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CHEMICAL ENGINEERING TRANSACTIONS
VOL. 43, 2015
A publication of
The Italian Association
of Chemical Engineering
Online at www.aidic.it/cet
Chief Editors: Sauro Pierucci, Jiří J. Klemeš
Copyright © 2015, AIDIC Servizi S.r.l.,
ISBN 978-88-95608-34-1; ISSN 2283-9216
On the Modelling of Particles Aggregation in Disperse
Reaction-Diffusion Systems
Arnold M. Brener*a, Ablakim S. Muratovb, Vladimir G. Golubeva
aState University of South Kazakhstan, Tauke Khan, 5, Shymkent, 160012, Kazakhstan
bUniversity “Miras”, Ilyaeva str., 3, Shymkent, 160012, Kazakhstan
amb_52@mail.ru
The work deals with the theoretical foundation and development of the engineering methods for calculating the
kinetics of the aggregation of insoluble products in chemical apparatus in terms of joint chemical reactions and
coagulation processes in the working zone of reactors. The models of aggregation of a disperse phase in
systems with chemical reactions have been developed. Regularities of aggregation process in systems with a
chemical source of first and second orders of insoluble phase monomers have been studied.
1. Introduction
Currently, the use of chemical apparatuses and reactors formation, aggregation and sedimentation of
insoluble phases in the working volume of the chemical apparatuses becomes more and wider, particularly in
the range of modern processes (Liu, 2008). In many cases processes of chemical technologies are
accompanied by formation of the new solid disperse phase. It can be phase transition, as in cases of
crystallization or desublimation, or it can be formation of low soluble substances during chemical reactions
(Voloschuk and Sedunov, 1975). As a whole, it is possible to allocate a lot of directions of a modern science
dealing with processes and apparatuses in which the problems of the kinetics and dynamical characteristics of
reactors with formation of a dispersed solid phase in a working zone are relevant (Blackman and Marshall,
1994). They are: 1) purification of gas pollution from fine particles and dust (Friedlander, 2000); 2) production
of nano-dispersed powders for constructional and functional bioceramics; 3) creation of sorbents, catalysts,
drugs and molecular grids with given structure (Bodmeier and Paeratakul, 1989); 4) creation of methods for
optimal engineering of technological processes dealing with a method of chemical sedimentation (Logan,
2012); 5) elaboration of polymeric films for molecular covering the products of chemical industry (Menon and
Pego, 2004). At the same time, the known theoretical models of aggregation of dispersed phase and
sedimentation are of little use for engineering calculations, as they are too complex and involve the need to
use a set of parameters some of which are difficult for finding (Spicer and Pratsinis, 1996).
In this paper, we propose a simple mathematical model, which opens, in our opinion, the prospects for
creating a calculation method of aggregation of insoluble solids in systems with chemical reactions of pseudo-
first and of second orders. As the result we can obtain expressions for calculating the evolution both of the
total number of clusters in the system and of the average cluster order with allowance for the kinetic
constants of the given chemical reaction and elements of the aggregation of the matrix.
2. First-order reaction
Let us consider the first-order reaction occurring in solution according to a conventional scheme
BA → (1)
In accordance with a relaxation approach (Brener, 2006), the kinetic equation for the reaction (1) can be
written as (Brener et al., 2009)
DOI: 10.3303/CET1543136
Please cite this article as: Brener A., Muratov A., Golubev V., 2015, On the modelling of particles aggregation in disperse reaction- diffusion
systems, Chemical Engineering Transactions, 43, 811-816 DOI: 10.3303/CET1543136
811
11
0
1 )()exp( dttA
tt
dt
dA t
c
c
−
−−=
τ
η (2)
For sufficiently small chemical relaxation time, we get the following expression for the concentration of the
reaction product (Boehm et al., 1998)
( )( )ctAB τ−−= exp10 (3)
Further, let us there exists a process of the nucleation of an insoluble dispersed phase of the reaction product
in the system
CB → (4)
Primary nucleation process is quite complex and theoretically poorly described, although his analysis is
involved by many researchers (Ernst, 1986). However, here we will describe the kinetics of this process,
approximately, namely, using the delay time of formation of the insoluble phase with respect to the time of
formation of the reactants for a certain period of nucleation. The time of formation of the equilibrium
concentration of the reaction product in the solution is not accounted. It seems acceptable for poorly soluble
substances (Slemrod, 1990).
Thus, the following kinetic equation can be written
−
−==
Δ
=
c
n
cn
tA
dt
dB
dt
Bd
dt
dC
τ
τ
τ
exp
)( 0 (5)
Here cτ is the typical relaxation time of chemical reaction, and nτ is the typical time of primary nucleation.
Smoluchowski equation for coagulation of the insoluble phase with account of the chemical source reads as
follows (Brener, 2011)
( )ttCtCNtCtCN
dt
dC
i
j
jijijij
i
j
jij
i Φ+−=
∞
=
−
−
=
− )()()()(2
1
1
,
1
1
, (6)
( )ttCtCN
dt
dC
j
jj 1
1
1,1
1 )()( Φ+−=
∞
=
(7)
Assume for simplicity, that during the primary nucleation only monomers of the insoluble phase can arise
(Wong et al., 2009), i.e.
( ) 0
1
=Φ
≠
t
i
i (8)
From this assumption it follows (Davies et al., 1999)
−
−+−=
∞
= c
n
cj
jj
tA
tCtCN
dt
dC
τ
τ
τ
χ exp)()( 0
1
1,1
1 (9)
Let us introduce the generating function of the form (Wattis, 2006)
)exp()(),(
~
1
iztCztC
i
i −=
∞
=
(10)
In the case of constant coagulation kernels, the kinetic equation in terms of the generating function takes the
form
( ) ( ) ( ) ( ) ( )iztztCtCztC
t
C
i
i −Φ+−=∂
∂
∞
=
exp,
~
0,
~
,
~
2
1
~
1
(11)
812
( ) ( ) ( )
−
−
−+−=
∂
∂
z
tA
ztCtCztC
t
C
c
n
c τ
τ
τ
χ exp,
~
0,
~
,
~
2
1
~
0 (12)
Let us introduce now the designations (Yu Jiang and Hu Gang, 1989a)
00
~
=
== zi CCM ;
0
1
~
=
∂
∂
==
z
i z
C
iCM (13)
Thus, the kinetic equation for a total concentration of clusters of different orders takes the form of (Yu Jiang
and Hu Gang, 1989b)
( )
−+−=
cc
cn tAM
dt
dM
ττ
ττ
exp
exp
2
1 02
0
0 (14)
For obtaining the analytical solution of Eq. (14) it is convenient to introduce transformed variable s and the
special control parameter E0
( )cts τ−= exp ;
( )
c
cnAE
τ
ττexp0
0 = (15)
Thus Eq. (14) can be rearranged to the form of Riccati equation (Bittanti et al., 1991)
0
2
0
0
2
EM
sds
dM c −=
τ
(16)
The solution of Eq. 16 can be expressed through Bessel functions 0I and 0K (Jin and Jjie, 1996)
U
dsdUs
M
cτ
2
0 −= ; ( ) ( )sEKUsEIUU cc ττ 002001 22 += (17)
The found solution has theoretical interest but it is unlikely useful for practical calculations (Leyvraz, 2003).
Figures 1, 2 depict some results of numerical experiments for calculating the evolution of the total clusters
concentration of various orders in the system during the aggregation in the system with monomers forming by
the first order reaction scheme.
Figure 1: Evolution of the total clusters concentration of insoluble phase in the first order system for a long
time (E0=1- 80; 2- 200; 3-800)
R
el
at
iv
e
cl
us
te
rs
c
on
ce
nt
ra
tio
n
Dimensionless time
1
2
3
813
Figure 2: Evolution of the total clusters concentration of insoluble phase in the first order system for
small time (E0=1- 80; 2- 200; 3-800)
While processing the results of numerical experiments, it was found that the time moments, at which extrema
of the total number of clusters in the system of first-order reaction can be observed, are determined on the
parameter 0E :
286,0
0max 437,0)/(
−= Et τ (18)
3. Second-order reaction
Let us consider the scheme of second-order reaction as follows
CBA →+ (19)
The total concentration of reactants reads
000 BA +=ρ (20)
After obvious rearrangements the kinetic equation takes the following form
( )C
dt
dC
k
dt
Cd
2022
2
−−= ρ (21)
Thus, we obtain the expression for the evolution of the product concentration
( )( )( )
( )( )tBAkAB
tBAkBA
C
00200
00200
exp
exp1
−−
−−
= (22)
If 00 BA = then
( )tk
tk
C
2
2
2
0
0
22 ρ
ρ
+
= (23)
If the reaction product is insoluble, it begins again the aggregation of primary monomers according to the
mechanism described above, but it occurs with the other type of chemical source of monomers:
1
2
3
R
el
at
iv
e
cl
us
te
rs
c
on
ce
nt
ra
tio
n
Dimensionless time
814
( ) ( )( )( )
( )( )( ) 200200
002
2
00002
1
1,1
1
)/exp(
exp
)()(
cn
cn
j
jj
tBAkAB
tBAkBABAk
tCtCN
dt
dC
ττ
ττ
−−−
−−−
+−=
∞
=
(23)
In order to analyze of this problem it can be used the same method as before. The kinetic equation for a total
concentration of clusters of different orders takes now the form
( )( )( )
( )( )c
c
tBAEAB
tBAEBA
M
dt
dM
τ
τ
/exp
/exp1
2
1
00000
000002
0
0
−−
−−
+−= (24)
Eq. (24) can be also resolved with the help of Bessel functions. This method however leads to the very bulk
relations. So, it is more expedient apparently to use numerical experiments as before. The more so, that in this
case we succeeded to distinguish also the convenient control parameter.
( )
−−=
c
nBAE
τ
τ
000 exp (25)
Figure 3 depicts some results of numerical experiments for calculating the evolution of the total clusters
concentration of various orders in the system with the second-order chemical source of monomers.
Figure 3: Evolution of the total clusters concentration of insoluble phase in the second- order system for
a small time (E0=1- 800; 2- 1000; 3-4000)
The graphs show that the order of the reaction does not affect essentially the qualitative character of the
evolution of the total number of clusters. It may be noted only a slower decline in the total number of clusters
after the peak of coagulation.
4. Conclusions
The paper discusses the modified equations for calculating the kinetics of aggregation of the dispersed phase
in the chemical reaction system based on the Smoluchowski equation for the binary coagulation. The basic
process control parameters of the aggregation process in the systems with a chemical source of insoluble
monomer phase of first and second-orders have been determined, and their validity has been confirmed with
the help of numerical experiment. It was shown that order of chemical reaction did not change the character of
aggregation process of insoluble product of the reaction. Particularly, in both cases the time moment, which is
corresponded to the peak of the total number clusters of insoluble phase in the system, is determined by the
1
2
3
R
el
at
iv
e
cl
us
te
rs
c
on
ce
nt
ra
tio
n
Dimensionless time
815
one control parameter. Relations for calculating this parameter however are different depending on the order
of chemical reaction.
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