1

SECTIONAL MODELING OF AEROSOL DYNAMICS IN MULTI-

DIMENSIONAL FLOWS

Shortened running title: Sectional Modeling in Multi-Dimensional Flows

By

D. MITRAKOS1,2, E. HINIS2, C. HOUSIADAS1

1 “Demokritos” National Centre for Scientific Research, Institute of Nuclear Technology

and Radiation Protection, Athens, 15310, Greece

2 National Technical University of Athens, Faculty of Mechanical Engineering

Athens, 15780, Greece

Aerosol Science and Technology, accepted, 2007

2

Abstract

The integration of computational fluid dynamics (CFD) with computer modeling of

aerosol dynamics is needed in several practical applications. The use of a sectional size

distribution is desirable because it offers generality and flexibility in describing the

evolution of the aerosol. However, in the presence of condensational growth the

sectional method is computationally expensive in multidimensional flows, because a

large number of size sections is required to cope with numerical diffusion and achieve

accuracy in the delicate coupling between the competing processes of nucleation and

condensation. The present work proposes a methodology that enables the implementation

of the sectional method in Eulerian multidimensional CFD calculations. For the solution

of condensational growth a number conservative numerical scheme is proposed. The

scheme is based on a combination of moving and fixed particle size grids and a re-

mapping process for the cumulative size distribution, carried out with cubic spline

interpolation. The coupling of the aerosol dynamics with the multidimensional CFD

calculations is performed with an operator splitting technique, permitting to deal

efficiently with the largely different time scales of the aerosol dynamics and transport

processes. The developed methodology is validated against available analytical solutions

of the general dynamic equation. The appropriateness of the methodology is evaluated by

reproducing the numerically demanding case of nucleation-condensation in an

experimental aerosol reactor. The method is found free of numerical diffusion and

robust. Good accuracy is obtained with a modest number of size sections, whereas the

computational time on a common personal computer remained always reasonable.

3

1. Introduction

Computer modeling of aerosol dynamics is of importance in a wide spectrum of

current applications, ranging form atmospheric chemistry and climate change, to a variety

of technological fields, like nuclear reactor safety or production of nano-sized materials.

In many circumstances the solution of the General Dynamics Equation (GDE) involves

multiple spatial dimensions and complex flows and so Computational Fluid Dynamics

(CFD) need to be combined with aerosol dynamics to accurately predict the behaviour of

the aerosol flow. CFD-based aerosol simulations have gained much attention in

experimental (Pyykönen and Jokiniemi 2000; Wilck and Stratmann 1997) or industrial

(Mühlenweg et al. 2002; Schild et al. 1999) aerosol reactors, to design and control the

system and improve the quality of the manufactured materials, usually nanoparticles.

However, despite the increase of the computational capabilities, computational fluid-

aerosol dynamics still remains a challenging task, especially when simultaneous

nucleation, condensational growth and coagulation take place within the flow. In such

cases, very dense spatial and/or temporal resolutions are required to describe

appropriately the coupling between processes characterized by largely different time

scales and the strong nonlinearities that are introduced (Pyykönen et al. 2002; Pyykönen

and Jokiniemi 2000). In such calculations the use of an efficient and accurate numerical

representation of the particle size distribution is a key issue.

Methods based on moments are widely used in aerosol dynamic simulations. The

main advantage of these methods is their low computational cost, because a small number

of additional equations, namely for the moments of the size distribution, need to be

solved. The basic problem of the moment methods is that they require some kind of

4

closure. Usually an assumption is made on the functional form of the particle size

distribution in order to achieve the closure of the transport equations (Modal methods,

Whitby and McMurry 1997). Several works exist in the literature where modal methods

are embedded in CFD codes for the simulation of multidimensional aerosol flows (Brown

et al. 2006; Schwade and Roth 2003; Stratmann and Whitby 1989; Wilck and Stratmann

1997). Obviously, modal methods do not allow for arbitrary evolution of the size

distribution because the functional form of the size distribution is specified beforehand

and remains fixed throughout the whole simulation. Moreover, the use of a constant

standard deviation may introduce inaccuracies in the calculation of coagulation and

condensational growth (Zhang et al. 1999). McGraw (1997) overcame the problem of

closure by proposing the quadrature-method of moments (QMOM). Because of its

potential the QMOM has been employed in several works, demonstrating its applicability

to more complex cases (e.g. Alopaeous et al. 2006, McGraw and Wright 2003). Also,

QMOM has been used in combination with CFD (Marchisio et al. 2003, Rosner and

Pyykönen 2002). The main advantage of the method is that no assumptions for the shape

of the distribution, or the form of the growth law, are required to satisfy the closure of the

moment equations. However, QMOM becomes quite challenging numerically in

multivariate cases (Rosner and Pyykönen 2002). A general drawback of the QMOM is

the non-uniqueness problem that arises in the reconstruction of the size distribution from

its moments.

The sectional approach (Gelbard et al. 1980) is conceptually straightforward and

offers more degrees of freedom, ensuring therefore greater generality. On the other hand,

the drawback is that the sectional method may be computationally very demanding

5

because its accuracy is directly related with the number of discretization sections (size

bins) that are used. A large number of sections is needed to reduce the numerical

diffusion inherent in the numerical solution of condensational growth, making the

sectional approach expensive to implement in CFD calculations (Mühlenweg et al. 2002;

Pyykönen and Jokiniemi 2000). A moving sectional method, where particle size bins are

allowed to move according to the growth law (Kim and Seinfeld 1990) can combat

numerical diffusion, but introduces serious complications in coupling with the other

aerosol processes, especially with transport. This limits the applicability of the moving

sectional approach, which is usually implemented in zero-dimensional aerosol

calculations along Lagrangian trajectories (Spicer et al. 2002; Tsantilis et al. 2002). Fixed

particle size grid is a more suitable approach for multidimensional, elliptic aerosol flows

and has been combined with CFD in multidimensional simulations in a number of works.

However, it has been used either under an approximate Lagrangian transport frame

(Johannessen et al. 2000; Pyykönen and Jokiniemi 2000) or in situations where growth is

only due to coagulation/agglomeration (Jeong and Choi 2003; Kommu et al. 2004I, II; Lu

et al. 1999; Mühlenweg et al. 2002; Park et al. 1999; Schild et al. 1999). When growth is

also due to vapor condensation then the coupling with nucleation makes the calculation

very demanding in terms of computational cost. Attempts to implement the sectional

method in a fully Eulerian, multidimensional computational framework on the basis of

standard CFD techniques revealed serious deficiencies (Pyykönen and Jokiniemi 2000;

Pyykönen et al. 2002). A methodology for implementing efficiently the sectional

approach in CFD-based aerosol calculations is therefore exceedingly needed in cases

6

characterized by strong coupling between nucleation and condensational growth. The aim

of our work is to propose such a methodology.

The reduction of the numerical diffusion is the greatest concern when

implementing a sectional method within a multidimensional CFD calculation. Several

methods have been proposed in the literature to combat numerical diffusion, while

keeping reduced the number of the sections needed. “Three point” discretization schemes

(Hounslow et al. 1998; Park and Rogak 2004) reduce numerical diffusion but introduce

oscillations and negative values of the size distribution that can lead to significant errors,

especially in cases where strong nucleation takes place. Lurmann et al. (1997) proposes to

solve the growth equation using a moving grid and then re-map the result on the initial

fixed grid, where all the other processes are treated. Re-mapping is done by using a cubic

spline interpolation scheme, which can introduce oscillations and negative values when

the size distribution is steep. Forcing these negative values to zero can result to

significant errors in the conservation of the particle number and volume. Jacobson (1997)

proposed the so-called moving center method in which the sections remain fixed through

the simulation but the characteristic particle size of each section is allowed to vary. The

particles grow according to the growth law and if the new size is out of the section all

particles are transferred into a next section. The size characterizing each section is

updated by averaging the transferred and preexisting particle volumes. Because particles

are not spitted in adjacent size bins numerical diffusion is combated efficiently. Zhang et

al. (1999) inter-compared several methods and qualified the moving center method as the

most appropriate for calculating the condensational growth of atmospheric aerosol in air

quality models. However, removing all particles from one size bin can result to

7

unrealistic empty bins and extra dents in the size distribution, as observed by Korhonen at

al. (2004) in nucleation simulations. Furthermore, because several particle sizes are

mixed through the transport processes, an averaging procedure must be performed

repeatedly to compute the characteristic particle size in every section, which introduces a

systematic bias in the results (Zhang at al. 1999). Yamamoto (2004) derived the

condensational growth equation in terms of the cumulative size distribution function,

which was solved with the help of semi-Lagrangian schemes. He showed that the use of

the cumulative size distribution function allows the conservation of the number of

particles and prevents overshoots, even in cases with sharp changes in the size

distributions.

The objective of the present work is to develop a methodology permitting to

incorporate efficiently the sectional approach into a multidimensional CFD calculation.

First, an efficient numerical solution of condensational growth is implemented by

combining the advantages of the schemes of Lurmann et al. (1997) and Yamamoto

(2004): a moving particle size grid is used and the results obtained are re-mapped on the

initial fixed grid. The interpolation during the re-mapping step is not performed directly

on the size distribution function, but on the cumulative size distribution, in order to

ensure the conservation of the particle number concentration. The developed aerosol

model includes nucleation, condensational growth, coagulation and all the major external

processes like transport, diffusion and external forces. The aerosol dynamics calculations

are one-way coupled with the CFD calculations, namely the velocity and temperature

fields are taken as input for the aerosol model while the output of the aerosol calculations

do not exert any influence on the flow. To cope with the strong coupling between aerosol

8

processes an operator splitting technique is employed (Oran and Boris 2001). The time

scales of the aerosol dynamics, and especially of nucleation and growth, highly differ

from these of the transport processes like convection and diffusion. Operator splitting is

generally more efficient in coupling multiple time scales than a global-implicit scheme,

which is based on a complete discretization of the equations. While the global implicit

approach is stable and more straightforward, since it treats all the processes together, it

can be very costly in multidimensional cases, when stiff processes are involved (Oran and

Boris 2000), as in the present case of simultaneous nucleation and condensation.

Recently, Mitsakou et al. (2004) and Kommu et al. (2004I, II) also used operator splitting

techniques to solve numerically the GDE with a sectional representation of the size

distribution.

The method we are proposing is validated extensively. Comparisons are made

with existing analytical solutions to assess the numerical solution of the GDE alone. To

assess the appropriateness of the overall method we reproduce numerically an aerosol

reactor case where all three major processes are coupled, i.e. nucleation, growth,

coagulation are present. As such, we selected the homogeneous nucleation experiments

performed by Ngyuen et al. (1987). These experiments are documented in detailed and

have been extensively analyzed in the literature. So, they are an appropriate basis for

benchmarking purposes.

9

2. Model

The temporal and spatial variation of the particulate phase is described by the

general dynamic equation (Friedlander 2000), which can be written as following:

( ) ( )( )ρ ρ ρ ρρ ρ∂ ∂ ∂ ∂+∇ ⋅ + −∇ ⋅ ∇ = + +

∂ ∂ ∂ ∂m m m m

th m p mnucl growth coag

n n n nn D nt t t t

u c (1)

In the above equation mn is the size distribution function, expressed per unit mass of gas,

ρ is the density of the gas and pD the particle diffusion coefficient, given by the Stokes-

Einstein equation. In cases where the particles are small, inertia can be neglected and the

particle velocity u can be taken equal to the gas velocity. The terms on the right-hand

side of Eq. (1) describe the variations due to homogeneous nucleation, condensational

growth and coagulation, respectively. Velocity thc corresponds to the transport velocity

of the particles due to external forces (thermophoresis is the only mechanism considered).

The thermophoretic velocity thc is calculated with the expression of Talbot et al. (1980),

using the local values (position dependent) of temperature and temperature gradient.

Equation (1) is coupled with the condensable vapor equation, which, neglecting

thermal diffusion, is given by:

( ) ( )m m mm v m

nucl growth

C C CC D Ct t t

ρ ρ ρρ ρ∂ ∂ ∂+∇ ⋅ −∇ ⋅ ∇ = − −

∂ ∂ ∂u (2)

where mC is the vapor mass fraction and vD is the vapor binary diffusion coefficient.

The first and second terms on the right-hand side of the equation above represent the

depletion of the condensable vapor due to homogeneous nucleation and condensational

growth, respectively.

10

In the present analysis two theories have been adopted for the prediction of

particle formation by homogeneous nucleation: the classical nucleation theory (Frenkel

1955) and the modified nucleation theory derived by Girshick at al. (1990). According to

the classical nucleation theory the nucleation rate is:

2 2 22 exp

3m crit

classicalp m m B

C dJm m k Tρ πσσ

ρ π

= −

(3)

4ln

mcrit

B

vdk T Sσ

= (4)

where critd is the embryos critical diameter,σ the surface tension, mv the molecular

volume, mm the molecular mass, pρ the density of the particles and Bk the Boltzman

constant. The symbols S and T denote the saturation ratio and the temperature,

respectively. According to the modified nucleation theory of Girshick et al. (1990) the

nucleation rate classicalJ is multiplied by a correction factor derived from a self-consistency

equilibrium cluster distribution, as following:

23 361 exp m

Girshick classicalB

vJ J

S k Tπ σ

=

(5)

Particles that are formed by homogeneous nucleation grow by condensational deposition

of the existing vapor on their surface. The rate of change of the particle diameter due to

condensational growth is given by a modified Mason equation (Mason 1971), which

accounts for both mass and heat transfer (see, for example, in Jokiniemi et al. 1994):

4/ /

p amass heat

p p mass FS heat FS

dd S Sdt d f F f Fρ

−= +

(6)

where

11

, 1( ) κ

= = −

v

mass heatv sat v g

R T L Lf fD p T R T T

(7)

In Eq. (6) massFSF and heat

FSF are the Fuchs-Sutugin correction factors for mass and heat

transfer, respectively:

2 2

11 , 1 1.71 1.333 1 1.71 1.333

gmass heatvFS FS

v v g g

KnKnF FKn Kn Kn Kn

++= =

+ + + + (8)

where the Knudsen number vKn ( gKn ) is defined as the ratio of the mean free path of the

vapor (gas) to the droplet radius. The term ( )exp 4 /σ ρ=a p v pS d R T in Eq. (6) accounts

for the Kelvin effect. In Eqs. (7), L is the latent heat of condensation of the vapor

species, vR is the gas constant, κ g is the thermal conductivity of the carrier gas, and

( )satp T is the equilibrium vapor pressure over a flat surface.

Calculation of coagulation is done on the basis of a modified Smoluchowski

equation, appropriate for a sectional representation of the particle size distribution, as

proposed by Jacobson et al. (1994). According to this formulation the variation of the

number concentration of thi − section can be approximated as following:

( ) ( ) ( )1

, , , ,1 1 1

, 1 ,BNi i

i ki j k j k j k i i i j i i j j

j k ji

dN vf K d d N N v N f K d d Ndt v

−

= = =

= − −∑∑ ∑ (9)

The first term on the right-hand-side of Eq. (9) accounts for appearance of particles in the

thi − size section due to collisions of smaller particles and the second term accounts for

depletion of particles in the thi − size section due to collisions with all other particles.

The coefficients , ,i j kf arise from the sectional representation of the size distribution, and

represent the fraction of the new particles formed from collisions of diameters jd and kd

that is partitioned into size section i. These coefficients are (Jacobson et al. 1994):

12

11

1

, , , , 1 1

;

1 ; 1 1 ; 0

k i j kk i j k B

k k i j

i j k i j k k i j k

i j k B

v v v v v v v v k Nv v v v

f f v v v v kv v v k N

++

+

− −

− − ≤ + < < − +

= − < + < >+ ≥ =

all other cases

, (10)

where v is the particle volume. The coagulation kernel K is calculated as:

B LSK K K= + (11)

considering the kernels associated with Brownian coagulation ( BK ) and laminar shear

coagulation ( LSK ). The Brownian kernel is determined from the standard Fuchs

interpolation formula (Fuchs 1964), which is valid from the continuum to the free

molecular regime:

( )( )

( ) ( )( )

, ,

1

, ,1/ 2 1/ 22 2 2 2

( , ) 2

8( )

2

B j k j k p j p k

j k p j p k

j k j k j k j k

K d d d d D D

d d D D

d d g g d d c c

π−

= + +

+ + + + + + + +

(12)

The laminar shear kernel is related with the velocity gradient in the direction normal to

the flow as following (see, for example, in Drossinos and Housiadas 2006):

31( , ) ( )6LS j k j k

uK d d d dy∂

= +∂

(13)

In Eq. (12) 1/ 2(8 / )i B ic k T mπ= is the mean particle velocity and ig the so-called Fuch's

length

( ) ( )3/ 23 2 213i i i i i i

i i

g d l d l dd l

= + − + − , (14)

where ,8 /( )i p i il D cπ= the mean free path of the aerosol particle.

13

3. Numerics

3.1. Internal aerosol processes

To combat numerical diffusion, while using a computationally economical particle

size grid resolution, we implemented a simple, hybrid method for solving condensational

growth. The method is based on a combination of a moving particle size grid and a fixed

particle size grid.

More specifically, the condensational growth equation is integrated, in each time

step, allowing the particles to grow to their actual sizes using the moving grid approach

(Gelbard 1990; Kim and Seinfeld 1990). The integration is performed using a fourth-

order Rosenbrock method with monitoring of the local truncation error to adjust the time

step (Press et al. 1994). Then the cumulative distribution function is calculated on the

moving grid using the following simple form, valid for a sectional formulation:

,1 1

i i

i i m i iC N n dρ= = ∆∑ ∑ (15)

where id∆ is the width of the thi − size section of the moving grid. The cumulative

distribution is then reallocated to the fixed size grid using a cubic spline interpolation. Let

*iC be the cumulative size distribution as inferred from the interpolation step. Then, the

new number concentration *iN can be simply calculated as:

* * *1i i iN C C −= − (16)

The use of a third order polynomial for the interpolation leads to significant reduction of

numerical diffusion. Number conservation and enhanced stability are achieved by taking

advantages of the mathematical properties of the cumulative size distribution. By

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definition, the cumulative size distribution must be monotonic and no negative.

Therefore, the following corrections are made:

* *1

* * * *max 1 1

* * * *max 1 1

0, ,

for = 1, 2, = , if and

for 2, 1, = , if

K total

i i i i

i i i i

C C N

i n C C C C

i n C C C C+ +

− −

= =

− ≥

= − ≤

(17)

where totalN is the total number concentration of the particles and maxn the number of the

used particle size sections. The above procedure ensures conservation of the total particle

number in the condensational growth calculations, conferring to the fixed grid approach

the appealing characteristics of the moving grid approach. Subsequently, we use the

abbreviation CICR (Cubic Interpolation Cumulative Re-mapping) to refer to the method

described above for the solution of growth.

The implementation of the two other internal processes, i.e. nucleation and

coagulation, is straightforward. Nucleation is resolved simply by adding particles in the

size section that contains the critical diameter. Coagulation is directly calculated on the

basis of Eqs. (9) and (10), using the semi-implicit scheme of Jacobson et al. (1994),

which does not require iterations and is unconditionally stable.

3.2. Coupling with CFD

The computation of fluid and aerosol dynamics is carried out in two stages. First,

CFD calculations are made to determine the gas velocity and temperature fields in the

reactor. These fields are used as input to the aerosol dynamics code. The CFD step is

performed using a general-purpose commercial CFD package (code ANSWER; ACRi

2001). The Nodal Point Integration (NPI) technique is used for the integration of the

equations (Runchal 1987a). The continuity, momentum and energy equations are solved

15

in a two-dimensional cylindrical uniform grid. Since the effect of heat release from

condensation on the gas temperature is neglected (one-way coupled system), only the

convective, diffusive terms are taken into account in the energy equation. The flow is

laminar. Standard numerical methods were selected in our CFD calculations. More

specifically, the hybrid scheme (Runchal 1972) was used for the discretization of the

convective terms, while the ADI method (Fletcher 1991) was used for the solution of the

algebraic system of equations. The sensitivity of the results to the scheme used in the

solution of the convection term is examined by using also the QUICK scheme (Leonard

1979). Practically, no difference was found in the overall calculations. The problem of

fluid flow and heat transfer in laminar flow aerosol reactors has been addressed

extensively in the literature. Parameters, concerning the fluid flow and heat transfer in the

reactor, that could affect the overall aerosol dynamics calculations have been examined

by employing both CFD-based and analytical methods (Pyykönen and Jokiniemi 2000;

Wilck and Stratmann 1997; Housiadas et al. 2002; Housiadas et al. 2000). Therefore, no

particular focus was given to the fluid flow and heat transfer problem in the reactor.

The aerosol dynamic calculations are then performed according to the following

methodology. The fully Eulerian, multidimensional problem described by Eqs. (1) and

(2) is numerically advanced in time using an operator splitting technique. The source

terms for each of the aerosol dynamics processes are explicitly calculated in each time

step by using the methods described previously and then combined (added) to give the

right-hand sides (overall source terms) for Eqs. (1) and (2). These source terms consist

input for the integration of Eqs. (1) and (2), which is performed by using an implicit finite

volume scheme (Patankar 1980). The convection terms are treated using the hybrid or the

16

power law scheme (Patankar 1980), while the ADI method is adopted for the solution of

the algebraic equations.

To accelerate the calculation an adaptive time step process is implemented as

following. At every computational grid point j the local relative change of the vapor

mass fraction, due to condensational growth, is forecasted using the time step of the

previous iteration 0∆t . If this change is higher than a pre-specified value (usually, a

relative change of 0.5% was specified in the runs of the present work) a local time step

1∆ jt is calculated on the basis of a pre-specified relative change for the number

concentration due to nucleation (usually 30%). Otherwise, the local time step is taken as

1 02∆ = ∆jt t . The global time step 1∆t with which the integration with the finite volume

method is performed is selected as the minimum of the local time steps 1∆ jt . The

doubling of the local time step in the locations where vapor depletion is small, and

consequently there is no important effect on the calculation of the saturation ratio, was

found to accelerate significantly the convergence of the solution. Steady state problems

are solved using a pseudotransient approach. Starting from arbitrary initial conditions, the

solution procedure marches along the time, exactly like in unsteady problems, until the

converged, steady-state solution is reached.

The spatial CFD and aerosol dynamics grids are independent. A spatially non-

uniform grid, finer at the nucleation zone is usually needed in aerosol dynamics

calculations. The temperature and the gas flow field at the aerosol dynamics grid nodes

are calculated by multidimensional linear interpolation (Press et al. 1994) on the output

data provided by the CFD calculations.

17

4. Validation tests

The algorithm and methods employed to calculate aerosol dynamics are assessed

by comparing with available analytical solutions of the general dynamic equation for a

number of idealised cases. The first comparison with theory is based on the analysis of

combined condensation and coagulation of Ramabhadran et al. (1976). In this analysis

the coagulation kernel K is assumed constant, while the growth rate is taken as a linear

function of the particle volume:

dv vdt

σ= (18)

The initial aerosol distribution function has the form of a first order gamma function:

00 2

0 0

( ) expN v vn vv v

= −

(19)

where 0N is the initial number concentration and 0v the mean volume of the initial

distribution.

Fig. 1 shows the evolution of the normalized number and volume concentrations

of the aerosol as a function of the dimensionless time. As can be seen, the particle

number decreases as the particles coagulate, whereas the aerosol volume increases due to

condensational growth. The change in the aerosol properties was numerically reproduced

using a size grid of 6 particle size sections/decade and a dimensional time step equal to

0.1. The numerical results are in excellent agreement with the analytical solution both in

terms of volume and number. Although the CICR method does not take into account the

conservation of volume, the results in fig. 1 indicate that, in practice, the errors

introduced in the calculation of the particle volume during the re-mapping step are very

small. Fig. 2 shows the calculated dimensionless size distribution function of the

18

particles. From the comparison with the analytical solution it is concluded that numerical

diffusion is satisfactorily combated even using a low resolution of 6 particle size

sections/decade.

Figure 1. Comparison between calculated and analytical normalized number and volume

concentrations, as a function of the dimensionless time tτ σ= , for a system with

0.3Λ = . (characteristic dimensionless quantity 0/ KNσΛ = ).

19

Figure 2. Calculated size distribution in comparison with the analytical solution at 5τ =

for the first analytical test. The initial size distribution of the aerosol is also shown

( 0τ = ).

The second validation test is concerned with the analytical solution of an evolving

size distribution under the influence of pure condensational growth (Seinfeld and Pandis

1998):

p

p

dd Adt d

= (20)

The aerosol has initially a lognormal size distribution:

2

00 2

ln ( / ) 1exp2ln2 ln

p pI

I pI

d dNndσπ σ

= −

(21)

20

where A is a constant and 0N is the initial total number concentration. A case with initial

median diameter 0.1 mpID µ= , initial geometric standard deviation 1.15Iσ = ,

10 2 11.0 10 cmA s− −= × and total simulation time 1 sfT = was chosen because it involves

steep evolution of the size distribution. Note that the same test conditions have been also

employed in previous assessment exercises (Test III in Yamamoto 2004). In the

calculations a particle diameter range from 0.06 mµ to 0.3 mµ was used to cover all the

sizes encountered during the simulation. Fig. 3 shows the volume distribution of particles,

as calculated using the CICR method (fig. 3a) and the moving center method (fig. 3b), as

a function of the number of size sections in which the size range is divided. As the

number of size sections increases the accuracy of the CICR method improves steadily,

converging to the analytical solution. Due to the spline interpolation used in the re-

mapping step and the imposed corrections (Eq. (17)) some oscillations can be artificially

introduced in the distribution. The results in Fig. 3a indicate that, in practice, such effects

are insignificant. In all investigated cases artificial peaks and deeps have been obtained

only at points where the distribution is vanishing (for example in the horizontal tail of the

distribution of fig. 3a). On the contrary, the moving center method predicts an oscillating,

unrealistic distribution. Moreover, it does not converge uniformly to the analytical

solution, but instead overshoots. This behavior is due to the frequent occurrence of empty

sections in the course of the calculation.

21

a). b).

Figure 3. Volume distribution at 1fT = s for various particle size grid resolutions, as

calculated using a) the CICR method, b) the moving center method, and comparison with

the analytical solution.

5. Results

To illustrate our model we reproduce a real nucleation-condensation case in an

aerosol reactor. We used the data reported (meticulously) by Ngyuen et al. (1987), as

obtained from homogeneous nucleation experiments. In these experiments dry air

saturated with DBP is conducted through a hot tube, whose temperature is kept equal to

that of the saturator, into a cooler tube, the “condenser”. A transition zone of 10 cm exists

between the hot tube and the condenser. The condenser (length of 52 cm, inner diameter

of 1 cm) is rapidly cooled, causing the gas to become highly supersaturated. Particles are

formed by homogeneous nucleation and grow mainly by condensation. Nguyen et al.

(1987) reported extensive measurements of both the number concentration and size

distribution of the formed particles at the outlet of the condenser, for different conditions

with respect to the flow rate and saturator temperature.

22

The model was applied to reproduce the fluid and particle dynamics in the

condenser. The analysis is made on a two-dimensional grid in cylindrical coordinates,

assuming symmetry with respect to the condenser axis. The flow is laminar and becomes

fully developed before the entrance of the transition zone. Therefore, at the tube inlet the

gas velocity is taken parabolic, and the temperature is taken uniform and equal to that of

the saturator. Previous works have showed the importance of choosing correctly the

boundary conditions in the analysis of nucleation experiments in laminar flow aerosol

reactors (Housiadas et al. 2000; Wilck and Stratmann 1997). The temperature values on

the wall nodes are set on the basis of a linear interpolation on the measured wall

temperature data, as provided by Nguyen et al. (1987). The wall boundary values for the

vapor mass fraction are those corresponding to equilibrium at the wall temperature. Also,

a zero particle number concentration is imposed at the wall (for all sizes), assuming the

tube wall to be a totally absorbing boundary. The physical properties of DBP are those

given by Nguyen et al. (1987) and the properties of the carrier gas (air) are taken all as

temperature dependent.

One-dimensional (1-D) calculations were also performed, based on a previously

developed one-dimensional, semi-Lagrangian model (Mitrakos et al. 2004). This model

was further elaborated by implementing the operator splitting technique and the

previously described CICR method for the solution of growth. This model is used to

further test the capabilities of the 1-D description of aerosol flow reactors and also to

intercompare easily the CICR method and the moving center method in a real nucleation-

condensation case.

23

Fig. 4 shows the bulk average number concentration of the particles at the exit of

the reactor, as calculated with our model. For comparison, the experimental data are also

shown, as well as simulation results obtained with other models. Our CFD calculations

were done with an equally spaced spatial grid consisted of 700 axial and 40 radial nodes.

In the two-dimensional (2-D) calculations aerosol dynamics are solved over a grid of 600

axial and 20 radial nodes, non-uniform in the axial direction (grid sizes ranging from

about 0.3 mm to 3 mm). As discussed before an adaptive time step was used, which,

typically, ranged between ~2 ms and 20 ms. In the 1-D calculations an adaptive axial grid

was used, similar to that used by Im et al. (1985) and Jokiniemi et al. (1994). The axial

step is derived from the maximum allowable relative changes for the particle number

concentration and the vapor mass fraction. The modified nucleation theory of Girshick et

al. (1990) was used in the present calculations for the estimation of the nucleation rate.

As commonly made in homogeneous nucleation analyses, the calculated nucleation rate

needs to be multiplied by a correction factor to get agreement with the experimental data.

In the present calculations the nucleation rate was multiplied by a factor C=5 ×10-4 for the

2-D runs and C=1.2×10-3 for the 1-D runs, which are both consistent with the factors

used by Pyykönen and Jokiniemi (2000) (C=3.2×10-4) and Wilck and Stratmann (1997)

(C=1×10-3). As the competition of nucleation and growth becomes stronger, or

equivalently as the number concentration and the subsequent vapor depletion become

more important the time step needs to decrease in order to accurately calculate the

nucleation rate. Hence, the running time of a simulation depends on the intensity of

coupling between these two processes. The running time is roughly proportional to the

resolution used in the discretization of the particle size grid. For a resolution of 10 size

24

sections/decade the CPU time using a personal computer (Pentium IV 3 GHz, 1 GB)

ranges from 25 minutes for the lower saturator temperature (left part of fig. 4) to 1 h and

40 minutes for the most demanding case, namely for the higher saturator temperature

(right part of fig. 4). Coagulation has an effect less than 0.5% in the final number

concentration of particles, but increases computational time roughly by a factor of three.

Therefore, in the runs coagulation was usually turned-off. The CPU time for the 1-D

calculation was trivial (less than a minute in all the cases).

Figure 4. Bulk average number concentration at the outlet of the reactor as a function of

the saturator temperature, as predicted from our models, in comparison with the measured

data and previous modeling results (System B of Nguyen et al. (1987) with gas flow rate

0.5 lt/min and condenser temperature 21.4 oC).

25

As the results of fig. 4 indicate, our model predicts with very good accuracy most

of the measured points and agrees closely with previous models. The previous numerical

results have been obtained with different computational approaches. The calculations of

Wilck and Stratmann (1997) are done with a fully 2-D Eulerian model, based on the

modal method. The calculations of Pyykönen and Jokiniemi (2000) are made with a

sectional, quasi-2-D model, based on a Lagrangian approach (called the “stream-tube”

approach). Instead, previous attempts to reproduce these data with a sectional, 2-D

Eulerian approach based on the full discretization of the governing equations failed (took

CPU days to converge, and gave unacceptably overestimated predictions; see, discussions

in Pyykönen et al. 2002; Pyykönen and Jokiniemi 2000; Stratmann and Witby 1989). Our

model reproduced successfully these experimental data, demonstrating the

appropriateness of the proposed methodology as to the way of using the sectional method

in a multi-dimensional, Eulerian computational framework.

As the results of Fig. 4 indicate, there are discrepancies between calculated and

experimental data for saturator temperatures higher than about 100˚C, for all the models.

Seemingly, all models underestimate the nucleation rate and overestimate the growth rate

over this range of saturator temperatures. The reason of this behavior is not yet clear.

Wilck and Stratmann (1997) tried to explain these discrepancies by artificially

suppressing the growth rate, but they did not reach to a definite conclusion. Finally, it is

interesting to notice that in this case the 1-D model gave satisfactory results for the

number concentration, in comparison with the 2-D approach, using practically the same

correction factor. This interesting behavior was reported in a previous communication

26

(Mitrakos et al. 2004). Similar trends in 1-D aerosol reactor simulations have been also

observed by other investigators (Jeong and Choi 2003; Park et al. 1999). The validity

limits of an 1-D approximation cannot be easily established. Moreover, as it will become

apparent below, the 1-D solution fails in predicting the size distribution, and, hence, the

use of the 1-D approximation is not generally recommended.

Figure 5. Bulk average number concentration at the outlet of the reactor as a function of

the saturator temperature, as calculated with various particle size grid resolutions (same

experimental case as in figure 4).

The characteristics of our model in terms of stability and accuracy are assessed by

examining the sensitivity of the results to the number of sections that are used to divide

27

the particle size spectrum. The influence of the particle size grid resolution on the

calculated particle number is illustrated in fig. 5, which shows the exit number

concentration of particles as calculated for various numbers of size sections. As can be

observed, the impact of the particle size grid resolution on the calculated number

concentration increases towards the right (higher particle number), i.e. as the competition

between nucleation and growth becomes stronger. The bias introduced by the CICR

method in the calculation of the particle total volume during the re-mapping step and the

unavoidable errors arising from the discretization of the particle size on the fixed grid are

the reasons for this trend. By construction, the CICR method is number conservative, but

not volume conservative. The small numerical errors on particle volume reflect on the

vapor depletion calculations, and have therefore a positive feedback on the error in the

nucleation rate calculation because of the extreme sensitivity of the nucleation rate on the

saturation ratio. The moving center method is less sensitive to this feedback effect

because is volume conservative. However, the moving center method was found to

predict unrealistic particle size distributions, as previously discussed in the validation

tests and as it will also become apparent latter. In fact, this was the motivating reason to

seek for a new numerical method (CICR). The remedy to the previously described error

propagation loop is the increase of the number of size sections. In this respect, the

proposed method performed satisfactorily. As the results of Fig. 5 indicate, the accuracy

of our calculations using 10 size sections/decade can be considered as adequate. Note that

this particle size grid resolution is similar to that used in previous analogous analyses

(Pyykönen and Jokiniemi 2000). Very good accuracy is achieved using 20

sections/decade. With this resolution, the computational time for the most demanding

28

run, namely, for the higher saturator temperature, reached 3h and 50 min on our

computer, therefore remaining tolerable in all cases.

Figure 6 shows the calculated size distribution at the exit of the reactor, for a test

for which Nguyen et al. (1987) reported detailed size distribution measurement data. The

calculations results of Nguyen et al. (1987) are also shown, which were obtained with a

particular semi-analytical model, tailored to this application, on the basis of the theory of

Green functions (Pesthy et al. 1983). Results for the calculated size distribution are not

provided by Pyykönen and Jokiniemi (2000) and Wilck and Stratmann (1997), and so it is

not possible to perform a direct comparison with other CFD-based computational

schemes. Following Nguyen et al. (1987) in this case we use the classical nucleation

theory. A nucleation rate correction factor C=2.1×104 was used in our calculations, which

is very close to that used by Nguyen et al. (1987) (C=4×104). The calculated size

distribution presents a similar dependence on the particle size grid resolution with that

previously discussed. Adequate accuracy is achieved and numerical diffusion is

efficiently combated even with 10 size sections/decade. The shape of the size distribution

remains smooth and a fast convergence is achieved as the number of size sections

increases. 20 sections/decade are adequate for an accurate representation of the size

distribution. The agreement with the experimental data is satisfactory. Our CFD-based

predictions are close to the theoretical, semi-analytical results of Nguyen et al. (1987).

Note, however, that the numerical model performs better than the semi-analytical over

the small size range (<0.8 µm).

29

Figure 6. Bulk average particle size distribution at the exit of the reactor, as calculated

with our models for various particle size grid resolutions and comparison with the

measured and calculated results of Nguyen et al. (1987) (System B with gas flowrate 1

lt/min, saturator temperature 80.5 oC and condenser temperature 8.7 oC). The comparison

between the CICR method and the moving center method is also shown.

In figure 6 the performance of the moving center method is also examined and

compared with the CICR method, using the 1-D model. A particle size grid of 20

sections/decade was used for the 1-D analysis, for both the CICR and the moving center

method. The reason we used the 1-D model to perform the comparison between the two

methods is that the implementation of the moving center method in a multidimensional

calculations is cumbersome (the characteristic size in each section differs among adjacent

computational cells, making the Eulerian calculation difficult to perform). The moving

center method also minimizes numerical diffusion, but in contrast with the CICR method,

introduces empty sections with a simultaneous overestimation of the peak. The net result

30

is that an unrealistic size distribution is inferred. This finding is in line what was

previously found in the analysis of idealized size distribution evolution cases (validation

tests).

Note that contrary to what was observed before, the correction factor required in

the 1-D run to match the measured number concentration (C=5.2×102) differs by two

orders of magnitude with that used in the 2-D calculations (C=2.1×104). In comparison

with the experimental data and the 2-D predictions the 1-D model overestimates severely

the particle sizes at the reactor outlet (the 1-D size distribution is displaced to the right).

So, in the 1-D analysis, even when the calculated number concentration is fitted to the

experimental one, using a correction factor, the size distributions is predicted incorrectly.

Therefore, the 1-D approximation should be used with caution in aerosol reactor

analyses. The proper choice is the use of a realistic multidimensional flow description.

6. Summary and conclusions

In this work we presented a methodology for the numerical simulation of aerosol

dynamics in a CFD-based multidimensional framework, using a sectional representation

of the size distribution. The emphasis was on computationally demanding cases,

characterized by strong coupling between homogeneous nucleation and condensational

growth. The model is fully Eulerian. The aerosol dynamics calculations are one-way

coupled with the CFD calculations, namely the output of the CFD analysis consists input

for the aerosol dynamics calculations. In order to deal with the multiple time scales that

are involved, due to the presence of simultaneous aerosol dynamics processes and

transport processes, an operator splitting technique is employed along with an adaptive

31

time step procedure that accelerates significantly convergence. For the solution of

condensational growth a novel method was developed, named CICR (cubic interpolation

cumulative re-mapping). The CICR scheme is based on a combination of moving and

fixed particle size grids and a re-mapping process for the cumulative size distribution,

carried out with cubic spline interpolation. The use of moving size grid in association

with high order interpolation minimizes the numerical diffusion inherent in the solution

of condensational growth. Moreover, the use of the cumulative particle size distribution

enables the scheme to be number conservative.

Validation tests are performed using available analytical solutions of the general

dynamic equation for idealized cases, characterized, however, by sharp changes in the

size distribution. The comparisons showed that numerical diffusion is efficiently

combated even with a relatively coarse particle size grid resolution (less than 10 size

sections/decade), while sustaining stability.

The appropriateness of the CFD-based methodology is evaluated by reproducing

the numerically stringent case of nucleation-condensation in an experimental aerosol

reactor. Previous works, which dealt with the same problem by using standard CFD

techniques, failed to incorporate a sectional size distribution in a fully Eulerian,

multidimensional CFD calculation. The proposed methodology enabled this association.

Satisfactory results are obtained in terms of both number concentration and size

distribution at the outlet of the reactor. The method was found to be relatively fast and

robust, and provided more realistic results than other sectional-based methods, in

particular the moving center method. In the current application, the use of 10 size

sections/decade removed practically numerical diffusion. Good accuracy in the delicate

32

coupling between the competing processes of nucleation and condensation and vapor

depletion is achieved with 20 size sections/decade. The computational time was

reasonable in all cases considered. On a common personal computer (3 GHz Pentium IV,

1 GB) the running times were of the order of CPU-hour (typically, 0.7 ms per node per

time step). We conclude that the sectional method can be efficiently used in

multidimensional CFD-based aerosol simulations, by employing the proposed

methodology.

33

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