Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
A DiffusionInertia Model for Predicting Aerosol Dispersion and
Deposition in Turbulent Flows
R. V. Mukin, N. I. Drobyshevsky, A. S. Filippov, V. F. Strizhov and L. I. Zaichik
Nuclear Safety Institute of the Russian Academy of Sciences,
Moscow, 115191, Russia
mukin@ibrae.ac.ru
Keywords: Aerosol particles, turbulent flows, deposition, duct, bend
Abstract
The objective of the paper is twofold: (i) to present a model (the socalled diffusioninertia model) for predicting dispersion
and deposition of aerosol particles in twophase turbulent flows and (ii) to examine the performance of this model as applied to
the flows in straight ducts and circular bends. The model predictions compare reasonable well with both experimental data and
Lagrangian tracking simulations coupled with fluid DNS or LES.
Introduction
The existing strategies of modeling turbulent twophase
flows can be subdivided into two groups depending on the
Lagrangian tracking and Eulerian continuum approaches
for handling the particulate phase. In the framework of the
Lagrangian method, the particles are assumed to encounter
randomly a series of turbulent eddies, and the macroscopic
particle properties are determined solving stochastic
equations along separate trajectories. As a consequence,
such a method requires tracking a very large number of
particle trajectories to achieve statistically invariant
solution. As the size of particles decreases, the
representative number of realizations should increase
because of the increasing contribution of particle
interactions with turbulent eddies of smaller and smaller
scale. Thus, this technique, especially when coupling with
DNS or LES for the computation of fluid turbulence,
provides a very useful research tool of investigating
particleladen flows, but it can be too expensive for
engineering calculations. The Eulerian method deals with
the particulate phase in much the same manner as with the
carrier fluid phase. Therefore, the twofluid modeling
technique is computationally very efficient, as it allows us
to use the governing equations of the same type for both
phases. In addition, the description of fine particles does
not cause great difficulties because the problem of the
transport of particles with vanishing response times
reduces to the turbulent diffusion of a passive impurity.
Overall, the Lagrangian tracking and Eulerian continuum
modeling methods complement each other. Each method
has its advantages and, consequently, its own field of
application. The Lagrangian method is more applicable for
nonequilibrium flows (e.g., highinertia particles, dilute
dispersed media), while the Eulerian method is preferable
for flows which are close to equilibrium (e.g., lowinertia
particles, dense dispersed media). Since the particulate
phase combines simultaneously the properties of
continuum medium and discrete particles, the situation
with these two approaches resembles the wellknown
"waveparticle" duality in the microword.
To simulate the dispersion of lowinertia particles in
turbulent flows, the Eulerian models of diffusion type
appear to be very efficient. In Zaichik et al. (1997),
Zaichik et al. (21' 4) a simplified Eulerian model called
the diffusioninertia model (DIM) was developed. This
model was based on a kinetic equation for the probability
density function (PDF) of particle velocity distribution
Zaichik (1997), Zaichik et al. (1999), Zaichik et al. I(2" '4 1 I
and was coupled with fluid RANS in the frame of oneway
coupling. The DIM was applied to simulate various
turbulent flows laden with lowinertia particles and this
was incorporated in the CFD code SATURNE for
modelling aerosol transport in ventilated rooms Nerisson et
al. (2'i '4). The Eulerian models of diffusion type were also
proposed in coupling with DNS and LES approaches for
calculating the turbulent carrier fluid Druzhinin (1995),
Ferry & Balachandar (2001), Rani & Balachandar (2003),
Shotorban & Balachandar (2006), Zaichik et al. (2009).
The advantage of the Eulerian diffusiontype models is that
the particle velocity can be explicitly expressed in terms of
the properties of the carrier fluid flow. By this means, one
avoids the need to solve the momentum balance equations
for the particulate phase, and the problem of modelling the
dispersion of the particulate phase amounts to solving a
sole equation for the particle concentration. By this means,
computational times are seriously shortened as compared
to full twofluid Eulerian models. The disadvantage is that
these are applicable only to the twophase flows laden with
lowinertia particles. For example, the DIM is valid when
the particle response time is less than the integral timescale
of fluid turbulence. Nevertheless, these models are capable
of predicting the main trends of particle distribution,
including the effect of preferential accumulation due to
turbophoresis, in a fairly wide range of particle inertia.
In this paper, we extend the DIM to include the backeffect
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
of particles on the fluid turbulence in the frame of twoway
coupling. Moreover, the socalled inertia and
crossingtrajectory effects are incorporated into the model
and the boundary condition for the particle concentration
equation is refined. This extended model is applied to the
threedimensional simulation of aerosol deposition in
straight ducts and circular bends when the transport of
particles is caused by the simultaneous action of diffusion,
turbophoresis, gravity, and centrifugal force.
Nomenclature
a0 dimensionless acceleration magnitude
D duct diameter
D, Brownian diffusivity
D, turbulent diffusivity of noninertial admixture
Dz,, particle turbulent diffusion tensor
De Dean number
dp particle diameter
F, acceleration of body forces acting on particles
(e.g., gravity)
J deposition flow rate
j+ deposition coefficient
Kn Knudsen number
k, Boltzmann constant
L turbulence spatial macroscale
M mass particle loading of the fluid
Rb radius of curvature of the bend
Ro curvature ratio
ScB Schmidt number of Brownian diffusion
ScT turbulent Schmidt number
St Stokes number
T, Eulerian integral timescale
T, Lagrangian integral timescale
TL eddyparticle interaction timescales
U average fluid velocity
U mean axial fluid velocity
U, fluid velocity seen by particles
u, wall friction velocity
(u'u') fluid kinetic stresses
V average particle velocity
V, relative velocity
v particle velocity
(v,'v) particle kinetic stresses
Y wallnormal coordinate
x spatial coordinate
Greek letters
a turbulence constant
7/ deposition efficiency
vf fluid kinematic viscosity
vT turbulent kinematic viscosity
penetration efficiency
r, Kolmogorov timescale
p particle response time
Tpo
ZT
TLu
Stokes particle response time
Taylor time microscale
nondimensional particle response time
particle volume fraction
rebound coefficient
autocorrelation function
Mathematical formulation
The governing equation for the concentration of
lowinertia particles is given by (see Zaichik et al. (2010))
F+ D+ 9 DU F=
at yx, xL Dt
C aoD, 0 a (,(u; )~f,) (1)
 +D J +Tp 
9x, 9x 9x x
By this means, for lowinertia particles, namely, when the
particle response time is shorter than the turbulence time
macroscale, the conservation equation set is reduced to the
diffusiontype equation for the particle concentration, and
hence one does not require solution to conservation
equations for the momentum of the particulate phase. This
approach is called the diffusioninertia model (DIM). In
the limit of zeroinertia particles ( r > 0 ), Eq. (1)
becomes the conventional diffusion equation
C9 9U, ( 9 0)
+ a Da+DT (2)
at Ox, 9x,[ 9x, 9x
with D, J = (u,'u )T being the diffusion tensor of
noninertial admixture.
In comparison with (2), Eq. (1) allows us to take into
account a number of effects caused by the particle inertia:
(i) the impact of gravity and other body forces, (ii) the
socalled inertial bias effect, i.e., the transport by reason of
the deviation of particle trajectories from the fluid
streamlines, (iii) the turbulent migration (turbophoresis)
due to the gradients of velocity fluctuations, and (iv) the
inertia and crossingtrajectory effects on particle turbulent
diffusivity.
The response time of aerosol particles is given by
r =0 1+KnA 4 +A, exp( 1 (1+0.15Reop87)1
P2
p pp (2)
18pff
where according to Talbot et al. (1980) A1 =1.20,
A2 = 0.41, and A = 0.88. The particle Reynolds number
appearing in (2) is evaluated as
d, [ V, 2 +2(12/f.)k] f +2 f
Rep = f'r 
Vf 3
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Paper No
The Brownian diffusivity is equal to
D, kT 1+Kn A +A2 exp
3 rpv, d Kndp
In a quasiisotropic approximation that corresponds to
averaging over different directions, Eq. (1) and the relative
velocity V, can be presented as
&0 aU F( DU7 =
 +++ Ir I
at ax, Px, Dt ) \
DB D Oc x, IxIq, )
[T )u
dtdxdx al
v, = vu,,
rF, DU,
Dt
(D1 8 +q. )q>]
Dm+q D m
O 6x
DT DTT,
TL
+ 2qu,
p 3 ,
3u
VT
D, = '
Sc,
pTfu
T11
S T +2T"
3
' Lp 3
Note that, in (3) as compared to (1), the space dependence
of the Brownian diffusivity is ignored.
Solution to Eq. (1) or (3) right up to the wall (y = 0) is
made difficult by the fact that the concentration of particles
can steeply rise due to turbophoresis when y > 0 .
Equations (1) and (3), as such, cease to be true in the
viscous sublayer of the nearwall region of turbulent flow,
where to determine the particulate kinetic stresses. To
solve the particle concentration equation up to the wall, we
use the method of wallfunctions that has extensively
employed starting from Launder & Spalding (1974) in
modeling singlephase turbulent flows. In accordance with
the wallfunction method, we invoke, as the boundary
condition, a relation between the flow rate of deposing
particles J4 and the particle concentration in the
nearwall region outside the viscous sublayer ,D
J. = I (yVIT +v) (5)
1+
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
velocity, body force acceleration, and fluid acceleration in
the nearwall region. The rebound coefficient, % ,
measures a probability of the rebound of a particle from
the wall and its return into the flow after collision. The
surface is perfectly adsorbing if x =0, and the particle
deposition is absent if Z = 1. The parameter b quantifies
the ratio of the 'convectionforce' and
'diffusionturbulence' components of the deposition rate.
Deposition is controled by the 'convectionforce'
mechanism when b > oo ( y > 0 ), and the deposition
)1 / rate tends to zero when b > oo ( > b) because the
action of this inhibits the motion of particles to the wall.
The deduction of the coefficient y is given in Zaichik et
al. (2010).
The 'diffusion' component of the deposition rate is found
(4) as a result of solving the diffusion equation in the viscous
sublayer for the fourthdegreelaw of rise in the turbulent
diffusivity at high Schmidt numbers Levich (1962),
Kutateladze (1973)
0.115u,
S VDF c (6)
OCB
where Sc, DB/v. is the Schmidt number of Brownian
diffusion.
The 'turbophoresis' component of the deposition rate is
obtained as a result of approximating a numerical solution
in the nearwall region Zaichik et al. (2010).
2104 25u,
V 1R (+l 325) max[0.8, min(1.320.27 nr, 1)]
(7)
The boundary condition (5) along with (6) and (7) is valid
for the particles with r+ =rpou2/ < 100 when the first
grid node is chosen outside the viscous sublayer
(y+ yu,/v > 20) where (O changes weakly with
variation in the normal distance from the wall.
In what follows let us consider the governing equations for
the carrier fluid. When the volume fraction of the
particulate phase is small ( << 1 ), its effect on the
continuity equation of incompressible fluid is negligible
and this is written as
'F U DU)
exp( b 2/) VCF
Y= b C
1+erf(b/l/2)' D T
Here VDT designates the 'diffusionturbulence'
component of the particle deposition rate caused by
diffusion VDF and turbophoresis VT. The quantity VCF
designates the 'convectionforce' component of the
deposition rate induced by an action of convection and
body forces in the nearwall region, where U,, F and
DU/Dt\, are the normalwall components of fluid
S=0.
The balance fluid momentum equation is given by
DU, 1 CP 8 C U,
Dt p 6x, x + x ,
Ui u +4,
A, = (v, u,)p)dv = ),
pf p p
where M =p /pf is the mass particle loading of the
fluid, and A quantifies the backeffect of particles on the
fluid momentum that is determined using (4).
Turbulent flow characteristics are simulated on the basis of
VDT = VDF +VTR
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a twoequation turbulence model incorporating the
equations of kinetic turbulence energy and its dissipation,
that is, the k turbulence model. In the frame of this
model, the fluid kinetic stresses are given by (see Zaichik
et al. (2010))
, 2k65, U SU,
3 ax x,
2 SU,
3 x ,
(1i+Af.1k ) k 2(1C2)
+ = C, ,
C +(lH, C)/C1 3C,
m +2 (If":)k, m =(+Mf/')H.
p
It is clear from (10) that, as distinct from the turbulent
viscosity coefficient of the standard k model
vr0 = Ck2 /, vj incorporates two additional effects:
(i) the presence of particles in the flow and (ii) the
nonequilibrium of turbulence that lies in a possible
inequality between the production and the dissipation. If
the particles are absent (M =0, H, =H, = e) and
the equilibrium between the processes of production and
dissipation takes place (H = ), v, reduces to vr0. In
the equilibrium approach (Hn = c,) which is valid, for
example, for modelling the turbulent nearwall flow, (10)
predicts
C, (1+ Alf.1)k2
+p
The turbulence energy balance equation is conclusively
given as (see Zaichik et al. (2010))
(l1+f.1)Dk ,)i + (l+Mf.) ] + (11)
Dt ox) yk 0 7 (11)
+(1+Af, )n (c + e + G )
By analogy with (11), the turbulence dissipation balance
equation is represented in the form
( +AfI ) De D 01* +(1+Af, T\r \e
( +(+f1 )1
Dt ax) f v[ a, x
+ [C,, (l+Afl,)n C,+2 ( +Gp + ). (12)
By this means, the standard k model is modified in
two aspects. Firstly, the modulation of turbulence due to
particles is taken into consideration. Secondly, instead of
standard expression for the eddy viscosity coefficient, v,
is assumed to be a function of the turbulence
productiontodissipation ratio nI,/c The values of
constants in (10)(12) are usually taken to be as follows:
C = 0.09, uk =1.0, a, =1.3, C,1 = 1.44, C2 = 1.92.
Moreover, C = 1.1, Sc = 0.9, and a = C12 = 0.3.
Calculation results and discussion
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
The DIM, consisting of the particle concentration equation
(3) and the boundary condition (5), is coupled with the
fluid balance equations (8), (9), (11), and (12). The model
advanced is evaluated against experiments and numerical
simulations of aerosol deposition in straight ducts and
circular bends. The surface is assumed to be perfectly
adsorbing, that is, the rebound coefficient ; is taken as
zero in (5).
Aerosol deposition in straight ducts
First we examine the performance of the model for the
deposition of particles in a vertical duct flow, when the
gravity force does not exert direct action on the deposition
rate. It is a common convention to describe the deposition
rate of particles from turbulent flow by the dependence of
the deposition coefficient j = Jm/ u, where D, is
the bulk volume particle fraction, on the particle inertia
parameter r In line with the primary mechanism
governing the process of deposition, the entire range of
particle inertia may be subdivided into three regimes: the
diffusion regime ( r <1 ), the turbophoresis regime
( 1 r, 100), and the inertia regime ( r > 100 ). The
deposition process of the diffusion regime is mainly
governed by Brownian and turbulent diffusion. In addition,
some driving forces that cause transport of submicron
particles (e.g., thermophoresis in nonisothermal flow) can
play a significant role. In the situation when the diffusion
mechanism plays the leading role, j, declines
monotonously with r, as a result of a decrease in the
Brownian diffusivity as the aerosol size increases. The
basis deposition mechanism of the turbophoresis regime is
the turbulent migration of particles from the flow core,
which is characterized by highlevel velocity fluctuation
intensity, to the viscous sublayer adjacent the wall. This
regime features a strong dependence of j, on r Kallio
& Reeks (1989) and McLaughlin (1989) were the first to
establish numerically the tendency of deposing particles to
accumulate in the viscous sublayer under the action of
turbophoresis; this effect was reproduced in numerous later
works. Highinertia particles ( r > 100 ) are weakly
involved in turbulent flow of the carrier fluid, which
causes the deposition coefficient j, in a vertical duct to
decrease with r .
102
104
106
Figure 1: The deposition coefficient in vertical duct flows.
(13) DIM: (1) Re =10000, (2) Re =20000, (3) Re =50000,
Paper No
(4) experiment by Liu and Agarwal (1974), (5) DNS by
McLaughlin (1989), (6) LES by Wang et al. (1997), (7)
DNS by Marchioli et al. (2003), (8) DNS by Marchioli et al.
(2007).
Fig. 1 presents the predictions of the deposition coefficient
for the pipe flow conditions which correspond to
experiments by Liu & Agarwal (1974). To focus attention
on the deposition mechanisms caused by the interaction of
particles with turbulent eddies, the gravity and lift forces
are neglected and hence F, =F 0 In Fig. 1, the
deposition coefficients obtained for duct flows using DNS
McLaughlin (1989), Marchioli et al. (2003), (2007) and
LES Wang et al. (1997) are shown as well. Note that, in the
diffusion and turbohoresis regimes, the deposition process
is mainly governed by the interaction of particles with
nearwall turbulent eddies. Therefore, the deposition rates
determined in round pipe and flat channel flows are hardly
distinguishable. As is clear from Fig. 1, the DIM properly
captures the dependence of j, on r, at r < 100. The
deposition coefficient predicted for highinertia particles is
found to systematically deviates from the measurements,
because the model does not predicts a decrease in j, with
r Thus, the DIM can be successfully employed in
predicting the deposition rate in the diffusion and
turbophoresis regimes.
Aerosol deposition in circular bends
In what follows we focus our attention on the deposition of
aerosol particles in bends. Hydrodynamic structure of these
flows is complex. It is characterized by the existence of
curved streamlines and recirculating regions. The key
nondimensional parameters that govern the flow are the
Figure 2: The streamlines of the mean flow in the
midplane of the bend at Re =10000 and De =4225.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Reynolds number defined as Re =DUm/V, and the
Dean number defined as De= Re/Rl/2 where
R 2Rb/D is the curvature ratio. For high Dean
numbers, the flow in the bend is mainly governed by the
centrifugal force which changes cardinally the flow pattern
as compared to that in the straight duct. Figs. 2 and 3 show,
respectively, the streamlines of the mean flow in the
midplane and the streamlines of the secondary flow for the
deflection angle of 900. The main features of the flow in
the bend consist in separating the mean flow from the inner
side, displacing it to the outer side, and generating the
secondary flow in the form of a symmetric pair of
counterrotating helical vortices.
The total process of aerosol deposition can be measured by
the penetration of particles which is defined as the ratio of
the particle flow rates in the outlet and inlet sections of the
bend, 4 = Go,,t,/G,,,, or by the deposition efficiency,
S=1 Fig. 4 presents the deposition efficiency
predicted in the 900 bend under the conditions
corresponding to experiment by Pui et al. (1987) for
Re =10000, De =4225, Ro=5.6, and pp/p, =755. The
inertia of particles is quantified by the Stokes number
defined as St=2rpUm/D In these circumstances, the
deposition of particles is caused by the simultaneous action
of diffusion, turbophoresis, gravity, and centrifugal force.
However, the dominating mechanism is the centrifugal
force due to the curvature of the main flow and the
formation of the secondary flow. As is clear from Fig. 4,
the effect of the Stokes number predicted by the DIM is in
good agreement with both experimental data Pui et al.
(1987) and simulations Breuer et al. (2006), Berrouk &
Laurence (2008).
Fig. 5 demonstrates the effects of and curvature ratio and
Stokes number on the penetration of particles in the 900
bend at Re =10000. Predictions are compared with
Figure 3: The streamlines of the secondary flow in the
bend for the deflection angle of 900 at Re =10000 and
De=4225.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
0 0.4 0.8 1.2 1.6 St
Figure 4: The effect of Stokes number on the deposition
efficiency in the 900 bend. (1) DIM, (2) experiment by Pui
et al. (1987), (3) LES by Breuer et al. (2006), (4) LES by
Berrouk & Laurence (2008).
experiments performed by McFarland et al. (1997) in a
wide range of curvature ratio. It is obvious that the
centrifugal effect increases as the curvature ratio decreases.
Therefore, the penetration falls with both increasing St
and decreasing R, As is clear, the DIM reasonably
reproduces these effects. Some distinction between the
predictions and the measurements is observed at small
Stokes numbers, when the DIM overestimates the
deposition rate.
Fig. 6 compares the deposition efficiency as a function of
bend angle with experimental data by Peters & Leith
(21'"4) at Re =203000 and Ro =5. These experiments
were carried out in bends of D =0.152 m at a mean
velocity of 20 m/s, and hence they were the first to be
directly applicable to industrial bends. As is clear from Fig.
6, the deposition efficiency increases for a given particle
size. Taking into consideration a great uncertainty of
measurements, Fig. 6 indicates that the DIM can
reasonably describe the deposition of aerosols at such high
Reynolds numbers which are typical of industrial
applications.
0 0.2 0.4 0.6 0.8 1.0 St
Figure 5: The effects of curvature ratio and Stokes number
on the penetration of particles in the 900 bend. (13) DIM,
(46) experiment by McFarland et al. (1997): (1, 4) Ro =4,
(2, 5) Ro =10; (3, 6) Ro =20.
Figure 6: The effects of particle size and bend angle on
the deposition efficiency. (13) DIM, (46) experiment by
Peters & Leith (21." 4): (1, 4) 450; (2, 5) 900; (3, 6) 180.
Conclusions
The paper is aimed at development and application of the
DIM for the simulation of dispersion and deposition of
aerosol particles in twophase turbulent flows. The model
stems from a kinetic equation for the probability density
function of velocity distribution of particles whose response
times do not exceed the integral timescale of fluid
turbulence. The salient feature of the DIM consists in
expressing the particle velocity as an expansion in terms of
the properties of the carrier fluid, with the particle response
time as the small parameter. By this means, the problem of
modelling the dispersion of the particulate phase reduces to
solving a sole equation for the particle concentration. Thus,
computational times are seriously shortened as compared to
full twofluid Eulerian models. The model presented is
capable of predicting the main trends of particle distribution
including the effect of preferential accumulation due to
turbophoresis.
The DIM has been incorporated in a CFD code and coupled
with fluid RANS in the frame of twoway coupling.
Simulations of aerosol deposition in straight ducts and
circular bends have been performed. The results of
deposition efficiency obtained using the DIM are found to
be in encouraging agreement with both experimental data
and Lagrangian tracking simulations coupled with fluid
DNS or LES.
Acknowledgements
This work was supported by the Russian Foundation for
Basic Research (grant number 090800084).
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