Probabilistic cellular automaton model for flocculation processes in heavy metals wastewater removal

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Probabilistic cellular automaton model for flocculation processes in heavy metals
wastewater removal
M´onica Alcal´a,
1
Javier Almaguer,
2,3
Arturo Berrones,
2,3,∗
Ricardo S´anchez,
2
and Eduardo Soto
1
1
Depto.de Ingenier´ıa Qu´ımica,Facultad de Ciencias Qu´ımicas,Universidad Aut´onoma de Nuevo Le´on,
Av.Universidad S/N 66450,San Nicol´as de los Garza,NL,M´exico
2
Posgrado en Ingenier´ıa de Sistemas,Facultad de Ingenier´ıa Mec´anica y El´ectrica,
Universidad Aut´onoma de Nuevo Le´on,Av.Universidad S/N 66450,San Nicol´as de los Garza,NL,M´exico
3
Centro de Innovaci´on,Investigaci´on y Desarrollo en Ingenier´ıa y Tecnolog´ıa,UANL,PIIT,Apodaca,NL,M´exico
The effect of agitation in a jar test apparatus for heavy metals wastewater removal in the presence
of a coagulant is modeled by a Markovian probabilistic cellular automaton.The existence of homo-
geneous stationary states of the model which admit large floc formation for a range of the agitation
speeds is analitically predicted by mean field calculations.The automaton is numerically studied via
Monte Carlo simulations,giving particle size distributions that closely resemble the experiments.
PACS numbers:46.65.+g,61.43.Hv
I.INTRODUCTION
The presence of heavy metal particles in wastewater is
a major threat for human health [1,2].Inadequate treat-
ment of industrial sludge may cause the contamination
of drinking water sources and it also represents a danger
for rivers and oceanic ecosystems.Despite of this,spe-
cially in the developing world the industry generates large
amounts of poorly treated wastewater [3,4].The design
of efficient and cost effective heavy metals wastewater
removal methods is therefore an important technological
issue [1,4].In this contribution we present the analy-
sis of an experimental system designed by us in which
the heavy metals removal can be efficiently performed.
A simple stochastic model is proposed to represent the
system.Mean field theory and Monte Carlo simulations
show that the model captures important features of the
experiment.
The system under study consist in a jar test apparatus
for the removal of heavy metals fromwastewater through
coagulation and flocculation.Coagulation and floccula-
tion consist on adding a floc-forming chemical reagent
to wastewater to enmesh or combine with nonsettleable
colloidal solids to produce a rapid settling floc.The floc
is subsequently removed in most cases by sedimentation.
In water treatment,the main use of coagulation and floc-
culation is to agglomerate solids prior to sedimentation
and rapid sand filtration.Chemical precipitation,which
is closely related to chemical coagulation,consists on
precipitation of unwanted ions from a water o wastew-
ater.In water treatment,the main coagulants used are
aluminium and iron salts or polyelectrolytes.Colloidal
particles have electrostatic forces that are important in
maintaining dispersion.The surface of a colloidal par-
ticle tends to acquire an electrostatic charge due to the
ionization of surface groups and the adsorption of ions

Electronic address:arturo.berronessn@uanl.edu.mx
fromthe surrounding solution.In most colloidal systems,
the colloids are maintained in suspension as a result of
the electrostatic forces of the colloids themselves.Since
most naturally occurring colloids are negatively charged,
the colloids remain in suspension due to the action of the
repulsive forces [2,5].In the jar test apparatus under
study an agitation can be applied to the solution.Un-
der agitation,the flocculation process consists of three
stages:growth,agglomeration and break.A possible
approach to model these type of systems in order to
predict the floc size distributions could be based on the
formulation of a transport equation for population bal-
ance [5–7].This type of formulations lead to complicated
integro-differential equations,for which efficient numer-
ical solution in different relevant experimental contexts
is an active research area [5–7].Other possibility to de-
scribe such a system is through Monte Carlo simulations
of Diffusion Limited Aggregation (DLA) type models [8–
10],which may offer a detailed description at relatively
large computation cost.In this contribution we present
a different approach,based on the simulation of the ex-
perimental system by a rather abstract but extremely
simple probabilistic model.The resulting method cap-
tures the observed phenomenology of the process at a
very modest computation cost.The model is also capa-
ble to reproduce the experimental floc size distributions.
These features makes our approach interesting for the
basic understanding of the factors relevant to the design
of wastewater management facilities.
II.EXPERIMENTAL SYSTEM
The system under study consists on a jar test appa-
ratus with agitation time and speed controls.The solu-
tions to treat were prepared with salts of zinc chloride,
nickel chloride,iron chloride and chromium (III) chlo-
ride in distilled water.The heavy metal concentration
for Zn(II) was of 750 mg/L,450 mg/L for Cr(III),200
mg/L of Fe(III) and 30 mg/L of Ni(II).A two level ex-
2
FIG.1:Microscopy of the flocculation process observed at
five different agitation times:a) 1 minute,b) 3 minutes,c) 5
minutes,d) 7 minutes and e) 10 minutes.The systemconsists
on 30 mg/L of Fe(III) in distilled water,with an iron chloride
salt used like flocculant at an agitation speed of 20 rpm.
perimental design was proposed,where the factors were
the type of metal and agitation time,performing each
experiment three times,and taking averages.The heavy
metals were removed by hydroxide precipitation adjust-
ing pH=10,while shaking at 150 rpm.After that,coagu-
lant chloride iron was added with gentile agitation at 20
or 40 rpm,the pH was adjusted again to pH=10.The to-
tal agitation time was of ten minutes,taking floc samples
every minute.The floc samples were analyzed by optical
microscopy to measure the size and the distribution floc
in order to establish the kinetics of flocculation.Fromthe
Fig.(1) it appears that a situation in which large flocs are
found with relatively high frequency emerges after some
agitation time.This kind of behavior is which is useful
for wastewater treatment,because large flocs favor sedi-
mentation.A particle size distribution that is represen-
tative of the considered experimental range is presented
in Fig.(2).This is the same case as in Fig.(1):30 mg/L of
Fe(III) in distilled water,with an iron chloride salt used
like flocculant.The distribution appears to be consis-
tent with a power law.For the larger agitation time the
tail shows a more rapid decay,indicating the presence of
larger shear forces.Notice however that the formation of
10 100
S
1
10
100
1000
f ( S )
FIG.2:Particle size distribution of the experimental system
with 30 mg/L of Fe(III) in distilled water after an agitation
time of seven minutes.The solid line represents the distribu-
tion with v = 20 rpm while the dotts are for the v = 40 rpm
case.The sizes are given in µm.
large flocs can not be expected for all the possible values
of the agitation speed.In particular,a very large agita-
tion speed induce the system to be dominated by break
processes.A very gentile agitation on the other hand,
makes the floc formation unacceptably slow.Additional
tests on the experimental system under large and small
agitation speeds have been also performed.It turns out
that for agitation speeds below 5 rpmno large floc forma-
tion is observed within the 10 minutes range.In this case
a floc size distribution concentrated at large floc sizes is
obtained only after several hours.For speeds larger than
120 rpm on the other hand,only small aggregates are
formed.
III.PROBABILISTIC CELLULAR
AUTOMATON MODEL
The proposed cellular automaton model is defined on
a one dimensional lattice with periodic boundary condi-
tions.Associated with any cell j in the lattice is a state
variable h
j
,that is interpreted as an abstract floc size.
The state in a cell can change by any integer value from
iteration to iteration following a set of local rules.The
integer units are called particles.The rules that dictate
the particle movement in the lattice depend on the aver-
age floc size,defined as follows.At an initial stage,M
particles are distributed over the N sites of the lattice,
giving an average floc size
¯
h = M/N.The subsequent
evolution of the automata is given by the rules:
• If h(j) ≤
¯
h then,with probability 1−v,the cell remains
unchanged,and with probability v receives particles
from its neighboring cells (from j −1,j +1,or both).
• On the other hand,if h(j) >
¯
h then,with probability
f(1 − v),the j − th cell,receives particles from its
neighboring cells (from j −1,j +1,or both),and with
probability 1 − f(1 − v) transfer the excess particles
3
(above average) to any of the neighboring cells j − 1
or j +1.
In this way the parameter v represents the degree of in-
stability of large clusters:at v → 1 only the flocs with
sizes below or equal to the average are stable.The pa-
rameter f on the other hand,acts like a stabilization
factor for large aggregates:depending on f it’s more or
less probable that a large floc grows in a given site from
on time step to the next,at v < 1.It is now shown
by a mean field analysis that the emergence of homoge-
neous stationary states with a positive probability of floc
formation with sizes larger than
¯
h is expected from the
proposed evolution rules.We first introduce the following
definitions.
• y
j
= h
j
−h ≡ deviation from the average number of
particles at site j.
• State +:y
j
> 0.State −:y
j
≤ 0.
• P
+
j
(t) ≡ P
t
(y
j
> 0).
• P

j
(t) ≡ P
t
(y
j
≤ 0).
• v ≡ P(+|−).
• b = f(1 −v) ≡ P(+|+).
• P
±
j
(t) > 0 positivity condition for any time and site.
• P
+
j
(t) +P

j
(t) = 1 normalization condition.
In the mean field approximation the automaton’s dynam-
ics is described by the Markovian model,
P
+
j
(t +1) = bP
+
j
(t) +vP

j
(t) (1)
P

j
(t +1) = (1 −b)P
+
j
(t) +(1 −v)P

j
(t) (2)
It’s convenient to introduce the ratio
θ
j
(t) =
P
+
j
(t)
P

j
(t)
(3)
for which the dynamics description is reduced to a single
equation,
θ
j
(t +1) =
v +bθ
j
(t)
1 −v +(1 −b)θ
j
(t)
(4)
From this equation follows that a homogeneous station-
ary state exists for any point in the f −v plane,
f(v;θ) =
1
θ

1
θ
1 −θ
1 −v
.(5)
The stability of these stationary states is studied by in-
troducing a small perturbation around a given stationary
value θ
s
,
θ(t) = θ
s
+δ(t) (6)
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
f
v
FIG.3:Stationary states for the ratios (from left to right)
θ
s
≡ θ = 0.01,0.1,0.2,0.5,0.7,0.9,0.99,0.999.
0
1
2
3
4
5
0
0.2
0.4
0.6
0.8
1

v
FIG.4:Relaxation times for the stationary states with ratios
(from left to right) θ
s
≡ θ = 0.2,0.4,0.999.
Introducing (6) in (5) it turns out that,
δ(t) =
￿
1 −
￿
1 +
1
θ
s
￿
v
￿
t
δ
0
,(7)
from which a characteristic relaxation time τ =
−1/ln(|r|) is given in terms of
r(v;θ
s
) = 1 −
￿
1 +
1
θ
s
￿
v,(8)
The isoprobability curves in the f −v plane for the ra-
tios θ
s
≡ θ = 0.01,0.1,0.2,0.5,0.7,0.9,0.99,0.999 are
given in the Fig.(3).The relaxation times for θ
s

θ = 0.2,0.4,0.999 are shown in the Fig.(4).It turns
out that the stationary states with smaller values of the
ratio θ
s
tend to have larger relaxation times.The gen-
eral behavior of the automaton is now studied through
Montecarlo simulations.In Fig.(5) the floc size distri-
butions for a lattice with N = 1000 sites,
¯
h = 4 and
v = f = 0.25 are plotted for two simulation times.The
graph in Fig.(5)a is at iteration 1000 and in Fig.(5)b is at
iteration 10000.The figure suggest that for the param-
eter values v = f = 0.25 and
¯
h = 4 a stationary state
is reached at a finite amount of time (around 1000 iter-
ations).The aggregates spread over a range of sizes up
to two orders of magnitude above the average floc size.
4
1 10 100
S
1
10
100
1000
f ( S )
( a )
1 10 100
S
1
10
100
1000
f ( S )
( b )
FIG.5:Particle size distributions at two different simulation
times (left:1000 iterations.Right:10000 iterations) for the
parameters v = f = 0.25.
50 100 150 200
S
0
10
20
30
40
50
60
70
f ( S )
( a )
0 50 100 150 200
S
0
10
20
30
40
f ( S )
( b )
FIG.6:Particle size distributions at two different simulation
times (left:1000 iterations.Right:10000 iterations) for the
parameters v = 0.01 and f = 1.
A plot of the histograms in the same lattice but with
v = 0.01 and f = 1 is presented in Fig.(6).The graphs
showhowthe systemslowly diverges fromit’s initial state
in which flocs sizes were concentrated around
¯
h = 4.The
histograms for a system with v = 0.9 and f = 0.1 is pre-
sented in Fig.(7).In this case the automaton quickly
converges to a state in which the majority of the aggre-
gates have a size only slightly above the average.
The numerical results agree with the mean field cal-
culations.In particular,because of mass conservation,
stationary states with very large flocs imply a small ratio
θ
s
.Both the mean field theory and the simulations indi-
cate that these type of stationary states are only attained
after long relaxations times for values close to f →1 and
v → 0.The shortest relaxion times are associated to
ratios θ
s
→ 1,with f → 0 and v → 1,however these
stationary states give flocs only barely above the average
size.The model also predicts the existence of stationary
states with cluster size distributions that span a range of
sizes well above the average.These states are associated
with intermediate relaxation times and ratios,which is
consistent with the experiments.
IV.CONCLUSION
A simple probabilistic model for floc formation in the
presence of agglomeration and break processes has been
presented.Mean field analysis and Monte Carlo sim-
0 2 4 6 8 10
S
0
100
200
300
400
500
600
f ( S )
( a )
0 2 4 6 8 10
S
0
100
200
300
400
500
600
700
f ( S )
( b )
FIG.7:Particle size distributions at two different simulation
times (left:1000 iterations.Right:10000 iterations) for the
parameters v = 0.9 and f = 0.1.
ulations show that this very economic model captures
essential features of the floc size distribution observed
in an experimental system for heavy metals wastewater
removal.
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