Journal of Colloid and Interface Science 417 (2014) 227–237
Contents lists available at ScienceDirect
Journal of Colloid and Interface Science
www.elsevier.com/locate/jcis
Microscale simulation of particle deposition in porous media
Gianluca Boccardo a, Daniele L. Marchisio a,⇑, Rajandrea Sethi b
a
b
DISAT – Dipartimento Scienza Applicata e Tecnologia, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy
DIATI – Dipartimento di Ingegneria dell’Ambiente, del Territorio e delle Infrastrutture, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy
a r t i c l e
i n f o
Article history:
Received 2 May 2013
Accepted 4 November 2013
Available online 22 November 2013
Keywords:
Porous media
CFD
Brownian deposition
Steric interception
Nanoparticle deposition
Pore-scale simulation
a b s t r a c t
In this work several geometries, each representing a different porous medium, are considered to perform
detailed computational fluid dynamics simulation for fluid flow, particle transport and deposition.
Only Brownian motions and steric interception are accounted for as deposition mechanisms. Firstly
pressure drop in each porous medium is analyzed in order to determine an effective grain size, by fitting the results with the Ergun law. Then grid independence is assessed. Lastly, particle transport in the
system is investigated via Eulerian steady-state simulations, where particle concentration is solved for,
not following explicitly particles’ trajectories, but solving the corresponding advection–diffusion equation. An assumption was made in considering favorable collector-particle interactions, resulting in a
‘‘perfect sink’’ boundary condition for the collectors. The gathered simulation data are used to calculate
the deposition efficiency due to Brownian motions and steric interception. The original Levich law for
one simple circular collector is verified; subsequently porous media constituted by a packing of collectors are scrutinized. Results show that the interactions between the different collectors result in behaviors which are not in line with the theory developed by Happel and co-workers, highlighting a different
dependency of the deposition efficiency on the dimensionless groups involved in the relevant
correlations.
Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction
The transport and deposition of colloidal particles in porous
media are important phenomena involved in many environmental
and engineering problems. Particle filtration [1], catalytic processes carried out through filter beds [2–4], chromatographic
separation [5,6] and aquifer remediation [7], just to cite a few
examples, all require the understanding of their fundamental
mechanisms. Among the above cited applications, of particular
interest is the injection of colloidal nanoscale zerovalent iron
(NZVI) as a technology for remediation of contaminated aquifer
systems [8–15]. In these applications, the possibility of employing
reliable mathematical models for the simulation of colloidal particle transport and deposition in porous media is particularly interesting, and often needed. In order to describe large spatial domains
the models have to ignore the micro-porous structure of the medium and must be derived from an homogenization procedure. The
final macro-scale models result in different submodels for the each
phenomenon involved and this work focuses on one of them: the
rate of particle deposition on the surface of the grains constituting
the porous medium. The commonly used theoretical framework for
treating deposition of colloidal particles onto stationary collectors
⇑ Corresponding author. Fax: +39 011 0904699.
E-mail address: daniele.marchisio@polito.it (D.L. Marchisio).
0021-9797/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.jcis.2013.11.007
is the classic colloid filtration theory (CFT) [16]. CFT describes fluid
flow and particle deposition in porous media and is based on seminal work by Happel [17], Levich [18] and Kuwabara [19]. Some of
these models have become very popular, as for example the Happel
model, that has been used in many works [20–22] even in derivative forms, some of which developed upon the sphere-in-cell model while retaining many of its features [23,24], while others only
use the simplicity of the spherical model as a starting point to develop more physical investigations [25,26]. Hydrodynamics and
particle deposition for a number of different systems, namely the
rotating disk, parallel-plate channel and for stagnation point flow
have also been studied [27].
Deposition is seen as occurring in two steps. The first step is the
transport of the colloidal particles from the bulk of the fluid to the
collector surface by advection and diffusion, and is quantified by
the collection efficiency, g0 ; the second step is the physico-chemical attachment of the particles to the collector, quantified by the
attachment efficiency, a, which represents the ratio of the number
of particles sticking to the collector after collision to the number of
particles colliding with the collector. The product of these two contributions is expressed by the total removal efficiency, g, which
includes both the transport and the attachment of the particles.
Among the different filtration theories, one which is very often
referred to is the model by Yao et al. [1], used to understand
particle deposition in packed beds [28]: its theoretical foundation
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G. Boccardo et al. / Journal of Colloid and Interface Science 417 (2014) 227–237
will be explained later on. Another class of widely used models are
the so called pore-network models, where the porous medium is
described by a graph-like geometrical construct [29–33].
These models predict particle deposition, or in other words particle removal from the suspension, through the analytical solution
for the fluid flow in simplified geometric models: Levich refers to
diffusion to a single solid particle descending freely in an infinite
medium at very low fluid and particle velocity (Stokes flow),
whereas the Happel model represents the porous medium with a
unit cell constituted by a spherical solid collector inside a fluid
shell, with their volume ratio chosen so that the porosity of this
single cell is equal to the porosity of the actual porous medium.
Generally, these models are developed for a single collector, while
the influence of the other grains of the porous medium on the flow
around this collector is accounted for through geometry-based, often empirical, corrective parameters [17]. Moreover, these solutions are valid only for clean-bed filtration (before any significant
deposition of particles on the collector occurs).
The difficulties in investigating these issues from the experimental point of view have prevented the development of accurate
corrections to account for the presence of many grains (collectors)
of irregular (i.e.: highly non-spherical) shapes and characterized by
wide grain size distributions. Nowadays, the advancement of detailed mathematical models based on computational fluid dynamics (CFD) offers an interesting alternative to experimental
investigation: some pore-scale simulations of physical packing of
spheres [34] and uniformly-sized flattened half-spheres have recently been performed [35].
The objective of this work is therefore to improve the current
understanding of particle transport and deposition in porous media, by means of more detailed CFD simulations. In the micro-scale
simulations small but representative portions of several porous
media are considered and the details of the porous structure included. Many different representations of grain packings are investigated, in order to explore the influence of porosity, grain size and
shape values. Some of the geometries used in this work were also
successfully used in a recent study of pore-scale flow of non-Newtonian fluids in porous media [36], and are similar to those employed by other authors in similar studies [37]. First fluid flow is
described by solving the continuity and Navier–Stokes equations,
since particles are assumed to follow the flow, and results are compared with theoretical predictions of flow in porous media. Then,
under the hypothesis of clean-bed filtration, particle deposition is
investigated with focus on Brownian and steric interception mechanisms. Particular attention is devoted to the quantification of the
effect of the irregularity of the grains and of the presence of multiple grains (or collectors) in the packing.
2. Governing equations and theoretical background
Although the length-scales involved in the investigation of flow
in the pores of porous media are very small (at the order of hundreds of lm), the continuum hypothesis for the fluids holds true.
As a first step the fluid velocity field can be determined in a CFD
simulation, by solving the continuity and Navier–Stokes equations,
that in the case of incompressible fluids read as follows:
@U i
¼ 0;
@xi
ð1Þ
@U i
@U i
1 @p
@ 2 Ui
þ Uj
¼
þm 2 ;
@t
@xj
q @xi
@xj
ð2Þ
where U i is the ith component of the fluid velocity, p is the fluid
pressure, and q and m are its density and kinematic viscosity,
respectively. These equations are generally applied with non-slip
boundary conditions ðU i ¼ 0Þ at the grain wall. The continuum approach can also be adopted for the particles, as long as their size
is smaller than the size of the pores. Moreover, we assume the concentration of particles in the liquid to be low enough to describe the
diffusive flux through Fick’s law. Thus, in an Eulerian framework the
transport of particles can be described by solving the well known
convection–diffusion equation:
@C
@C
@
@C
;
þ Uj
¼
D
@t
@xj @xj
@xj
ð3Þ
where C is the particle concentration and D is the particle diffusion
coefficient due to Brownian motion, generally estimated with the
Einstein equation [38]:
D¼
kB T
;
3pldp
ð4Þ
where kB is the Boltzmann constant, T is the temperature, l ¼ qm is
the dynamic viscosity and dp is the particle size. As mentioned it is
assumed that particles follow the fluid: a condition typically verified for solid particles (of density of about 5000–10,000 kg m-3)
moving in water at room temperature with size equal to or smaller
than 1 lm or, more properly, when the suspended particles’ Stokes
number, identifying the ratio between the characteristic relaxation
time of the particle and that of the fluid is very small ð 1Þ, as it is
the case under all the operating conditions under scrutiny in this
work.
On the other hand hydrodynamic interactions, namely the increase in viscous drag force on the particles in the vicinity of the
collector, are also present. As a result, their velocity decreases
due to the additional friction between the fluid and the wall. This
hydrodynamic retardation phenomenon may cause a deviation of
some significance in the particles’ path from the streamlines of
the undisturbed flow near the collector when the distance between
the particle and the collector is of the order of 2 or 3 particle radii
[39]. Moreover, the particles’ mobility near the wall is also reduced, decreasing the particle diffusivity coefficient which is not
constant anymore but depends on the distance to the wall. Corrective factors for both particle velocity and their diffusivity coefficient, accounting for these hydrodynamic interactions, have been
obtained in a number of earlier studies [39–41]. In this work we
will be considering the case where chemical conditions are favorable to particle attachment (as will be explained later on), thus
accounting for the contribution of all near-wall, sub-particle size
attractive and repulsive forces (e.g.: London attraction forces) in
a single attachment efficiency parameter. In this framework, it
would not be possible to include explicitly the effects of hydrodynamic interaction separate from the calculation of these forces as,
in the absence of any attractive force, the particles would not be
able to reach the collector lacking the means to overcome the viscous repulsion [22]. An useful approximation in this case is the
Smoluchowski–Levich approximation, in which it is assumed that
the hydrodynamic retardation experienced by the particle while
approaching a solid wall is balanced by the attractive London
forces [18,41]. In fact, Van der Waals, electrical double layer and
hydrodynamic interactions are excluded from the calculations.
Neglecting the effects of the latter makes it possible to assume that
the particles move with velocity equal to the fluid (as seen earlier)
and that the diffusion coefficient is independent of the particle’s
position in the domain. This approximation is only valid when
the particle size is smaller than the thickness of the particle diffusion boundary layer, which sets an upper limit equal to, in the
majority of cases, a few hundred nanometres of size [42,43]. This
range covers a large part of the cases explored in this study, and
an assessment of the possible deviations due to the use of this
approximation in this work will be done in the results section.
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G. Boccardo et al. / Journal of Colloid and Interface Science 417 (2014) 227–237
By numerically solving Eqs. (1)–(3) on realistic geometries, fluid
flow and particle transport and deposition can be accurately described. Since this description will be used in this work to improve
on existing theories for macro-scale models, it is useful to review
some of these before treating the simulation results. Flow in saturated porous media is characterized by the permeability, k, expressed by Darcy’s law [44],
DP l
¼ q;
L
k
ð5Þ
which is valid only for very small values of the fluid superficial
velocity q [45]; k is the medium permeability, DP is the pressure
drop across the porous medium and L is its length. As well
known, Eq. (5) fails to predict non-linearities arising at
larger q [46] that can be described by the Forchheimer equation
[47,36]:
DP l
¼ q þ bqqjqj;
L
k
ð6Þ
where b is the so called inertial flow parameter and, like k, is independent of fluid properties and only depends on the microstructure
of the porous medium.
Alternatively, porous media are frequently described as packed
filter beds. In the case of grains of spherical shape, two such
correlations can be used (the Blake–Kozeny equation and the
Burke–Plummer equation), combined together into the well-known
Ergun’s law [48]:
g0 ¼
rate of particles colliding the collector
;
pd2
qC 0 4 g
where C 0 is the concentration of particles at a long distance upstream the collector.
The particle deposition rate coefficient in Eq. (9), kd , contains
the deposition efficiency, g0 , a parameter accounting for all the
non-idealities in the transport of particles to the collector and
the correlations for its calculation typically include three mechanisms: Brownian diffusion, interception and sedimentation. Interception is a steric effect occurring when particles moving with
the fluid along a streamline come into contact with the solid grains
due to their finite size, and as such is the endpoint for all deposition mechanisms. Interception of the smallest particles is enhanced
by Brownian diffusion, which is the dominant effect for nanoparticles. Interception of the largest particles is enhanced by sedimentation, which acts when their density is higher than that of the
carrier fluid and will not be considered in this preliminary work.
The theoretical models mentioned above provide with the
means to determine the single collector efficiencies pertaining to
each of the transport mechanisms individually, namely Brownian
motion, gB , and interception, gI , resulting in analytical solutions
only in the case of spherical collectors.
The simplest case, pertaining brownian diffusion in creeping
motion, was developed by Levich ([18]) and builds upon the aforementioned Smoluchowski–Levich approximation, resulting in:
2
DPq dg e3
1e
þ 1:75;
¼ 150
ðdg G0 Þ=l
G20 L ð1 eÞ
gB ¼ 4:04Pe3 ;
ð7Þ
where G0 is the mass flux ðG0 ¼ qqÞ, dg is the grain size (an equivalent diameter calculated from the specific surface assuming spherical shape) and e is the porosity. The Ergun law is often expressed in
terms of the dimensionless pressure drop, DP , and the modified
Reynolds number, Re , as follows:
DP ¼
150
þ 1:75;
Re
ð8Þ
where of course DP ¼ DPqdg e3 =ðLG20 ð1 eÞÞ and Re ¼
dg G0 =ðð1 eÞlÞ. This modified Reynolds number differs from the
classical Reynolds number (Re ¼ dg G0 =l) just for the presence of
the porosity.
As far as particles are concerned, the deposition or ‘‘collection’’
performance of a porous medium can be expressed, in one-dimensional systems, as follows ([1,28,20]):
q
b
d CðxÞ
3 q 1e
b
b
¼
ag0 CðxÞ
¼ kd CðxÞ;
dx
2 dg e
ð9Þ
where x is the spatial coordinate that identifies the main particle
flow direction in the medium. Eq. (9) assumes that the deposition
b , collection
rate is linearly dependent upon particle concentration C
efficiency g0 and attachment efficiency a. This latter parameter,
depending on the balance between repulsive and attractive forces
between colloidal particles and the solid grains can assume values
0 6 a 6 1. As mentioned earlier, in this study this factor is assumed
equal to unity, as is the case where chemical conditions are favorable, thus neglecting the effects of any energy barrier to particle
attachment, since the focus is on the transport of particles from
the fluid to the grain surface. At last it is interesting to point out that
Eq. (9) does not contain a dispersion term since this is usually considered at larger length-scales. The collection efficiency, g0 , of a single spherical collector is defined as the ratio between the rate of
collision of particles with the collector and the rate of particles
flowing towards it ([1])
ð10Þ
ð11Þ
where the Peclet number is defined as Pe ¼ qdg =D. Later, Yao added
the case for interception [49], resulting in:
3
2
gI ¼ N2R ;
ð12Þ
where N R is the ratio dp =dg .
Subsequently then, as previously mentioned, many works improved on these relations, mainly to account for the influence of
the other collectors in the system on the velocity field. For example, taking the steps from the analysis made by Kuwabara [19],
Kostandopoulos et al. [50] derived the following equations for
the single (spherical) collector efficiency:
2
gB ¼ 3:5gðeÞPe3
ð13Þ
and
gI ¼
3
N2R
gðeÞ3 ;
2 ð1 þ N R Þs
ð14Þ
where the exponent s ¼ ð3 2eÞ=3e and gðeÞ is the Kuwabara porosity function:
!
e
:
gðeÞ ¼
1
9
2 e 5 ð1 eÞ3 15 ð1 eÞ2
ð15Þ
Another approach [51] builds upon the Happel model following the
Levich procedure, obtaining for the diffusive efficiency:
1
2
gB ¼ 4As3 Pe3 ;
ð16Þ
where As, like gðeÞ, is a function of bed porosity:
As ¼
2ð1 c5 Þ
;
2 3c þ 3c5 2c6
ð17Þ
1
where c ¼ ð1 eÞ3 . A similar law for interception, which improves
Yao’s result, also exists [43]:
3
2
gI ¼ AsN2R :
ð18Þ
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G. Boccardo et al. / Journal of Colloid and Interface Science 417 (2014) 227–237
From these, it is possible to obtain the total efficiency of the single spherical collector with the relation [42]:
g0 ¼ gB þ gI ;
ð19Þ
3. Test cases and operating conditions
In this work, Eqs. (1)–(3) are solved, in steady state conditions,
on two sets of two-dimensional geometric porous medium models
and results are then interpreted by using the theoretical models reported in Eqs. (5)–(19). The first set is composed of four randomly
arranged distributions of non-overlapping identical circular elements, with their placement calculated via an appropriate algorithm (see Fig. 1a). The second set is constituted of eight realistic
geometries with irregularly shaped and polydispersed grains (see
Fig. 1b). The various cases for each of these sets represent porous
media characterized by different values of mean grain diameter,
dg , and porosity, e. A total of 12 geometries were necessary in order
to provide for a large enough field of investigation.
The first four synthetic cases were made via a simple bruteforce algorithm that sequentially placed the items in random locations, each time checking for possible contacts with the other items
already in place, repeating the step if necessary: thus preventing
any overlap between them. The source for the eight realistic geometries were instead as many SEM (Scanning Electron Microscopy)
images of real sand samples, which were treated in order to be
transformed in geometric structures compatible for the use in a
CFD code. The rationale behind the choice of these two different
types of models (i.e., irregular versus circular grains) is to investigate two different situations: one that more closely resembles reality, and a more simplified one in which there is no effect of grain
polydispersity and irregularities. In this way, it will be possible
to study the influence of grain size distribution and shape on particle transport and deposition.
Since porosity (e) is an important parameter with respect to
packed bed collector efficiency, we considered a wide range for this
variable (0.3–0.5) in order to gain better insight with respect to its
influence on particle removal efficiency. A similar choice (although
in a complementary range 0.28–0.38) was made in recent works
[52] which also highlighted the strong influence of porosity and
grain size distribution on particle deposition. The final 12 geometries resulted also in nominal grain size dg from 200 lm up to
650 lm, as detailed in Table 1. Fig. 2 reports instead the nominal
grain size distribution for some selected cases. The nominal grain
Table 1
Porosity, nominal grain size (lm) and domain extension (mm) for the synthetic
geometries (first four rows) and the realistic geometries (subsequent eight rows).
Case
Geometry type
e; ðÞ
dg ;
S1
S2
S3
S4
R1
R2
R3
R4
R5
R6
R7
R8
Synthetic
0.418
0.464
0.404
0.458
0.339
0.352
0.433
0.489
0.326
0.405
0.469
0.488
205
206
309
308
328
358
413
381
644
514
471
367
Realistic
lm
Lx Ly ; mm
2:05 2:05
2:24 2:24
3:07 3:07
3:36 3:36
2:69 2:32
2:69 2:23
2:69 2:32
2:69 2:32
6:65 5:75
6:65 5:74
5:56 5:75
4:75 5:74
size was estimated for those two-dimensional models from the
specific surface area, s, of the porous medium: dg ¼ 4=s.
Table 1 also reports the spatial extent in the flow direction (Lx )
of the 12 geometries, excluding the premixing and postmixing
zones added to improve the accuracy of the solution of the flow
field and to reduce the impact of the backflow. In fact in order to
minimize anomalous flow recirculations caused by the irregular
nature of the packings, and to get a more accurate solution for
the flow field (i.e.: less influenced by boundary effects), the length
of these geometries was extended by adding two new ‘‘ buffer’’
zones, before the inlet and after the outlet, as shown in Fig. 1.
After having built the 12 geometries, GAMBIT 2.4.6 was used to
generate suitable meshes for the CFD code. Two-dimensional,
quadrilateral, unstructured meshes were generated, minimizing
cell skewness and aspect ratio. Then the meshes were refined uniformly and on the grain borders to fully resolve the momentum
and particle concentration boundary layers. In Fig. 3 an example
of a typical mesh near the border of a grain, as it undergoes the
refining process, is reported.
One parameter was used to assess the grid independence of the
results for the flow field. In order to obtain this single parameter
for each geometrical model and for all the flow rates investigated
(with particular attention to the momentum boundary layer near
the grain border), pressure drop data from the simulations were
analyzed using the Ergun law, reported in Eq. (8). These results
were then fitted to this law, in order to obtain a new equivalent
or effective grain size, dg . As it can be seen in Fig. 4, where the var
iation of dg increasing the mesh cell number is reported (starting
from the original cell number), grid independent flow field results
Fig. 1. Sketch of the synthetic (left) and realistic (right) porous medium models investigated in this work.
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G. Boccardo et al. / Journal of Colloid and Interface Science 417 (2014) 227–237
Fig. 2. Grain size distribution for geometries (left to right and top to bottom): R1, R2, R6, R8.
Fig. 3. Particular of a mesh, showing the series of grid refinements.
Table 2
Final number of mesh cells for the synthetic geometries (first four rows) and the
realistic geometries (subsequent eight rows).
Fig. 4. Effective grain size, dg , with varying number of mesh cells, cases R1 () and
S3 ().
are obtained for the second (uniform) refinement. It is therefore
possible to conclude that the corresponding grid fully describes
the momentum boundary layer.
The boundary layer of particle concentration required a finer
mesh (near the grains) due to the resulting Schmidt number [53],
especially for the larger particles. Grid-independence for particle
transport and deposition will be, for clarity, discussed later on,
Case
Mesh cells (thousands)
S1
S2
S3
S4
R1
R2
R3
R4
R5
R6
R7
R8
894
1.058
1.525
1.840
893
884
816
642
1.950
2.254
1.828
1.931
and Table 2 reports the final number of mesh cells for the 12 investigated cases.
The system is considered isothermal at T ¼ 293 K, and the fluid
3
is Newtonian (density q ¼ 998; 2 kg m and dynamic viscosity
1 1
l ¼ 0; 001003 kg m s ). The flow is considered laminar, and
thus the CFD code did not solve additional turbulence models. This
is justified given the range of velocities explored: in fact, the super-
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G. Boccardo et al. / Journal of Colloid and Interface Science 417 (2014) 227–237
ficial velocities, q, ranged from 106 to 101 m s1 (roughly corresponding to Re ranging from 104 to 10).
As for the boundary conditions for these cases, an inlet zone
was set on one side of the geometry, where water flowed into
the domain at a fixed velocity, and the outlet at the opposite side.
This setup aims to represent a situation in which the fluid is moving predominantly in one direction, and as such the simplest and
most appropriate way to reconstruct the large-scale system with
a smaller, representative domain is to align the flow direction with
one of the three main cartesian axis, resulting in the two sides
orthogonal to that axis to be respectively the inlet and outlet
boundaries of the domain. It has to be noted that gravity was not
considered in our model and its effect was not accounted and
solved for in the CFD calculations. On the two remaining sides a
condition of symmetry was set. This condition ensures that all
property fluxes are equal to zero along the specified border, resulting in no fluid flow across these boundaries. Regarding the interpretation of the results, those of most interest refer to the
pressure drop, which was calculated as the difference between
the inlet and outlet boundary values. After normalizing on the
length of the domain (DP=Lx ), the results for the cases at small
superficial velocity was used to estimate the permeability of the
porous media k ¼ ql=ðDP=Lx Þ. Pressure drop results were also
compared with the predictions of the Ergun law, as explained
earlier.
As for particle deposition, only Brownian motions and interception were considered, for different populations of monodispersed
particles, with diameters, dp , equal to 1, 10, 100, 200, 500, 625,
750, 875 nm and 1 lm. The presence of the particles flowing into
the domain with the fluid was simulated by representing them
with a scalar, namely the normalized particle concentration, C. Initially the particle deposition velocity was calculated with a twofluid Eulerian multiphase model, however, in the range of operating conditions considered in this work particle velocity always relaxed to the fluid velocity. This result was to be expected since, as
explained earlier, both Stokes numbers for all considered cases are
very low, and viscous drag forces causing hydrodynamic retardation were not considered in our simulation framework, thus the
assumption of one-way coupling between particle and fluid, with
the velocity of the former always relaxing to that of the latter;
moreover, the typical particle concentration is low enough not to
have any two-way coupling. Brownian motions were accounted
for by the diffusive term of Eq. (3), with the diffusivity coefficient
calculated with the Einstein–Stokes relationship reported in Eq.
(4). While diffusivity of particles could be anisotropic and in general will vary between the bulk of the fluid and near the borders
of the solid grains, here it will be considered constant.
The boundary conditions for this normalized concentration scalar were set to one at the inlet, with diffusive flux at the outlet set
to zero. Regarding the initial conditions, in all cases scalar concentration was set to zero in all the domain, as it provided for a
smoother and faster convergence of the numerical solution with
respect to other initialization values.
The last fundamental boundary condition is imposed at the
grain borders. As described earlier, since in this work we are interested in evaluating the deposition efficiency, g0 , all the repulsive
forces responsible for the reduction of the attachment efficiency,
a, were ignored (making a ¼ 1), which translates into setting a particle concentration at the border of the grains equal to zero. To account for interception caused by steric effects due to the finite size
of the particle, a user-defined function (UDF) was used to set to
zero the particle concentration in all the mesh cells whose centroids’ distance from the grain wall was less than the particle radius. In practice, the UDF shifts the grain boundary by a distance
equal to the particle radius. It has to be noted that the lengths involved are very small, representing yet another reason for the need
for a finer grid at the grain border. The final grid sizes, reported in
Table 2, are however reasonable, since interception is only significant for large particles.
From these simulations the value of particle concentration in
the microscale domain is obtained. Our objective is to extract from
this information, i.e. Cðx; yÞ, a macroscopic model with which to
calculate the particle deposition rate without describing the details
of the pores. As it has been said, the fluid moves mainly from the
inlet to the outlet, and it is thus along this direction that a meaningful particle concentration gradient will be identifiable. More
specifically, it is useful to express the results as the difference in
concentration between the porous inlet and outlet zones. First
then, the concentration value at any point along the porous bed
is averaged in the direction orthogonal to the flow ð0 < y < Ly Þ:
1
b
CðxÞ
¼
Ly
Z
y
Cðx; yÞdy:
ð20Þ
0
The concentration at the inlet and outlet boundaries, equal to
b ¼ Lx Þ, is readily available from our CFD simulab ¼ 0Þ and Cðx
Cðx
tions. From these, a simplified description which ignores pore details but results in the same overall deposition rate can be
obtained simply by using the following deposition efficiency (in
the case of a = 1):
g0 ¼
ln CC0
L ;
3 1e
2
e
ð21Þ
x
dg b ¼ 0Þ and C ¼ C
b ðx ¼ Lx Þ are obtained from CFD
where again C 0 ¼ Cðx
simulations. A preliminary test case was set up, in order to validate
the methodology used both in the CFD simulations and in the analysis of the results. The system chosen as a benchmark was Levich
case of diffusion on a free-falling solid sphere, for the simplicity
of its geometrical model and for its solution resulting surely from
an analytical derivation. Results from simulations replicating this
model and ran in operating conditions inside the range of validity
of the Levich model were then compared with the theoretical predictions of Eq. (11). The outcome was positive, as the coefficients
of the power law describing the CFD results showed a remarkable
accordance with Eq. (11), thus ensuring the appropriateness of
the methodology used in this work for treating this kind of systems.
Summarizing, the following strategy was used. First, simulations with only the Brownian diffusion mechanism accounted for
were run, resulting in the Brownian deposition efficiency gB . Secondly, both Brownian motion and interception mechanisms were
considered resulting in the overall deposition efficiency g0 . Subsequently gI was calculated by inverting Eq. (19). Eventually the rela
tionship between gB (and gI ) and q; e; dg and dp , was assessed by
running simulations at several superficial velocities, q, for the porous media (characterized by several porosity, e, and effective grain
size, dg , values) previously described and with different particle
sizes, dp .
4. Results and discussion
In this section the results obtained from simulations run on the
grids reported in Table 2 are discussed. First, values of pressure
drops in the domain were calculated and used to estimate the permeability of the system. As well known, Darcy’s law has an upper
range of validity (in terms of Reynolds number) beyond which the
relationship between pressure drop and fluid velocity is not linear
anymore: typical values of Re for this transition are of the order of
unity [45]. These considerations are confirmed in Fig. 5, where values of the group ql=ðDP=Lx Þ (equal to permeability for low Re) are
reported in a logarithmic scale against q and labeled with the
corresponding Re values. As it can be clearly seen this group is
233
G. Boccardo et al. / Journal of Colloid and Interface Science 417 (2014) 227–237
Table 3
Original and fitted equivalent grain size, for synthetic geometries (first four rows) and
realistic geometries (subsequent eight rows).
Fig. 5. Values of the group ql=ðDP=Lx Þ, case R1 () and S1 ().
characterized by a constant value up to Re 1 (corresponding to q
of about 102 m s1 for these two cases), after which it undergoes a
sharp decrease, for both the realistic and the synthetic geometries.
The next step in analyzing fluid flow results is to compare pressure drop values obtained from simulations with Ergun’s law predictions calculated with the nominal grain size, dg . As it can be seen
in Fig. 6 (left), the qualitative behavior of DP versus Re follows the
theoretical predictions, with a large displacement with respect to
the theoretical law. By fitting the Ergun law (see Fig. 6, right),
the modified or effective grain diameter, dg , is calculated. As already explained this is done by fitting the value of dg , appearing
in Eq. (7) in order to reduce the distance between the CFD simulation data and the Ergun’s law predictions to less than 10% in value,
moving from Fig. 6 (left) to Fig. 6 (right). In Table 3 the values of the
original grain diameter dg and the equivalent diameter resulting
from fitting the Ergun law are compared. While there is a differ
ence between dg and dg in the case of the eight realistic geometries,
values are much closer for the four synthetic geometries. Finally,
qualitative description of the flow field can be found in Fig. 7.
After having investigated the flow in the system, particle deposition was simulated as previously explained. Contour plots of
particle concentration for one case are reported in Fig. 8. As
explained, from these results the overall deposition efficiency,
g0 , is calculated with Eq. (21), and plotted versus fluid velocity
(106 < q < 102 m s1 corresponding to 104 < Re < 1) and
particle size in Fig. 9.
Fig. 9 clearly shows how the deposition rate (and consequently,
g0 ) progressively decreases as the superficial velocity is increased
(for small particles). This is due to the time scale of convective
transport becoming smaller with respect to diffusive transport;
in fact, particles at higher velocities have less time to reach the
grain borders (and thus to be removed) from the bulk of the fluid.
The behavior changes for particles of about 500 nm and larger, as
g0 initially decreases but then stays approximately constant. This
is due to the effect of steric interception, which has a higher impact
on deposition velocity only for larger particles (on the contrary of
Brownian diffusion which acts predominantly at smaller particle
sizes) and for higher fluid velocities, when the effect of Brownian
Label
dg
dg
S1
S2
S3
S4
R1
R2
R3
R4
R5
R6
R7
R8
205
206
309
308
328
358
381
413
644
514
471
367
133
143
315
291
215
210
179
178
224
198
215
123
deposition becomes negligible. The same data (for the same
parameters) is represented in Fig. 10, where the deposition efficiency, g0 , is now plotted versus particle size, dp , at different superficial velocities q. As already discussed, for the lower superficial
velocities the particle deposition efficiency always decreases as
the particle size is increased, while for higher velocities the particle
deposition initially decreases for small particle sizes and instead
sees a positive slope for larger particles. In fact, for q larger than
103 m s1 a minimum for g0 is detected at about 300–500 nm.
Similar trends were experimentally observed (although in a
slightly different porosity range) by other authors [52].
These results and the general considerations just made are consistent with the underlying theory and are valid both for the realistic and the synthetic geometries. There are, however, some
differences between these two cases. In order to better highlight
these differences, let us consider results for Brownian deposition
efficiency only, gB . In Fig. 11 the aggregate results of the overall
deposition efficiency, g0 , normalized by the porosity-dependent
function, As, of Eq. (17) are reported versus Pe ¼ qdg =D for all the
synthetic geometries. This normalization is done to compare data
from different geometries.
First of all, it is interesting to highlight that when data corresponding to different synthetic geometries obtained under different superficial velocities and particle sizes are plotted versus the
Peclet number, a single master curve is obtained, which is consistent with the theoretical predictions of Eq. (16). By fitting the results of our CFD simulations with an equation of the form:
1
gB ¼ C 1 As3 Peb ;
ð22Þ
the following values C 1 ¼ 0:487 and b ¼ 0:552 are obtained.
While the value of C 1 is quite different from the theoretical [51] value, a small difference between b ¼ 0:552 and the 2=3 exponent
of the theoretical law, reported in Eq. (16), is detected. Regarding
these deviations, it is interesting to note a recent work [54] which
has shown in detail some of the shortcomings of correlation
Fig. 6. Comparison of CFD results (symbols) with Ergun’s law (continuous line).
234
G. Boccardo et al. / Journal of Colloid and Interface Science 417 (2014) 227–237
Fig. 7. Contour plots of fluid velocity (m s1 ) for cases (left to right and top to bottom): R2, R4, S1, S2, R5, R6, for superficial velocity q ¼ 106 m s1 .
equations with a power-law dependence on Peclet number and
of employing Eulerian models with Dirichlet boundary conditions
for concentration (as is the case in this work), arising predominantly at very low fluid velocities. These findings can help to
explain the presence, in Figs. 11 and 12 of values of g over unity.
Moreover, since when one single circular collector is considered,
as in the benchmark case of Levich, the original exponent is obtained (b ¼ 2=3) from our simulations, the difference can be
attributed to the presence of many collectors in the porous medium, due to the interactions of the particle concentration boundary
layers.
These two values of C 1 and b are also used to assess grid independence of these results. As grid independence for flow field pre
dictions was evaluated with the effective grain size, dg , the grid
independence of particle deposition predictions was assessed with
the invariance of the fitted C 1 and b values to further grid refinements. Results showed that using the meshes of Table 2 grid independence was achieved.
A different scenario emerges when looking at results of gB versus Pe for the eight realistic geometries, shown in Fig. 12.
Comparison of Figs. 11 and 12 reveals two main differences: the
first lies in the different coefficients for the power law gB versus Pe,
whereas the second one is the separation between the curves corresponding to different geometries, highlighting the imperfect collapse of the data into one single master curve for realistic grains. In
fact, the fitted master curve for the synthetic grains (see Fig. 11)
corresponds to a coefficient of determination (R2 ) of 0.9789,
whereas that of the realistic grains (see Fig. 12) shows a much lower coefficient of determination, equal to 0.9141.
These deviations call for the search for another parameter (besides porosity and grain size) to be included in the predictive laws
for Brownian particle deposition. In this work, possible influences
of the tortuosity of the porous media and of other forms of the
porosity-dependent term As were analyzed. In the first case, tortuosity of each system was estimated, via CFD simulations, by means
of analyzing the residence time distribution function of an inert
tracer flowing through the medium. In the second case, a search
for a new porosity-dependent function was performed, with the
aim of eliminating the offset seen in Fig. 12. This was done in practice by trying to find a constitutive equation that can correctly predict the value of gB for different systems, at different values of Pe.
Unfortunately, for both cases, no clear relationship linking tortuosity to deposition efficiency, nor a new and more adequate form of
As, could be found. It is also interesting to highlight here that the
use of the Kuwabara function, gðeÞ, of Eq. (15) instead of the porosity dependent function, As, of Eq. (17) led to very similar results.
Regarding the coefficients of the power law, and again referring
to an equation of the form of Eq. (22), the values of C 1 ¼ 0:1926
and b ¼ 0:52 were found. As it can be seen, the coefficient b
has a similar value with respect to results for the synthetic grids,
while the value of C 1 lies even further from the theoretical one.
Even in this case the discrepancies can be attributed to the complex interactions between collectors that result in a very different
overall behavior.
G. Boccardo et al. / Journal of Colloid and Interface Science 417 (2014) 227–237
235
Fig. 11. Brownian deposition efficiency gB versus Pe for synthetic geometries,
compared with the theoretical law (Eq. (16), continuous line).
Fig. 12. Brownian deposition efficiency gB versus Pe for realistic geometries,
compared with the theoretical law (Eq. (16), continuous line).
Fig. 8. Contour plots of normalized particle concentration for case R8, for
q ¼ 106 m s1 (first row), q ¼ 105 m s1 (second row), q ¼ 104 m s1 (third
row), for dp ¼ 1 nm (first column) and dp ¼ 100 nm (second column).
Fig. 9. Particle deposition efficiency for case R1, at different fluid velocities and
particle sizes.
Fig. 10. Particle deposition efficiency for case R1, at different fluid velocities and
particle sizes.
Moreover, it has to be taken into account that our simulations
were conducted with the same assumptions of the Smoluchowski–
Levich approximation, valid only for particles smaller than a few
hundred nanometers. Including the drag correction in the
calculations would decrease the rate of collection with effects of
increasing magnitude with increasing particle size [22,55]. Quantitative comparisons between the inclusion of the drag corrections,
or lack thereof, can be found in Rajagopalan and Tien [22] and
Prieve and Ruckenstein [42]. The use of the Smoluchowski–Levich
approximation in the case of large particles would lead to an overestimation of the collector efficiency. This is clearly shown in the
work of Tufenkji and Elimelech [20] which included the contribution of hydrodynamic retardation, obtaining a different exponent
of the Peclet number in the equation for the Brownian deposition
efficiency and also added a dependency on N R , absent in the original Levich–Yao formulation. These findings are related to the influence of hydrodynamic interactions. Nonetheless, it has to be noted
that the range of particle sizes explored in that work was considerably wider (up to 10 lm), with hydrodynamic interactions influencing more the deposition efficiency for larger particles. In fact,
when comparing their results with works not fully including
hydrodynamic effects, [22] the maximum of the difference (about
50%) was found at dp ¼ 2 lm, with much lower discrepancies at
smaller particle sizes. While it would not be possible to provide
an accurate estimate for the deviation between our results (based
on the Smoluchowski–Levich approximation) and a more complete
model, due to the presence of attractive forces which where also
not explicitly included in our simulations and that would balance
out, in a measure, the hydrodynamic retardation effect, [20,41]
the indications found in these references call for greater care when
interpreting results for particles in the 1-lm range, and the possible dependence on a new term in the brownian deposition efficiency equation (N R ) warrants for a deeper investigation of these
phenomena.
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G. Boccardo et al. / Journal of Colloid and Interface Science 417 (2014) 227–237
0.01
5. Conclusions
1.77E-4
1.77E-3
1.77E-2
0.177
1.77
ηI
0.001
0.0001
1e-05
0.002
0.003
0.004
0.005
0.006
NR
Fig. 13. Interception deposition efficiency gI results for case R3 varying the
parameter N R and for a range of Reynolds numbers (specified on the key) compared
with the theoretical relationship Eq. (18) (continuous line).
0.01
1.77E-4
1.77E-3
1.77E-2
0.177
1.77
1.77E-4
1.77E-3
1.77E-2
0.177
1.77
ηI
0.001
0.0001
1e-05
0.002
0.003
0.004
0.005
0.006
NR
Fig. 14. Interception deposition efficiency gI results for case R3 varying the
parameter N R and for a range of Reynolds numbers (specified on the key) compared
with the new modified relationship, obtained from fitting the aggregate results of
all the geometries (continuous and dashed lines).
Finally, interception results are also presented. In this case there
are no clear differences between results of synthetic and realistic
geometries. Fig. 13 shows the results of gI at different values of
N R , compared with the theoretical predictions of Eq. (18). For clarity of exposition, only results pertaining to case R3 are reported in
Fig. 13, but very similar results were obtained for the other 11
cases. In this case it is clear that while the theoretical law restrains
the interception effect as being dependent on the parameter N R ,
defined as the ratio between grain diameter and particle size, our
results show large deviations between cases corresponding to different fluid velocities. In this case, searching for a new and corrected predictive equation clearly points towards adding a
dependence of gI on the fluid velocity (or more properly, in order
to retain a law dependant on dimensionless parameters, the Reynolds number), which would result in a form similar to the original
law:
gI ¼ C 2 AsN2R Rec :
ð23Þ
A least-squares fitting of these results returns a best value for C 2
and c respectively equal to 1.116 and 0.145. Even if the value for c
does not show a strong correlation between Re and gI , it is
nevertheless possible to obtain more accurate predictions of the
interception effect on deposition with the new law, as shown in
Fig. 14. Again, only results pertaining to case R3 are reported, for
a clear comparison with Fig. 13, but it has to be noted that
this least-squares fitting process, resulting in the C 2 and c
reported above, has been performed not on this single geometry
but on the aggregate of all the particle deposition efficiency results
for all the geometries considered in this work. At last we highlight
that interpreting the data with the other laws available in the
literature reported in Eqs. (12) and (14), resulted in similar
conclusions.
Particle transport and deposition in porous media was investigated in this work by means of detailed microscale CFD simulations. Firstly results were carefully analyzed to assess grid
independence. When considering one single collector with Brownian motion as the dominant mechanism for deposition, the original
correlation proposed by Levich was obtained: gB ¼ 4:04Pe2=3 , thus
proving the consistency of the methodology adopted with the
underlying theory. When considering complex porous media constituted by different circular collectors an analogous dependency,
characterized by a different slope: gB ¼ 0:487As1=3 Pe0:552 , was
found. In this case porosity, e, and effective grain size, dg , were
capable of fully characterizing the observed behavior. In the case
of irregular collectors a similar dependency was again observed
(gB ¼ 0:1926As1=3 Pe0:52 ) although porosity, e, and effective grain
size, dg , seem not to be capable alone to fully characterize the porous medium. In fact, comparing Figs. 11 and 12, the former shows a
collapse of the results on a single master curve dependent on these
two parameters, while the latter shows a clear separation of the results pertaining to each geometry. This is further highlighted by
the very different coefficients of determination showed by the
two obtained master curves, respectively equal to R2 ¼ 0:9789
and R2 ¼ 0:9141. This deviation arising in the case of realistic
geometries would seem to indicate that another parameter should
be included in the laws predicting Brownian particle deposition.
We are confident that the effect of the neglected hydrodynamic
retardation on our results is small for nanometric particles, but
might be more important (up to 50%), for micrometric particles.
Therefore results for the largest particles investigated in this work
should be treated with great care. When interception is also considered, the interaction between the particle boundary layers of
the different grains create a complex dependency resulting in the
following relationship: gI ¼ 1:116AsN2R Re0:145 , valid for both circular and irregularly shaped collectors.
Acknowledgments
This work was financially supported by the EU research project
AQUAREHAB (FP7, Grant Agreement No. 226565) and by the Italian
Ministry of Research and Higher Education through the PRIN project 2008 ‘‘ Disaggregazione, stabilizzazione e trasporto di ferro
zerovalente nanoscopico (NZVI)’’. The contributions of Francesca
Messina and Matteo Icardi are also gratefully acknowledged.
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