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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