE contains sources and sinks due to nucleation, agglomeration (Smolu- chowski, 1917) and breakage in some cases (Das, 2016). In addition, because

35

the particles sizes can increase or decrease by surface reaction, thus without change of the number of particles, a conservative convective flux in size space also appears in the PBE (Hulburt & Katz, 1964).

As for any non-linear convective effect (Ferziger & Peri´ c, 1996), a stable numerical discretisation of this surface growth/loss term results from a com-

40

promise between stability and numerical diffusion. In the case of physical problems for which nucleation is associated to fast agglomeration, the error accumulated in the PSD when solving for particles surface change may not be a serious issue. However, focussing on specific physical phenomena, or flow zones, where nucleation and growth dominate, the verification of chemi-

45

cal kinetics or other models against experiments requires a precise treatment.

This is particularly the case in the careful validation of the basic mechanisms driving sooting flames (Desgroux et al., 2017), or more generally in any flow where agglomeration is slower than surface growth or loss, as previously dis- cussed by Park & Rogak (2004).

50

As noticed above, many numerical approaches already exist and some have been specifically developed with success to minimise numerical errors associated to particles surface evolution (Kumar & Ramkrishna, 1996a,b, 1997; Tsantilis et al., 2002; Park & Rogak, 2004; Nguyen et al., 2016; Sew- erin & Rigopoulos, 2017). The objective of this work is to further progress

55

along these lines, by discussing an alternative strategy to solve for particles growth/loss in the PBE.

The PSD, number of particles per unit of flow volume and per unit of particle size, is first decomposed into the total number of particles per unit of flow volume times the size probability density function (PDF) per unit of

60

particle size. The total number of particles per unit of flow volume varies with nucleation and a sink due to agglomeration or disappearance. The transport equation for the probability density function of size is then derived from the population balance equation of the PSD. A dual discretisation of size space is proposed, based on both fixed sections (i.e., sections boundaries are fixed)

65

and a fixed number of stochastic particles. The sections of size are defined as in well established fixed-sectional methods and allow for computing the am- plitude of the agglomeration and nucleation sources from physical modeling.

Each stochastic particle carries information on size independently of sections, with a distribution of sizes over the stochastic particles representative of the

70

PDF. Growth is then applied to every stochastic particle in the form of a simple linear problem, free from any artificial diffusion in size space.

A Monte Carlo procedure with random selection and movement of the stochastic particles between sections is designed to simulate agglomeration and nucleation. The number of particles selected is calibrated by the sources

75

and sinks computed over the sections. Because the number of stochastic particles stays finite, a residual exists in the form of a roundoff error between the exact continuous PSD and the distribution reconstructed from the PDF discretised over the stochastic particles. To overcome this issue, the evolution of this residual is solved with a fixed-sectional method and cumulated till the

80

roundoff error becomes large enough to be redistributed over a finite number of stochastic particles.

Using a moderate number of stochastic particles, the CPU time stays of the order of the one required by fixed-sectional methods, but particles sur- face change (growth/loss) is solved with a very limited amount of numerical

85

diffusion of the PSD in size space.

The hybrid sectional Monte Carlo solution discussed in this work stands as a numerical method to solve for a PDF balance equation. It should not be confused with direct Monte Carlo solutions of non-inertial particles dynamics, which aim at performing a direct numerical simulation of the elementary

90

physical phenomena acting on the particles (Falope et al., 2001; Li et al., 2001; Oullion et al., 2009; Pesmazoglou et al., 2016). The presented method also differs from constant number Monte Carlo methods (Smith & Matsoukas, 1998) or differentially weighted Monte Carlo approaches (Lee et al., 2015;

Patterson et al., 2011; Zhao & Zheng, 2013). In these methods, collision

95

frequencies and corresponding agglomeration source terms are calculated over

all stochastic particles or strategies like majorant kernels are proposed to

avoid calculating collision probabilities between all stochastic particles. CPU

cost depends mainly on the number of stochastic particles in these former methods. In the proposed hybrid approach, agglomeration source terms are

100

calculated from the particle number density discretised over the sections, as in fixed-sectional methods and therefore for a much lower CPU cost (which depends primarily on the number of sections), allowing a future application to the simulation of real systems.

The paper is organised as follows, the formulation of the problem is

105

first given in the subsequent section. Then, the attempt to set up a hy- brid stochastic/fixed-sectional approach is reported in detail. A set of well- established canonical test cases retained for evaluating the method are pre- sented before discussing the results obtained. Time evolutions of PSD are compared with the analytical solutions and simulations using fixed-sectional

110

methods. Discretisation errors are quantified from the moments and the Wasserstein metric of size distributions. Convergence and response to the resolution parameters are also studied.

2. Problem formulation

The Particle Size Distribution n(v; x, t), number of particles of charac- teristic size v (in terms of volume or mass, v is a continuous independent variable), per unit of flow volume and per unit of characteristic size of an aerosol submitted to simultaneous nucleation, surface variation and agglom- eration, is governed by a Population Balance Equation (PBE) (Ramkrishna, 2000; Solsvik & Jakobsen, 2015):

∂n(v ; x, t)

∂t + u · ∇ n(v; x, t) + ∂

∂v [G(v )n(v; x, t)] = ˙ h(v _o ; x, t) (1) + 1

2 Z v

0

β(v − v, ¯ v)n(v ¯ − ¯ v; x, t)n(¯ v; x, t)d¯ v − n(v ; x, t) Z ∞

0

β(v, ¯ v)n(¯ v; x, t)d¯ v , where usual notations are adopted. G(v) > 0 is the surface growth rate or

115

G(v) < 0 the surface loss rate. ˙ h(v _o ) > 0 is the nucleation rate or ˙ h(v _o ) < 0 the disappearance rate, seen at size v _o . The integral source term on the RHS accounts for agglomeration following the continuous counterpart of Smolu- chowski equation (Smoluchowski, 1917), with β(v, v) the collision kernel for ¯ two particles of volume v and ¯ v. The PSD evolution is thus driven by an

120

integro-partial-differential equation of the hyperbolic type.

The surface variation rate G(v) stands as a convective term in the particle

size space. Resolution of G(v) is challenging, similarly to the non-linear flow

convective term in physical space (Ferziger & Peri´ c, 1996), which motivates the present study.

125

Further quantities related to the PSD are usually introduced. N i (x, t) is defined as the number of particles of characteristic size v _i per unit of flow volume

N _i (x, t) = Z

I

n(v; x, t)dv , (2)

where the interval I v

≡ [v _i ^inf , v _i ^sup ] defines the i-th fixed-section of size. The total number density per unit of flow volume is the sum over all sizes or over the M sections considered

N _T (x, t) = Z ∞

v

n(v; x, t)dv = X M

i=1

N _i (x, t) . (3)

Similarly, the nucleation source per unit of flow volume is H ˙ _o (x, t) =

Z

I

h(v; ˙ x, t)dv . (4)

The Smoluchowski agglomeration sources/sink (Eq. (1)),

˙

a(v; x, t) (5)

= 1 2

Z v

0

β(v − v, ¯ v)n(v ¯ − v; ¯ x, t)n(¯ v; x, t)d¯ v − n(v; x, t) Z ∞

0

β(v, ¯ v)n(¯ v; x, t)d¯ v , leads to the definition of the agglomeration source for the i-th section

A ˙ _i (x, t) = Z

I

˙

a(v ; x, t)dv , (6)

and A _T is the total sink due to agglomeration over all particles, thus the sum

of A _i (x, t) over all sections A ˙ _T (x, t) =

Z ∞

v

˙

a(v; x, t)dv = X M

i=1

A ˙ _i (x, t) . (7)

All these quantities allow for combining the PBE with the evolution of the probability density function of the characteristic particle size.

3. Hybrid Stochastic/Fixed-Sectional method 3.1. Control parameters and statistical description

To benefit from a description in which surface growth or loss is cast

130

into the form of a linear term, instead of directly solving for the population balance equation, it is proposed to consider both N _T (x, t), the total number of particles per unit volume, and P (v ^∗ ; x, t), the probability density function (PDF) of the particles characteristic size, where v ^∗ ∈ [v _o , ∞ ] denotes the sample space variable associated to v, seen as a random variable.

135

The relation between n(v; x, t), the particle number density per unit size, N _i (x, t), the number density of particles whose size is in the section I _v

(v ∈ I _v

) at the flow position ‘x’ at time ‘t’ (Eq. (2)), and P (v ^∗ ; x, t), the PDF of the particles sizes reads:

Z

I

n(v ^∗ ; x, t)dv ^∗ = N _i (x, t) = N _T (x, t) Z

I

P (v ^∗ ; x, t)dv ^∗ , (8)

where Z

I

P (v ^∗ ; x, t)dv ^∗ (9)

is the probability to find particles of sizes v ∈ I _v

. Because (8) should be valid whatever I v

,

n(v ^∗ ; x, t) = N _T (x, t)P (v ^∗ ; x, t) . (10) The function

δ(v − v ^∗ ) = lim

dv→0 1/dv if v ∈ [v ^∗ − dv/2, v ^∗ + dv/2] (11)

= 0 otherwise , (12)

is introduced and P (v ^∗ ; x, t) = δ(v(x, t) − v ^∗ ), where · denotes a statisti- cal average (Lundgren, 1967; Dopazo, 1979; Kollmann, 1990; Dopazo et al., 1997).

The nucleation term in the PBE (Eq. (1)) may be written ˙ h(v _o ; x, t) =

140

H ˙ _o (x, t)δ(v _o − v ^∗ ), with ˙ H _o (x, t) defined by (4) in the limit where the size of the interval I o goes to zero. Similarly, the agglomeration term may be written ˙ a(v ^∗ ; x, t) = ˙ A _i (x, t)δ(v _i − v ^∗ ), with ˙ A _i (x, t) defined by (6) in the limit where I _v

goes to zero. Then the PBE formally becomes

∂n(v ^∗ ; x, t)

∂t + u · ∇ n(v ^∗ ; x, t) + ∂

∂v ^∗ [G(v ^∗ )n(v ^∗ ; x, t)] (13)

= H ˙ _o (x, t)δ(v _o − v ^∗ ) + ˙ A _i (x, t)δ(v _i − v ^∗ ) . The total number density N _T evolves according to

∂N _T (x, t)

∂t + u(x, t) · ∇ N _T (x, t) = ˙ H(v _o ; x, t) + ˙ A _T (x, t) , (14) with ˙ A _T (x, t) given by (7). From (10) the PDF evolves as

∂P (v ^∗ ; x, t)

∂t =

1 n(v ^∗ ; x, t)

∂n(v ^∗ ; x, t)

∂t − 1

N T (x, t)

∂N _T (x, t)

∂t

P (v ^∗ ; x, t) . (15) Introducing (13) and (14) in this relation, the PDF evolution equation is obtained

∂P (v ^∗ ; x, t)

∂t + u(x, t) · ∇ P (v ^∗ ; x, t) =

z }| (i) {

− ∂

∂v ^∗

G(v ^∗ )P (v ^∗ ; x, t)

+ H ˙ _o (x, t)

N T (x, t) δ(v _o − v ^∗ ) − P (v ^∗ ; x, t)

| {z }

(ii)

+ 1

N _T (x, t)

A ˙ i (x, t)δ(v i − v ^∗ ) − A ˙ T (x, t)P (v ^∗ ; x, t)

| {z }

(iii)

. (16)

In this balance equation, as in the PBE, the change of particles sizes at the

145

rate G(v _i ) is a convective term in size space (term (i)). The term (ii) on the

RHS is nucleation, which is decomposed into two parts preserving the normal- isation of the PDF. The first, proportional to δ(v _o − v ^∗ ), increases the prob- ability to find the smallest particles at the nucleation rate ˙ H o (x, t)/N T (x, t), while the second decreases, at the same rate, the probability for all sizes. A

150

similar formulation is found for agglomeration (term (iii)), with the proba- bility evolving at the positive or negative rate ˙ A i (x, t)/N T (x, t), associated to a correction proportional to − A ˙ _T (x, t)/N _T (x, t) > 0, so that the PDF nor- malisation is preserved. Indeed, when two particles of characteristic sizes v _i and v j agglomerate, the probability of their respective initial size decreases

155

( ˙ A _i (x, t) < 0 and ˙ A _j (x, t) < 0), to increase the probability of their new size v _k ( ˙ A _k (x, t) > 0). However, because the total number of physical particles decreases in this process, the probability of all sizes benefit from an increase proportional to − A _T , the overall particle sink.

The solutions of the equations (14) and (16) provide all the necessary

160

information to simulate the nucleation and the growth of an ensemble of particles transported in a flow. The particle size distribution N _i (x, t) can then be recovered from (8).

Because the focus is on the numerical solving of terms controlling the PSD shape, a perfectly stirred/homogeneous reactor is considered (u = 0).

165

However, the straightforward addition of the convective flow velocity will be discussed in the conclusion.

3.2. Hybrid Stochastic/Fixed-Sectional solution

The probability density function P (v ^∗ ; t) can be discretised over a set of N _P stochastic particles, ¹ each carrying information on the particle size, i.e.

170

v = v ^k for k = 1, · · · , N P and P (v ^∗ ; t) = (1/N P ) P N

k=1 δ(v ^k (t) − v ^∗ ). The total number of stochastic particles N _P is fixed.

The v-space is also discretised in M fixed sections, to define a mesh provid- ing a distribution of ∆v i = v _i ^sup − v _i ^inf , for i = 1, · · · , M . Uniform, geometric and exponential sectional grids will be tested thereafter. The characteristic

175

size v ^k of a stochastic particle can take any value between the considered size bounds [v _o , v _M ], independently of the fixed sectional mesh.

Within this set of N _P particles, an integer number n _P

(t) of stochastic par- ticles features sizes so that v ^k ∈ I _v

≡ [v _i ^inf , v ^sup _i ]. This number of stochastic particles relates to the PDF and to N _i (t), the number densities of the physical

1

Space is omitted in this subsection for brevity.

particles (Eq. (8)), according to:

Z

I

P (v ^∗ ; t)dv ^∗ = n P

(t)

N _P = N i (t)

N _T (t) . (17)

To simulate the PDF time evolution through v ^k (t), the stochastic particles time evolution, a fractional-step method is followed. Starting at time t ⁿ , surface growth/loss is first applied to advance the solution to time t ⁿ⁺

=

180

t ⁿ + δt/2. This is applied in a deterministic way to every k-th particle, as a simple linear process proportional to G(v ^k (t)), which is the major advantage of the proposed approach. Then from the time t ⁿ⁺

, the solution is advanced to t ⁿ⁺¹ = t ⁿ⁺

+ δt/2 by applying nucleation and agglomeration effects, which are simulated by moving the stochastic particles between the defined

185

sections. The number of stochastic particles randomly selected to be removed from a section and dispatched over the others, are calculated according to the nucleation and agglomeration rates controlling the PDF evolution (Eq. (16)).

At every instant t ⁿ , δt is determined so that stability is secured, different amplitudes of δt may be required in practice to advance from t ⁿ to t ⁿ⁺

190

(growth/loss) and from t ⁿ⁺

to t ⁿ⁺¹ (nucleation and agglomeration).

3.2.1. Surface growth/loss

During surface growth or loss, the size of the k-th stochastic particle evolves according to:

dv ^k (t)

dt = G(v ^k (t)) , k = 1, · · · , N _P . (18) Each stochastic particle then carries information on an updated size v ^k (t ⁿ⁺

).

The total number density stays constant during growth (dN _T (t)/dt = 0).

Once Eq. (18) is solved for each particle, an updated distribution of the

195

stochastic particles is available and the PDF P (v ^∗ ; t ⁿ⁺

) is known along with n _P

(t ⁿ⁺

) the number of stochastic particles in every section.

3.2.2. Nucleation and agglomeration

Nucleation and agglomeration are subsequently applied, which impacts on

the number density N T (t) and on the PDF through the change of n P

(t ⁿ⁺

)

for each interval I _v

. Starting from N _T (t ⁿ ) = N _T (t ⁿ⁺

), the number density

dv

(t)

dt = G(v

(t))

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PDF Stochastic particles

Residual number density Number density distribution

Surface growth/loss

Nucleation &

agglomeration

N

Stochastic particles M Fixed sections

Sources computed from fixed sections

Stochastic particles Residual number density

Total number density

Random reallocation of stochastic particles according to nucleation & agglomeration

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Updated number density distribution

Updated PDF Updated Residual number density

Updated total number density

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(t

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P(v^⇤;tⁿ) = 1 NP

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X

k=1

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P(v^⇤;tⁿ)dv^⇤

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X k=1

⇣

v^k(tⁿ⁺¹²) v^⇤⌘

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N

(t

)

<latexit sha1_base64="+txQ+1Io0eowRRZWUrjsBS3Viy0=">AAACAXicbVDLSsNAFL2pr1pfVTeCm2ARKkJJ6kLdFdy4kir2IW0aJtNJO3QyCTMToYSI4K+4caGIW//CnX/j9LHQ1gMXDufcy733eBGjUlnWt5FZWFxaXsmu5tbWNza38ts7dRnGApMaDlkomh6ShFFOaooqRpqRICjwGGl4g4uR37gnQtKQ36phRJwA9Tj1KUZKS25+76pz49Ki6iT8uO0LhBM7TcppeuTmC1bJGsOcJ/aUFCrnjzBC1c1/tbshjgPCFWZIypZtRcpJkFAUM5Lm2rEkEcID1CMtTTkKiHSS8QepeaiVrumHQhdX5lj9PZGgQMph4OnOAKm+nPVG4n9eK1b+mZNQHsWKcDxZ5MfMVKE5isPsUkGwYkNNEBZU32riPtI5KB1aTodgz748T+rlkn1SKl/rNO5ggizswwEUwYZTqMAlVKEGGB7gGV7hzXgyXox342PSmjGmM7vwB8bnDwLvl3A=</latexit>

@N_i^R(t)

@t = RHS(3-point sectional)

<latexit sha1_base64="XdVGLR39+niRHxGoCzi3DDl1c6Y=">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</latexit>

nPi(tⁿ⁺¹²) =b nPi(tⁿ⁺¹²)e+{ nPi(tⁿ⁺¹²)}

<latexit sha1_base64="JAzC1bbxZWapw9KUpAbDdsXG65o=">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</latexit>

H ˙

(t

) A ˙

(t

) A ˙

(t

)

<latexit sha1_base64="qkTDlAecZR21455qEWwcy5mZvRc=">AAACTnicfVHLSgMxFL1TX7W+qi7dBEVQlDJTF+pOcdNlBeuDtg6ZNKPBTGZI7ghlGPD/3Ig7P8ONC0U003bhC08IHM499yY5CRIpDLruk1MaG5+YnCpPV2Zm5+YXqotLpyZONeMtFstYnwfUcCkUb6FAyc8TzWkUSH4W3BwV9bNbro2I1Qn2E96N6JUSoWAUreRXeacXI2n48QZeZmqrE2rKMi/P6nm+STrbw0UGpkNf/GcaeU7+8FT86ppbcwcgv4k3ImsH+3dQoOlXH+04lkZcIZPUmLbnJtjNqEbBJM8rndTwhLIbesXblioacdPNBnHkZN0qPRLG2m6FZKB+7choZEw/CqwzonhtftYK8a9aO8Vwr5sJlaTIFRseFKaSYEyKbElPaM5Q9i2hTAt7V8KuqQ0C7Q8UIXg/n/ybnNZr3k6tfmzTuIAhyrACq7ABHuzCATSgCS1gcA/P8ApvzoPz4rw7H0NryRn1LMM3lMqf2fuysQ==</latexit>

P(v^⇤;tⁿ⁺¹) = 1 NP

NP

X k=1

v^k(tⁿ⁺¹) v^⇤

<latexit sha1_base64="2XKH93f/GHg9tWH3jXikEiNPoqw=">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</latexit>

nPi(tⁿ⁺¹) =NP

Z

Ivi

P(v^⇤;tⁿ⁺¹)dv^⇤

<latexit sha1_base64="xk8FpdJALTVLNcSGd7/xebSSt1A=">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</latexit>

N

(t

) = N

(t

) Z

I_vi

P (v

; t

)dv

<latexit sha1_base64="EnQRbR1CUKcXENWt514dZWTQihg=">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</latexit>

Ni(tⁿ⁺¹) =NT(tⁿ⁺¹) Z

I_vi

P(v^⇤;tⁿ⁺¹)dv^⇤

<latexit sha1_base64="i7Uleclq7UhiCnoy5+Drzy3ejJU=">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</latexit>

N

(t

) = { n

(t

)}

N

N

(t

)

<latexit sha1_base64="lOX5uQhUoSIvx9r1bhyIZFNHFX8=">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</latexit> dNT(t) dt=˙Ho(tn+12)+˙AT(tn+12) <latexit sha1_base64="wpb2BYbYFme+m1A49BAoZ3v8reg=">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</latexit>

Figure 1: Flowchart of the hybrid stochastic/fixed-sectional method.

evolves from t ⁿ⁺

to t ⁿ⁺¹ with dN T (t)

dt = ˙ H _o (t) + ˙ A _T (t) . (19) Once N _T (x, t ⁿ⁺¹ ) is known solving (19), the PDF equation (16) is advanced in time with nucleation and agglomeration:

200

P (v ^∗ ; t ⁿ⁺¹ ) = α _H

δ(v _o − v ^∗ ) + α _A

δ(v _i − v ^∗ )

+ (1 − α H

− α A

)P (v ^∗ ; t ⁿ⁺

) , (20)

with α _H

the relative increase of P (v _o ; t ⁿ⁺

) by nucleation (and decrease of

P (v ^∗ ; t ⁿ⁺

) for v ^∗ 6 = v _o ), α _A

the relative increase/decrease due to agglomer-

ation and α _A

the total agglomeration sink defined by Eq. (16):

α _H

=

H ˙ o (t ⁿ⁺

)

N _T (t ⁿ⁺¹ ) · δt , (21)

α _A

=

A ˙ _i (t ⁿ⁺

)

N _T (t ⁿ⁺¹ ) · δt , (22)

α _A

=

A ˙ _T (t ⁿ⁺

)

N _T (t ⁿ⁺¹ ) · δt . (23)

According to Eq. (17), integrating over I _v

and multiplying by N _P the PDF evolution given by the relation (20) leads to the evolution of the number of stochastic particles per section. This discretised time evolution is organised as:

n _P

(t ⁿ⁺¹ ) = n _P

(t ⁿ⁺

) + ∆n _P

(t ⁿ⁺

) , (24) with increments ∆n _P

(t ⁿ⁺

) in the form of real numbers, which will need to be transformed subsequently into integer numbers of particles in the Monte

205

Carlo algorithm. From (20),

∆n _P

(t ⁿ⁺

) = (α _H

+ α _A

) N _P − (α _H

+ α _A

)n _P

(t ⁿ⁺

)

+ α _R

N _P , (25)

∆n _P

(t ⁿ⁺

) = α _A

N _P − (α _H

+ α _A

)n _P

(t ⁿ⁺

)

+ α R

N P for i 6 = o . (26)

The terms proportional to α _R

are the accumulation of the round-off error, which goes to zero for N _P → ∞ . Cumulated over the iterations, this will impact on particles when α R

≥ 1/N P . At every iteration, ∆n P

(t ⁿ⁺

) is thus decomposed into its integer and fractional (or decimal) parts. The fractional part { ∆n _P

(t ⁿ⁺

) } is defined from the nearest integer b ∆n _P

(t ⁿ⁺

) e ,

{ ∆n _P

(t ⁿ⁺

) } = ∆n _P

(t ⁿ⁺

) − b ∆n _P

(t ⁿ⁺

) e . (27) The integer part b ∆n _P

(t ⁿ⁺

) e sets the variation of the number of stochastic particles within a section during the reallocation step corresponding to nu- cleation and agglomeration. The following Monte Carlo algorithm is applied:

• If b ∆n P

(t ⁿ⁺

) e is negative, a random number −b ∆n P

(t ⁿ⁺

) e of stochas-

210

tic particles is picked among the n _P

(t ⁿ⁺

) present in I _v

.

• All the picked particles from all I _v

intervals (i = 1, · · · , M) constitute an ensemble P (t ⁿ⁺

) of particles whose characteristic size needs to change.

• If b ∆n _P

(t ⁿ⁺

) e is positive, b ∆n _P

(t ⁿ⁺

) e particles are taken from P (t ⁿ⁺

)

215

and allocated to I v

at the representative size v ^? _i (t ⁿ⁺

), defined to con- serve mass, as discussed in the next subsection.

The larger the total number of stochastic particles N _P , the smaller the relative contribution of the decimal part { ∆n _P

(t ⁿ⁺

) } to ∆n _P

(t ⁿ⁺

). This residual decimal part defines N _i ^R (t ⁿ ), a residual number density of physical particles in the section v _i , which is computed at time t ⁿ following (17)

N _i ^R (t ⁿ ) = { ∆n _P

(t ⁿ⁻

) }

N _P N _T (t ⁿ ) , (28)

where ∆n P

(t ⁿ⁻

) denotes ∆n P

of the previous iteration in time. The growth/loss of the physical particles represented by this number density resid- ual N _i ^R (t ⁿ ) is not included in the stochastic particles and needs a separate solving, between t ⁿ and t ⁿ⁺

(i.e., simultaneously with growth/loss for the stochastic particles Eq. (18)) This is done with a sectional method based on the 3-point discretisation for particle growth/loss (Park & Rogak, 2004) (Appendix A). Then, N _i ^R (t ⁿ⁺

) is known and α _R

is obtained from

α _R

= N _i ^R (t ⁿ⁺

)

N _T (t ⁿ⁺¹ ) , (29)

and applied to Eqs. (25) and (26) to compute ∆n P

(t ⁿ⁺

). For sufficiently large values of N _P , typically 10 ⁵ as shown thereafter, the residual number density of particles is expected to be negligible and will not perturb much

220

the accuracy of the method. Then, α R (v i ) can be set to zero in the relations (25) and (26). However as shown below, accounting for the contribution of the residual part allows for reducing N _P (such as 10 ³ or less) and therefore the CPU time.

Optionally, a trigger can also be set so that when the number of stochastic

225

particles present in a given section becomes too small, the surface growth/loss

is then fully solved through the evolution of N _i ^R (t). In practice, a trigger of

5 particles per section is used and has been found to be sufficient to avoid

any noise on the tails of distributions.

Table 1: Growth parameters

Case 1(a) 1(b)

Initial 1 for 0.2 ≤ v; 0 else δ(1)

Growth kernel 0.05 v

Agglo. kernel 0 0

Number of sections 20 40

Grid type unif. ∆v = 0.2 geo. F

= 2

Size range 0 – 4 0.7 – 7.3 · 10

Figure 1 displays a flowchart summarising the method.

230

3.2.3. Agglomeration source

The method proposed in Kumar & Ramkrishna (1996a) is retained for computing the agglomeration source ˙ A _i (t) of Eq. (22). For any colliding par- ticles of volume v in section i and ¯ v in section j, the collision kernel β(v, v) ¯ is assumed fixed to β(v i , v j ) = β i,j . Particles formed by agglomeration are

235

distributed in the sections in a manner that conserves the zeroth and first moments of the PSD, namely number and mass. This method avoids the evaluation of the double integrals of the collision kernel and is therefore com- putationally efficient (see Kumar & Ramkrishna (1996a) for more details).

The agglomeration source used in (22) reads

240

A ˙ _i (t) =

k≤j≤i

X

j,k v

≤v

+v

≤v

1 − δ _j,k 2

ηβ _j,k N _j (t)N _k (t)

− N _i (t) X M

k=1

β _i,k N _k (t) , (30)

with

η =

 

 

 



v ^? _i+1 − (v _j ^? + v _k ^? )

v ^? _i+1 − v _i ^? if v ^? _i ≤ v ^? _j + v ^? _k ≤ v ^? _i+1 , v ^? _i−1 − (v _j ^? + v _k ^? )

v ^? _i−1 − v _i ^? if v ^? _i−1 ≤ v _j ^? + v _k ^? ≤ v _i ^? ,

(31)

In the hybrid stochastic/fixed-sectional approach, the characteristic volume v ^? _i must be representative of the average mass contained in the i-th section.

v ^? _i is calculated dynamically, depending on the volumes of the stochastic

particles contained in both the section and the residual terms resulting from

the roundoff,

v _i ^? (t ⁿ⁺

) = (N _T (t ⁿ )/N _P ) P ⁿ

^(t)

k=1 v _i ^k (t ⁿ⁺

) + N _i ^R (t ⁿ⁺

)v ^? _i (t ⁿ )

(N T (t ⁿ )/N P )n P

(t ⁿ⁺

) + N _i ^R (t ⁿ⁺

) , (32) where v _i ^k = v ^k if v ^k ∈ I _v

and v _i ^k = 0 otherwise, n _P

(t ⁿ⁺

) is the number of stochastic particles in the i-th section (Eq. (17)) and N _i ^R (t ⁿ⁺

) is the residual number density of the particles in the section after applying surface gross or loss. v ^? _i needs to be updated again after reallocation of the particles due to agglomeration, to provide v _i ^? (t ⁿ⁺¹ ) from (32) with N _T (t ⁿ⁺¹ ), v _i ^k (t ⁿ⁺¹ ),

245

N _i ^R (t ⁿ⁺¹ ), v ^? _i (t ⁿ⁺

), n _P

(t ⁿ⁺¹ ).

Once v ^? _i determined, the particles reassigned in the i-th section are dis- tributed in this section following a two-step process:

• First, the b ∆n _P

(t ⁿ⁺

) e particles are allocated randomly within the section at sizes v ^k (t ⁿ⁺

), which are samples of a random variable v fol- lowing a target piecewise linear distribution defined by the probability density function,

p(v | v _i ^inf , v _i ^sup , w _i , w _i+1 ) = 2 w _i (v _i ^sup − v) + w _i+1 (v − v _i ^inf )

(w _i + w _i+1 )∆v _i ² . (33) In this distribution, the weights, w i , are calculated from the variations of the number densities at v ^? _i ,

250

w i = ∆n i (t ⁿ⁺

) + ∆n _i (t ⁿ⁺

) − ∆n _i−1 (t ⁿ⁺

)

v _i ^? − v _i−1 ^? (v _i ^inf − v ^? _i−1 ) , w _i+1 = ∆n _i (t ⁿ⁺

) + ∆n _i+1 (t ⁿ⁺

) − ∆n _i (t ⁿ⁺

)

v _i+1 ^? − v _i ^? (v _i ^sup − v ^? _i ) , with ∆n _i (t) = ∆n _P

(t)N _T (t)/(N _P ∆v _i ) (Eq. (17)). Such random piece- wise linear distribution secures a continuous distribution of the stochas- tic particles. However, it does not guarantee strict volume/mass con- servation by itself.

• Mass conservation is achieved in a second step by calculating a correc-

Table 2: Agglomeration parameters

Case 2(a) 2(b)

Initial e

e

Agglo. kernel 1 v

+ v

Number of sections 40 40

Grid type: Exponential, α 1.17 1.25

Size range 6.7 · 10

– 209 6.7 · 10

– 2006

tive factor K _i

K _i = v ^? _i (t ⁿ⁺

) (1/n P

(t ⁿ⁺

)) P ⁿ

^(t

⁾

k=1 v ^k _i (t ⁿ⁺

)

, (34)

then,

v _i ^k (t ⁿ⁺¹ ) = K _i v _i ^k (t ⁿ⁺

) , (35) and mass is conserved through the reallocation process.

255

Nucleation size is set as the lower boundary of the smallest size section v _o . As the numerical steps corresponding to nucleation/agglomeration and growth are sequential in the present model, it is necessary to account for a dispersion of effective nucleation sizes due to particle growth during the nu- cleation/agglomeration time step. For b ∆n _P

(t ⁿ⁺

) e > 0, the b ∆n _P

(t ⁿ⁺

) e

260

particles are therefore allocated randomly following a target uniform distri- bution between v _o and v _o + G(v _o )δt.

3.2.4. Time steps

As stated earlier, a fractional-step method is followed. The notation δt used above was schematic to explain the algorithm structure. The charac- teristic time step size of the first growth/loss sub-step (Fig. 1) is calculated following a usual Courant Friedrichs Lewy (CFL) condition (Ferziger & Peri´ c, 1996), based on the velocity G(v) and sections discretisation

δt _G = C min [∆v ₁ / | G(v ₁ ) | , · · · , ∆v _M / | G(v _M ) | ] . (36) Calculations have been performed with C = 0.01, to fully secure stability for both stochastic and sectional parts.

265

The characteristic time step size of the nucleation-agglomeration sub-step

of the algorithm is determined to limit the relative change of the distribution

δt _A = (γ + σ) N _T

H ˙ _o + ˙ A _T

+ P M i=1

A ˙ _i

, (37)

with σ = 0.02 in the simulations presented thereafter. If particle nucleation dominates, as in the beginning of a calculation with a negligible initial dis- tribution mostly present in the smallest section with also very few exchange of particles between sections, larger time steps may be allowed to let N _T increase faster until the exchange of particles between sections becomes sig-

270

nificant, then γ = 1 is used in (37). This specific ‘nucleation dominated’

regime is considered reached at a given time in a simulation if

H ˙ _o (t) + ˙ A _T (t)

> 100 · X M

i=1

A ˙ _i (t)

, (38)

N ₀ (t)/N _T (t) > 0.99 . (39)

Otherwise, γ = 0 is imposed in (37) to solve for the more general regimes of PSD evolution.

For the test cases considered in this work, δt _G ≤ δt _A and one or several

275

surface growth/loss sub-iterations can be applied between two agglomera- tion/nucleation sub-iterations. δt G is then further adjusted so that δt A is one of its multiple, still verifying the stability condition.

4. Canonical test cases

Four main representative cases for which analytical solutions exist are

280

considered. Sectional methods, based on two discretisations of the growth term, and the hybrid stochastic/sectional approach discussed above are ap- plied to simulate these canonical problems.

The number of sections set to discretise the normalised problems is fixed to 20, 40 or 80 depending on the case, for various size ranges (see the details in Tables 1 to 4). Following the literature, three types of grid discretisation are used: uniform, geometric and exponential. Defining v _i ^inf the inferior boundary of section i, the uniform grid reads

v _i ^inf = v ₀ ^inf + i∆v , (40)

Table 3: Growth/Loss & agglomeration parameters

Case 3(a) 3(b) 3(c) 3(d) 3(e)

Initial e

e

e

e

e

Growth kernel v v v v − v

Agglo. kernel 0.1 1 10 1 1

Number of sections 40 40 40 80 40

Geometric grid, F

2 2 2 √

2 2

Size range 6.7 · 10

– 7.3 · 10

8.3 · 10

– 9.1 · 10

6.7 · 10

–73

the geometric grid is constructed as in Park & Rogak (2004) following v _i ^inf = v ₀ ^inf F _s ⁱ , (41) and the exponential grid as in Rigopoulos & Jones (2003) ²

v ^inf _i = v ^inf ₀ + v ^inf ₀ 1 − α ⁱ

1 − α . (42)

The values of F _s and α are given in Tables 1 to 4.

The first main case features only growth and is broken into two subcases

285

1(a) and 1(b). Following Sewerin & Rigopoulos (2017), case 1(a) of Table 1 considers the advection of a unit step distribution, whose exact solution is a pure advection of the step function at the constant normalised speed G = 0.05.

Case 1(b) in Table 1 is from Park & Rogak (2004) and represents the

290

pure growth of a set of mono-disperse particles. The initial particle size distribution is a delta function, which is translated in size space at a speed proportional to the particle volume G(v) = v. Cases 1(a) and 1(b) are quite stringent, since numerical diffusion can transform the expected delta function into a poly-disperse distribution.

295

The second case in Table 2 is pure agglomeration. Case 2(a) is with a fixed agglomeration frequency, as proposed in Rigopoulos & Jones (2003).

From an initial exponential distribution exp( − v), the analytical solution of

2

Exponent i is used instead of i − 1 of Rigopoulos & Jones (2003) as i begins at 0 in

the present case.

the time evolution of the PSD was discussed in Scott (1968), n(v; t) = 4

(t + 2) ² exp

− 2v t + 2

. (43)

Case 2(b) is with a non-uniform collision frequency. Here, the Golovin sum kernel β(v _i , v _j ) = (v _i + v _j ) is retained to mimic the expected increase with volume of the collision frequency between two particles of characteristic sizes v _i and v _j . Starting from the same initial exponential distribution, the an- alytical solution of the PSD reads (Scott, 1968; Rigopoulos & Jones, 2003)

n(v ; t) =

1 − θ θ ^1/2

· exp ( − v(θ + 1)) v · I ₁

2vθ ^1/2

, (44)

where θ = 1 − exp( − t) and I 1 denotes the first order Bessel-I function.

In a third series of cases, agglomeration with either surface growth or loss is considered (Table 3). Starting from an initial exponential distribution, the time evolution follows (Ramabhadran et al., 1976):

n(v; t) = 4

(2 + β ₀ t) ² exp

− 2v exp( − t) 2 + β ₀ t − t

. (45)

The value of the size-independent collision kernel β _o is varied by two orders of magnitude (β _o = 0.1 case 3(a), β _o = 1 case 3(b), 3(d) and 3(e), β _o = 10 case 3(c), Table 3). In case 3(d), in the comparison between the hybrid method and the fixed-sectional one, the latter benefits from twice the number of

300

sections. In case 3(e), surface loss is applied instead of surface growth with G(v) = − v. The initial distribution and collision kernel are the same as in case 3(b).

Case 4 is the evolution of an initial exponential distribution submitted to nucleation and growth (Table 4), evolving into a uniform distribution for

305

large times. The normalised nucleation kernel ˙ H(v _o , t) is fixed to unity in this last test case.

5. Results

Analytical solutions are compared with simulation results. Simulations were run for all cases of Tables 1 to 4 with the hybrid stochastic/fixed-

310

sectional approach and with the standard fixed-sectional method. Convection

in size space was solved using either a 2-point or a 3-point algorithm (Park

Table 4: Nucleation & growth parameters

Case 4

Initial 10

δ(v

) Growth kernel v

Number of sections 40 Geometric grid, F

2

Size range 0.7 – 7.3 · 10

& Rogak, 2004) (see also Appendix A). Only test case 1(a) was not run with the 3-point algorithm, which is not designed to solve for growth on a uniform grid.

315

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

𝘷

−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

𝘯( 𝘷)

Figure 2: Particle Size Distribution n(v; t). Growth: case 1(a) of Table 1. Dashed line:

initial distribution (jump is between centred sections values). Normalised time t = 60. Line with empty diamonds: 2-point sectional method. Line with full circles: hybrid method, N

= 10

(values are shown at v

(Eq. (32)).

In case 1(a), the hybrid method perfectly reproduces the analytical so- lution (Fig. 2), with the pure convection of the step function. As expected, applying growth directly on the stochastic particles enables to convect the distribution in size space with no numerical diffusion. Similar results were obtained by Sewerin & Rigopoulos (2017) using explicit adaptive grid method

320

(EAGM), while in Fig. 2, the 2-point fixed sectional approach yields results

close to those of fully upwinded orthogonal collocation finite element method

(OCFEM), see Fig. 7 of Rigopoulos & Jones (2003).

10¹ 10³ 10⁵ 10⁷ 10⁹

𝘷

0.0 0.2 0.4 0.6 0.8 1.0

𝘯( 𝘷). 𝘷

Figure 3: Size Distribution n(v; t) · v. Growth: case 1(b) of Table 1. Triangle: Initial distribution. Plus symbol: Analytical solution. Line with diamonds: sectional 2-point method. Line with circles: sectional park 3-point method (Park & Rogak, 2004). Diamond symbols: hybrid method without residual term, α

= 0, N

= 10

.

Case 1(b) of Table 1 features an initial monodisperse distribution submit- ted to growth only, with a particle surface growth rate proportional to the

325

particle volume. The expected solution is thus a translation of the distribu- tion in size space. Figure 3 shows that the fixed-sectional methods (line with diamonds and circles) would need much more advanced numerics to capture this extreme case. However, the hybrid method operating here without any residual, returns the exact solution. In specific aerosol flow zones, where nu-

330

cleation and pure growth can dominate the physics, the spurious spreading of the distributions observed with sectional methods could strongly impair the calibration of the physical models.

The pure agglomeration case 2(a) of Table 2, with a fixed agglomeration frequency, is simulated with the hybrid approach for N _P = 10 ³ , 10 ⁴ and 10 ⁵ .

335

The initial distribution and the solutions at two successive times are shown in Fig. 4. To assess the impact of the residual number density (Eq.( 28)), Figs 4(a), 4(b) and 4(c) are obtained forcing α _R = 0 (Eq. (29)). As N _P is decreased from 10 ⁵ to 10 ³ , the effect of the roundoff then becomes visible.

The introduction of the procedure discussed above to deal with the residual

340

part, allows for alleviating this effect to better match the solution (Fig. 4(d)).

The case 2(b) of Table 2 with the Golovin agglomeration kernel (Eq. (44))

10⁻¹ 10⁰ 10¹ 10²

v

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

n(v)

(a) N

= 10

, α

= 0

10⁻¹ 10⁰ 10¹ 10²

v

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

n(v)

(b) N

= 10

, α

= 0

10⁻¹ 10⁰ 10¹ 10²

v

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

n(v)

(c) N

= 10

, α

= 0

10⁻¹ 10⁰ 10¹ 10²

v

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

n(v)

(d) N

= 10

, α

by (29).

Figure 4: Particle Size Distribution n(v; t). Size independent agglomeration: case 2(a) of Table 2 (Eq. (43)). Points: initial distribution. t = 10, dashed line: analytical solution, crosses: hybrid method. t = 20, solid line: analytical solution, plus: hybrid method.

(a)-(c): without residual term, α

= 0. (d): with residual term (Eq. (29)).

is also perfectly reproduced (Fig. 5).

The cases 3(a), (b) and (c), with both surface growth and agglomeration of the particles, are shown in Fig. 6. Here comparisons are made between

345

the exact solution given by the relation (45), the solution with the hybrid

method and the solutions with the fixed-sectional method using the classical

2-point and 3-point discretisation for growth (Park & Rogak, 2004). Three

collision parameters are applied (β _o = 0.1, 1.0 and 10) and results are given