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MODELING AND SIMULATION OF IRON/SAND FILTERS
Titre en français : Modélisation et simulation des filtres à eau faits de particules de fer et de sable
Short title: Modeling and simulation of Iron/sand filters
Togue Kamga Fulbert(a), Noubactep Chicgoua (b,c), Woafo Paul*(a)
(a) Laboratory of Modelling and Simulation in Engineering and Biological Physics, Faculty of Science,
University of Yaounde I, Box 812 Yaounde, Cameroon
(b) Angewandte Geologie, Universität Göttingen, Goldschmidtstraße 3, D - 37077 Göttingen, Germany
(c) Kultur und Nachhaltige Entwicklung CDD e.V., Postfach 1502, D - 37073 Göttingen, Germany
* corresponding author: pwoafo1@yahoo.fr9
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Abstract:
This article develops a mathematical model of Iron (Fe0) /sand filter taking into account the loss of porosity
during the filtration process. The mass transport advection-dispersion equation is derived to predict,
through numerical simulation, the spatiotemporal distribution of pollutants and the flow rate in the filter as
well as the water quality at the exit of the filter. Simulation results demonstrated that by mixing sand and
iron particles in proportion of 40 vol% Fe0, the filter can be used continuously for 83 months.
Author keywords: Iron/sand filters; Mathematical model; Loss of Filter porosity; Advection-diffusion
equation; Numerical Simulation; Distribution of pollutants; Filter service life.
RESUME :
Cet article développe un modèle mathématique de filtre à base de particules de fer (Fe0) et de sable en
tenant compte de la perte de porosité au cours du processus de traitement de l’eau. L’équation
d’advection-dispersion est établie pour prédire, par simulation numérique, la répartition spatiotemporelle
des polluants et le débit d’eau à travers le filtre ainsi que la qualité de l’eau à la sortie du filtre. Les
résultats des simulations montrent qu’en mélangeant le fer et le sable dans une proportion de 40% en
volume de fer, le filtre peut être utilisé en continu pendant une période de 83 mois.
Mot clés : Filtres fer/sable ; Modèle mathématique ; Perte de porosité du filtre ; Equation d’advection-
dispersion ; Simulation numérique ; Répartition des polluants ; Durée de vie du filtre.
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1.INTRODUCTION
Water pollution mostly results from (i) natural mobilization/leaching, (ii) mining activities, (iii) industrial
discharge, (iv) agricultural discharge (e.g. fertilizers, herbicides, pesticides), (v) solid waste disposal, and
(vi) unplanned urban and rural development. Poor water quality has damaging impacts on human health,
aquaculture and ecology. In general, water quality problems for human health fall into two major
categories: (i) microbiological contamination responsible for immediate acute diseases and (ii) chemical
contamination, which poses cumulative and chronic health risks to human beings and aquatic life
(Ahamed et al., 2009).
The scientific community is on schedule to develop universal water treatment units for domestic use
(Clasen et al., 2009). The suitability of metallic iron (Fe0) as universal filter material has been recently
discussed (Noubactep et al., 2010). It is based on the fact that corroding iron has the ability to remove all
soluble species by an unspecific mechanism. There are however two main modifications in the design of
Fe0/sand filter in comparison to the Kanchan Arsenic Filter (KAF) developed and distributed in Nepal by
Ngai et al. (2007): (i) the Fe-oxide-producing unit (Fe0/sand bed) is placed after the conventional Biosand
filter (BSF) and (ii) the Fe0/sand bed contains an optimal proportion of Fe0 mixed to sand to extent the
filter service life.
In this paper, an approximate and feasible mathematical model that allows for the prediction of
concentration profiles of aqueous pollutant is developed. The prediction is based on time and height
measurements in a Fe0/sand bed according to the variation of the pore space within the filter as
expansive corrosion proceeds. The service life of the filter is operationally defined as the time to reach (i)
75 % porosity loss and (ii) more than 75% removal from a 100% influent contaminant concentration. This
assumption is justified by the fact that chemical pollutants typically occur in trace amounts, rarely
considerably larger that the maximum contaminant levels (Schwarzenbach et al., 2010). It is found that
the pollutant removal percentage and the service life of the filter depend on the relative proportions of
sand and iron in the filter.
2 PHYSICAL MODEL DESCRIPTION
Let us consider a layer of water containing contaminants as shown in Fig.1. H is the height and D the
diameter of the bed. Water which is considered as an incompressible and Newtonian fluid flows toward
the bottom of the column (porous medium) by gravity. The column which contains a mixture of spherical
particles (sand + metallic iron) of equal size constitutes the filter.
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2.1 Filtration velocity and Variation of porosity of the filter
In the case the water flows towards the column by gravity, the filtration velocity, follows the Darcy
law (Marle, 2006). Therefore the flow velocity modulus in the filter is a temporal function given as
2 3
0
2( ) 1180 (1 )
watergd hq t
H
ρ ϕ
μ ϕ
⎛ ⎞= ⎜− ⎝ ⎠+ ⎟4 (1)
with ( )0( ) 1 . .Fe pt n aϕ ϕ= − t , 302dpa D Hδε= , 53.8x10δ −= . ρ is the density of water, d0 is the
diameter of iron and sand particles, H is the height of the cylinder, is the porosity based on porosity
losses kinetics (Mackenzie et al., 1999), the height of water,
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waterh μ the dynamic viscosity. nFe is the
number of iron particle, a
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p is the porosity losses rate and t the operation time. is the fraction volume of
particles. This expression of the porosity has been derived recently by the authors (Noubactep et al.,
2010).
2.4.2 Mass transport equation for pollutants in the filter
The spatiotemporal variation of pollutants in the filter obeys the following equation
0
1
0
2
2
.
1
.
, 1 b
n
f
C
Dz z
C C
t z
q
R
n K C
with R
ϕ
ρ
ϕ
−
∂
∂
∂ ∂= −∂ ∂
⎛ ⎞⎜ ⎟⎝ ⎠
= +
(2) 13
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This equation describes a microscopic mass balance using Fick’s law adapted to macrodispersion
(Rooklidge et al., 2005). The hydrodynamic dispersion coefficient, Dz, includes turbulence effects caused
by species and gravity. bρ is the iron bulk density of the porous media, n the sorption intensity parameter
and
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fK the sorption capacity parameter (Williams et al., 2003). The main feature of this equation (2) is
the time dependence of the flow velocity q and the porosity φ. This time dependence is seen hereafter to
be an important factor that makes the theoretical prediction similar to what is found from the experimental
investigation. This equation (2) can be coupled to transfer equation that describes the transfer of water
and its pollutants to the filter (Peel et al., 1980; 1981). However, this equation is not necessary here since
it is assumed that the pollutant concentration at the entrance of the filter is assumed known and constant.
The main concern is to find out how the effects of time dependence of the porosity affects the
concentration of pollutants at the exit or at any point of the filter.
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The initial and boundary conditions related to the transport equation are C (z, 0) = C0 if z=0, if not C (z,
0) = 0, i.e. initially uncontaminated column; C (0, t) = C0, where C0 is the concentration of pollutants at the
entrance of the filter and H the depth of the filter. The following values of the parameters are used. ρ=
1000 kg/m3, g= 9.81 m.s-2, d0=1.2 mm, μ = 0.01 g/s.cm, hwater=20 cm, H=50 cm, =0.64, D=50 cm (this
gives a cylindrical bed volume equal to 98 L), Dz=0.14 cm2/min (Williams et al. 2003), ρb=7800 kg/m3,
n=0.32, Kf=278 L/kg. We remind the reader that the sand density is equal to 2650 kg/m3. The
determination of nc, which is the threshold number of iron particles necessary to fill completely the initial
volume of pores, is obtained as the ratio between total volume of pores in the filter and the volume of
expansive corrosion product minus the mean value of the iron particle (Caré et al., 2008, Noubactep et
al., 2010). With the above values of it is found that nc≈20 millions (which corresponds to about 141 kg of
iron).
3 RESULTS AND DISCUSSIONS
3.1 Temporal distribution of solutes
Equation (2) is solved numerically. The spatial derivative is discretized using the backward finite
difference scheme while the time derivative is handled using the fourth order Runge-Kutta computer
routine written in FORTRAN. The value of concentration of the pollutant at the entrance of the filter is C0=
0.35 mg/L. Figure 2a presents the time variation of the pollutant concentration at the filter exit for different
values of the number of iron particles. It is found that the pollutant concentration at the exit increases with
time and decreases when the number of iron particles increases. If one assumes that 25% of the pollutant
concentration is tolerable, then the service time of the filter is found to be more than 83 months for 20
millions of iron particles. In contrast, Figure 2b shows that for 20 millions of iron particles and assuming
that the porosity is constant (as it considered in most of the scientific papers), the service time of the filter
is approximately 17.2 months. This demonstrates that the long-term performance of the filter is assured
by the iron corrosion product formed continuously over the iron surface. To end this section, we note that
the numerical simulation can help to find out how the pollutant concentration evolves over the filter length
showing the action of the zero valent iron in the removal process of pollutants (Youwen et al., 2005).
3.2 Validation of the model implementation
To validate the correct implementation of the model, the experimental breakthrough curve obtained by
Sanghamitra and Gupta in 2005 is used (see Fig. 2 of this reference). The transport Eq. (2) is then solved
numerically with input parameters provided by the same authors (C0=2 mg/L, n=0.53, Kf=2.67L/g,
V=0.41 m/min) For three different bed depths, Figure 3 shows the results obtained from our simulation
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for pollutant in the influent against time (in hours, h). Our Figure and that of Sanghamitra and Gupta
(2005) reveal agreement for the treated pollutant system. Therefore, our model can be used to predict the
breakthrough curves. By taking into account the porosity loss which is one factor that affects the
performance and lifetime of the granular iron media, the new model developed in this work can therefore
be used to predict long-term performance of filters.
4 CONCLUSION
From an analysis on a typical cylindrical sand/iron filter and the kinetics of iron corrosion in water, an
equation describing the spatiotemporal variation of pollutants concentration in the filter has been
developed. The numerical simulation has been used to monitor the spatial and temporal variation of the
pollutants concentration. It has been found that by taking into account the time variation of the porosity
due to the reduction of porosity by the corrosion products, the mathematical model gives results similar to
what is obtained from experimental investigations.
This study constitutes a step for a more complete design of an affordable filter that can be for
community access to safe drinking water. But, this requires further studies, as for instance, long term
experimental data of fixed bed column operations on pollutants removal by sorption onto iron/sand filter,
after the model has taken into account the transfer processes between the liquid phase (water) and the
surface of the filter (entrance side).
ACKNOWLEDGMENTS
The authors thanks to R. Ntchantcho, Hydrogeologic Centre at the National Institute of Geological and
Mining Research, Yaoundé-Cameroon, for insightful comments and fruitful suggestions.
REFERENCES
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MARLE, C.M. (2006) Henry Darcy and Fluid Flows in Porous Media. Oil & Gas Science and Technology –
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NGAI T.K.K., R.R. SHRESTA, B. DANGOL, M. MAHARJAN and S.E. MURCOTT (2007) Design for
sustainable development – Household drinking water filter for arsenic and pathogen treatment in
Nepal. Journal of Environmental Science and Health, Part A, 42, 1879 –1888.
NOUBACTEP C., S. CARÉ, F. TOGUE-KAMGA, A. SCHÖNER and P. WOAFO (2010) Extending service
life of household water filters by mixing metallic iron with sand. Clean – Soil, Air, Water, 38 (10), 951–
959.
PEEL R.G., A. BENEDEK (1980) Dual Rate Kinetic Model of Fixed Bed Absorber. J. Environ. Eng. ASCE,
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PEEL R.G., A. BENEDEK, C.M. CROWE (1981) A Branched Pore Kinetic Model for Activated Carbon
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SANGLAMITRA K. and A.K. GUPTA (2005) Analysis and modeling of fixed bed column operations on
AS(V) removal by adsorption onto iron oxide-coated cement (IOCC). Journal of Colloid and Interface
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Figure Caption
Figure 1. Schematic of analyzed system, including initial condition in the column.
Schéma du système analysé, y compris la condition initiale dans la colonne
Figure 2. Temporal distribution of the pollutants concentration at the filter exit (z=H) for different values of
the number of iron particles, for time dependent porosity (Figure 2a), and for a constant porosity (Figure
2b). nc is the threshold number of iron particles necessary to fill completely the initial volume of pores.
Distribution Temporelle de la concentration du polluant à la sortie du filtre (z=H) pour différentes valeurs
du nombre de particules du fer, pour une porosité dépendante du temps (Figure 2a), et pour une porosité
constante (Figure 2b). nc est le nombre de particules de fer nécessaire pour remplir le volume initial des
pores.
Figure 3. Breakthrough profiles for As (V) adsorption on iron oxide-coated cement (IOCC) at three
different bed heights (for the validation of our model).
Profil de l’adsorption de As (V) sur du ciment enduit d’oxyde de Fer à trois hauteurs différentes du lit (pour
la validation de notre modèle).
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vz
C=C (z, t)
r
Z=0, CA=C0, t≥0
Z
hwater
H
Sand-Iron
Mixture
Figure 1.
10 20 30 40 50 60 70 80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
C
/
C
o
Time (months)
nfe=nc
nfe=(0.98)nc
nfe=(0.96)nc
nfe=(0.94)nc
75% removal
1
2
3
(a)
Figure 2 a.
14 15 16 17 18 19 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
C
/
C
o
Time (months)
nfe=nc
nfe=(0.98)nc
nfe=(0.96)nc
nfe=0.94)nc
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(b)
Fig. 2 b
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0 10 20 30 40 50 60 70
0
0.5
1
1.5
C
/
C
o
Time (h)
Bed depth=10 cm
Bed depth=20 cm
Bed depth=30 cm
Figure 3.