 MULTI-SCALE APPROACH TO POPULATION BALANCE
MODELLING OF DISPERSE SYSTEMS
Béla G. Lakatos
Institute of Process Engineering, University of Pannonia, Egyetem Street 10, Veszprém, Hungary
Keywords: Disperse systems, Multi-scale modelling, Population balance, Crystallization, Two-population model.
Abstract: A three-scale model is presented and analysed using the multi-scale methodology of complex systems. The
micro-scale model is formulated as a set of stochastic differential equations for the individual disperse
objects and it is shown that the population balance equation, containing also terms describing collision
interchange of extensive quantities between the disperse elements is a meso-scale model of disperse
systems. The macro-scale model is formulated by means of the moments of internal quantities. As an
example a two-population model, governing the coupled behaviour of crystals and fluid elements is
presented for describing micromixing in solution crystallization.
1 INTRODUCTION
Disperse systems of chemical engineering, contain-
ing large numbers of individual interacting dispersed
objects such as solid particles, liquid droplets or gas
bubbles, or often combinations of those are nonequi-
librium and (usually) nonlinear multi-phase systems.
Their characteristic property is that, not depending
on the nature of disperse elements, a number of in-
teracting size and time scales can be distinguished
and identified therefore these systems, in principle,
belong to the class of complex multi-scale systems .
The multi-scale structure of disperse systems has
been considered relating to different modelling and
computational problems (Eberard et al., 2005; Ing-
ram and Cameron, 2002; Li et al., 2004;
Wei, 2007),
and also in context of population balance models.
(Mazzotti, 2010; Lakatos, 2010). However, the mul-
tiscale nature of the population balance models of
disperse systems has not been analysed in details
yet.
In this paper the population balance models of
disperse systems are analysed applying the multi-
scale methodology of complex systems. The micro-
scale model is formulated as a set of stochastic dif-
ferential equations for the individual disperse objects
with collision interactions and it is shown that the
population balance equation, containing also terms
describing interchange of extensive quantities bet-
ween the disperse elements is a meso-scale model of
disperse systems. The macro-scale model is for-
mulated using the moments of internal quantities. As
an example a two-population model, governing the
coupled behaviour of crystals and fluid elements is
presented for describing micromixing in reaction
crystallization.
2 MICRO-SCALE MODEL
Consider a large population of interacting disperse
objects, solid or fluid particles moving stochastically
in a continuous carrier phase. Let us assume that 1)
the extensive quantities carried by the disperse ele-
ments, such as mass of chemical species and heat are
distributed homogeneously inside those, or internal
motion of those is irrelevant regarding the behaviour
of system; 2) the continuous phase is modelled on
kinetic scale, i.e. by means of concentrations of che-
mical species; 3) collision interactions of disperse
elements may cause their coalescence, aggregation,
breakup as well as interchange of extensive quanti-
ties between the colliding elements. Therefore the
micro-scale of the disperse system can be assigned
by the individual disperse elements.
Assuming that
p
x
and
p
u
denote, respectively,
the space coordinates and velocities along those of a
disperse element, v
p
denotes its volume,
p
c
stands
for the vector of concentrations of K0 relevant
chemical species inside the disperse elements and T
p
denotes its temperature. Then the state of a disperse
186
Lakatos B..
MULTI-SCALE APPROACH TO POPULATION BALANCE MODELLING OF DISPERSE SYSTEMS.
DOI: 10.5220/0003620701860191
In Proceedings of 1st International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH-2011), pages
186-191
ISBN: 978-989-8425-78-2
c
2011 SCITEPRESS (Science and Technology Publications, Lda.) element at time t is given by the vector
()
8
,,,,
+
K
ppppp
Tv Rcux
, and, introducing the sim-
plified notation
(
)
pppp
Tv ,,cχ =
, the micro-scale
model of disperse system, completed with the model
equations of the continuous phase is given by the
following set of stochastic differential equations
()
(
)
dtttd
pp
ux =
(1)
()
(
)
()
()( )
+
++=
X
pppppu
pppp
dddv
tddtdttdm
p
χuxu
Wσfuαu
u
,,,
,
N
Ξ
(2)
()
(
)
()
()( )
+
+=
X
ppppp
ppp
ddd
tddttd
p
χuxχu
Wσχβχ
χ
χ
,,,
,
N
Ξ
(3)
where
f
are deterministic forces,
(
)
tW is a
multivariable Wiener process, α and β are determin-
istic functions, σ
u,p
and σ
χ,p
are the diffusion mat-
rices, function
N
determines the conditions of col-
lisions between the disperse elements while func-
tions
()
,
Ξ
describe the velocity, volume, concentra-
tion and temperature changes induced by collisions.
The set of differential equations (1)-(3) describes
the behaviour of the population of disperse elements
entirely by tracking the time evolution of the state of
each disperse object individually. The first terms on
the right sides of Eqs (2) and (3) describe the
deterministic and stochastic disperse element-con-
tinuous environment interactions, i.e. motion of the
disperse elements induced by the continuous carrier
as well as the mass and heat exchange between the
disperse elements and continuous environment. The
integral terms in Eqs (1)-(3) represent jump-like sto-
chastic changes of the internal properties induced by
collisions, i.e. jump-like changes of the velocities,
volumes, concentrations and temperatures of the
disperse elements.
The system of stochastic equations (1)-(3) in-
duces a Markov process
() () ()
{
}
0
,,
t
ppp
ttt χux
with
continuous sample paths and finite jumps (Gardiner,
1983, Sobczyk, 1991). Taking it into consideration,
a multidimensional population density function
()
t,,, χux
()
tn ,,,
ˆ
χux is defined where the vari-
ables
(
)
ppp
χux ,,
are measured on the
()
χux ,, co-
ordinates, and
()
χuxχux dddtn ,,,
ˆ
provides the num-
ber of disperse elements at time
t in the domain
()
χχuuxxχux ddd +++ ,,;,, . The population
density function
()
tn .,
ˆ
provides the state function of
population.
Then, in analogy with the transition probability,
a transition measure and, in turn, the conditional
transition measure
(
)
.,.,.;
ˆ
tsP
c
can be derived (Laka-
tos, 2010) by means of which variation of the state
function of population of disperse elements is de-
scribed by the transformation
()
()
(
)
()
()
()
X
XX
ˆ
,,;
"'"'"',",","
ˆ
,',','
ˆ
",",",',,
;',',',
ˆ
1
,,,
ˆ
ˆˆ
>
×
×
=
∫∫
χux
χχuuxxχux
χuxχuxχux
χuxχux
st
ddddddsn
snt
sP
s
tn
c
N
(4)
where
)
)
=
X
ˆ
,,,
ˆ
χuxχux dddsnsN
(5)
denotes the number of disperse elements in the given
domain at time
s.
In Eq.(4), expression
()
()
""",",","
ˆ
1
χuxχux dddsn
sN
(6)
is interpreted as the probability that there exists a
disperse element in the state domain
(
,","," χux
)
"","","" χχuuxx ddd
+
+
+
possibly interacting with a
disperse element of state
(
)
',',' χux and the result of
this interaction event is expressed by the conditional
transition measure
c
P
ˆ
.
Eq.(4) is an integral equation formulation of the
population balance equation of interacting disperse
elements. It provides a global description of the
population of disperse elements but appears to be
unpractical in computations since identification of
the multivariate conditional transition measure
(
)
.,.,.;
ˆ
tsP
c
is a crucial problem. However, Eq.(4) is
an important intermediate state in developing the
integral-differential equation form of the population
balance equation.
3 MESO-SCALE MODEL
Let us now define an
ε
-environment around the
position vector x in the physical space as
{
}
ε
ε
<
=
),(:)( yxyx dU
(7)
where
ε>
0 and d(x,y) denotes t he distance of two
vectors.
Assuming that
1) The
ε
-environment contains sufficient number of
disperse elements for defining a population density
function as
MULTI-SCALE APPROACH TO POPULATION BALANCE MODELLING OF DISPERSE SYSTEMS
187 () ( )
()
∫∫
=
x
yuχuyχx
ε
U
ddtntn
U
,,,
ˆ
,,
(8)
by means of which
()
()
χχx
x
dtn
t
,,
,
1
N
(9)
is interpreted as the probability that there exists a
disperse element in the
ε
-environment in the domain
(χ,χ+
dχ) where
()
t,xN provides the total number
of these elements in the
ε
-environment at time t.
2) The transport of the populations of disperse
elements is governed by the convection-dispersion
model, then
the system is governed by the spatially distributed
multi-dimensional population balance equation
()
()
[]
() ()
[]
()
[]
()
[]
()
[]
()
[]
tntn
tntn
tntn
tn
t
tn
,,,,
,,,,
,,,,
,,
,,
χxχx
χxχxG
χxuχxD
χx
χx
χχ
xxxxx
2K
a
1
b
2K
i
MM
M
B
+
+
++
+
+
=
(10)
The first term on the right hand side of Eq.(10)
represents the source of the disperse elements born
in the continuous phase. The next two terms de-
scribe the transport of the population density func-
tion in the physical phase while the fourth term
represents the rate of change of population density
function due to continuous phase-disperse elements
interactions
()
[]
()
= tn
dt
d
tn ,,,, χx
χ
χxG
χχχ
(11)
The next terms, in turn, represent the rates of change
of the population density function because of the
direct mass and/or heat exchange between the dis-
perse elements, breakage and aggregation/coales-
cence of disperse elements induced by collision
events.
The second integral in the expression
()
[]
()
()( )
()
+
+
×
×
+
=
∫∫
∫∫
+
"'
)'(
)(
1
"',",,,',,)(
"
)'(
)(
1
,,
0
1
0
,
0
1
0
'
,
χχ
ωp
χχ
ωp
χχχxχxω
χχ
ωp
χχ
ωp
χ
χ
cc
ω
cc
χ
mm
mm
υυ
υ
υ
υυ
υ
δ
π
υυ
δ
π
00
00
m
m
v
v
v
v
t
dddtntvndF
t
tvn
N
N
2K
i
M
(12)
()( )
"',",,,',,)( χχχxχxω
ω
dddtntvndF
υ
υ
υ
×
provides the rate of increase of the number of dis-
perse elements having internal variables χ due mass
and heat interchange between colliding disperse
elements with volumes
v and υ having internal
variables χ’ and χ” while the first integral describes
the rate of the number of decrease due to similar
events. In Eq.(12) ω denotes a random vector with
conditional probability distribution function
(.)
υ
ω
F
describing the extent of equalization of intensive
variables under the condition that a disperse element
of volume
υ is colliding with one of volume υ,
()
υ
π
,v
is the frequency of collisions of such disperse ele-
ments, while
)(tN
expresses the total number of dis-
perse elements in the
ε-environment of x. In Eq.(12),
the components of parameter vector
υ
p take the
form
(
)
υ
υ
+
=
vvp .
The rate of increase of the number of disperse
elements because of breakage is expressed by the
second integral of the term
()
[]
()
()
()
()
()( )
()
()
()
()
()( )
"',
ˆ
,",,
ˆ
,',
"'',
1
"',
ˆ
,',,
ˆ
,,
',"
1
,
ˆ
,,
0
",'
'.
00
υυυυ
πυυυφ
υυυπ
υυφ
υ
υ
ddtntn
Sv
t
ddtntvn
vSv
t
tvn
mm
mm
vv
v
vbb
v
vv
bb
χxχx
χxχx
χx
×+
×=
∫∫
∫∫
N
N
1
b
M
(13)
where
(
)
"'
υυ
b
S
is the probability of breakage of dis-
perse element of volume
υ’ collided with one of vol-
ume
υ”, and
(
)
',
υ
φ
v
b
denotes the ratio of disperse
elements of volume
v resulted from breakage of
disperse elements of volume
υ’. The first integral
provides the rate of decrease due to similar events.
Here it was assumed that the effect of a breakage
event on the extensive quantities carried by the
disperse elements is negligible. Note that writing Eq.
(13) the notation
(
)
χχ
ˆ
,v
=
was used.
When the breakage of disperse elements occurs
because of their collisions with some solid surface
or, as in the case of fluid droplets and bubbles in-
duced by turbulence then Eq.(13) can be written in
the form
()
[]
()
()
()
∫∫
×=
Θ
θ
πθυφ
v
vbb
vSvtvn
0
,
,",
ˆ
,, χx
1
b
M
(14)
()() ()
()
()
()()
',
ˆ
,',
'',",
ˆ
,,
,'
υθυπ
θυυφυθ
θθ
Θ
θ
ddFtn
SvddFtvn
v
v
v
bb
m
χx
χx
∫∫
×+
where
θ is a random variable with the probability
distribution function
F
θ
(.) characterizing the fre-
quency of random events inducing breakage of the
disperse elements.
SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and
Applications
188 In the case of aggregation or coalescence, the
rate of increase of the number of disperse elements
is given by the second integral of the term
()
[]
()
()
()( )
()
()
()()
υυυπ
δ
υυ
υυ
δπ
υ
υυ
υυ
υ
υ
υ
dtntvn
vS
ddtntvn
vS
tvn
v
v
a
v
v
a
mm
mmm
,"
ˆ
,,,'
ˆ
,,
"'
)'(
,
'
ˆ
,'
ˆ
,,,
ˆ
,,
"
)'(
,
,
ˆ
,,
.
0
0
',
χxχx
χχ
p
χχ
p
χχxχx
χχ
p
χχ
p
χx
cc
χ
cc
χ
×
+
+
×
+
=
+
∫∫
∫∫∫
00
00
2K
a
M
(15)
where
()
υ
υ
,vS
a
denotes the probability of agglo-
meration or coalescence of the colliding elements
having volumes
v-υ and υ. Here it is assumed that
agglomeration or coalescence of two disperse ele-
ments leads to full equalization of their intensive
quantities therefore ω=
1.
4 MACRO-SCALE MODEL
A mathematical model of a disperse system, not
depending on the method of development must
satisfy the requirements of the first principle models,
i.e. the balances of conservative extensive quantities
have to be fulfilled as concerns the whole system.
From one side this requirement provides strong lim-
itations on the forms of the constitutive expressions
of Eqs (10)-(15). From the other side, the population
density function playing a centred role in the meso-
scale model (10)-(15) is not an extensive quantity by
itself. Extensive quantities for the population of dis-
perse elements needed for the balances are formu-
lated only by the multivariate joint moments of
internal variables defined as
() ( )
=
X
χχxx dtnTcct
m
k
K
k
l
mkkl
K
K
,,...,
1
1
1,...,
υμ
(16)
As a consequence, for instance, the total mass in-
volved in the population of disperse elements is
given as
() ()
tvtm ,, xx
ρ
=
(17)
where
()
tv ,x
denotes the mean volume of the po-
pulation expressed as
() ()
=
X
χχxx dtn
t
tv ,,
)(
1
,
υ
N
(18)
while
ρ is the density of the disperse elements.
Similarly, the total heat involved in the popu-
lation is expressed as
() ()
=
X
χχxx dtTn
t
C
tTC ,,
)(
,
N
ρ
ρ
(19)
while
C is the heat capacity of the disperse elements.
Finally, the mean value of
k
th
species is given as
()
()
()
=
X
χχx
x
x dtnc
t
tc
kk
,,
,
1
,
N
(20)
As an example the heat balance from Eq.(10)
takes the form
(
)
() ()
[]
() ()
[]
+
=
XX
B χχxχχx
xuxD
x
xxxxx
dtnTCdtn
dt
dT
C
tQtQ
t
tQ
,,,,
,,
,
ρρ
(21)
Where
(
) ()
tTCttQ ,)(, xx
ρ
N=
(22)
The third term on the right hand side represents
the population-continuous phase interaction, while
the last term describes the heat effect of source of
disperse elements.
5 TWO-POPULATION MODEL
The multidimensional population balance equation
(10)-(15) is cognitive-type model of populations of
disperse elements. When modelling real processes
appropriate reductions of this cognitive-type model,
the so called purpose-oriented models are applied.
These models contain necessary and sufficient infor-
mation to provide an adequate description of the
process to be modelled. Here as an example a two-
population model is presented aimed to describe
micromixing in reaction crystallization.
Crystallization from solution is an important
fluid-solid disperse system in which the disperse
phase is formed by solid particles. Let us assume
that the crystallizer is isothermal and the
supersatura-tion is generated by the chemical reac-
tion
+ CBA
. When the continuous phase can
be treated in the
ε
-environment of x as a homo-
geneous continuum with respect to scalar quantities
then the composition and temperature environment
is the same for all crystals. Consequently, crystals
born and growth in the same composition and tem-
perature environment hence the behaviour of crys-
tals can be modelled adequately by using the popula-
tion balance equation (10). This crystallizer is con-
sidered perfectly mixed on micro-scale.
When, however, solution exhibits varying in time
spatial inhomogeneites of scalars even on micro-
scale then these changes modulate not only the pro-
MULTI-SCALE APPROACH TO POPULATION BALANCE MODELLING OF DISPERSE SYSTEMS
189 cesses in solution (mixing, reaction, nucleation) but
also the crystal-solution and crystal-crystal interact-
tions since crystals moving randomly in the fluid
phase meet diverse composition and temperature re-
gions even inside the
ε
-environment of x. This phe-
nomenon can be modelled making use of the gener-
alized coalescence/re-dispersion (gCR) model de-
veloped in the context of multi-scale structure of
(Lakatos, 2008; Lakatos et al., 2011).
In the gCR model, the Kolmogorov-scale eddies
of solution are treated as a large population of fluid
elements having identical volume
v
η
. Then, two dif-
ferent interacting populations are identified in the
crystallizer, i.e. the population of crystals and that of
fluid elements. Since, however, kinetic processes are
also determined by micro-scale phenomena, the rate
expressions of crystallization kinetics may be in-
fluenced significantly by stochastic interactions of
these two populations. Therefore, the mathematical
model of the crystallizer consists of two population
balance equations and of averaged kinetic and cons-
titutive equations describing their interactions.
Since the concentration of species in crystals is
negligible then the population density function of
crystals is given as
() ()
()
()
=
X
χχc dtxnTTdvtvxn ,,,,
δδ
(23)
while that of fluid elements is defined as
()
()
(
)
()
=
X
χχcc dtxnTTvvdtxp ,,,,
δδ
η
(24)
where
T denotes the temperature of process.
Assuming that 1) the motion of populations in
the vessel is described by the 1D axial dispersion
model, 2) no breakage of crystals occurs then the
population balance equation for crystals is given as
()
()()
[]
()()
()( ) ()
τξδτξ
ξ
τξ
ξ
τξ
τξτξ
∂τ
τξ
,,
,,,,1
,,,,
,,
2
2
ann
BtvvBt
vnvn
Pe
v
vnvG
t
vn
++
=
+
(25)
while that for fluid elements takes the form
()
()
()( )
[]
()
[]
()()
ξ
τξ
ξ
τξ
τξ
τξτξ
τξ
ττ
τξ
+
=
+
,,,,1
,,
,,,
,,
,,
2
2
cc
c
c
c
c
c
c
pp
Pe
pt
c
pR
t
p
d
dp
c
3
i
M
(26)
where
Lx=
ξ
is the dimensionless axial coordi-
nate scaled with the length of vessel
L while
tt
=
τ
denotes the dimensionless time scaled with
the mean residence time
t
. In Eqs (25)-(26),
R
denotes the mean rate of consumption of solute in
solution due to nucleation and growth of crystals,
G
,
n
B
and
a
B denote, respectively, the
rates of crystal growth, nucleation and agglomera-
tion, all averaged over the ensemble of fluid ele-
ments in the
ε
ξ
-environment of ξ. These kinetic
equations take the forms as follows.
()()
=
m
c
cc
0
dpvccGG
ec
p
τξ
,,,,
1
N
(27)
where c
c
and c
e
denote the solute and the equilibrium
saturation concentrations,
()
()()
dvdvnp
ccBB
m
ecn
np
n
cc
c
τξτξ
,,,,
,
1
0
×
×=
∫∫
0
NN
(28)
where
N
p
and N
n
stand for the total numbers of fluid
elements and crystals in the
ε
ξ
-environment of ξ.
The rate of agglomeration is given by a simpli-
fied form of Eq.(15) since the system is assumed to
be isothermal, and there is no mass exchange bet-
ween the crystals:
()
()
()( )()
()
()
()()()
cc
cc
c
c
ddvpvnvvn
ccS
ddvpvnvn
ccSB
m
m
vvveca
p
vv
eca
p
a
',,,',,',
,
2
1
',,,',,,
,
1
0
','
',
0
τξτξτξ
π
τξτξτξπ
×
×+
×
×=
∫∫
∫∫
0
0
N
N
(29)
In general case, the model equations of the two-
population model (25)-(29) can be solved only by
numerical method, but when the exponents
b and g
are positive integers and the agglomeration kernel
provides closed moment terms then, since the mo-
ment terms of micromixing operation are always
closed (Lakatos, 2011), a closed finite set of moment
equations can be obtained. Indeed, assuming the fol-
lowing forms for the intrinsic kinetic rates
1
a
S ,
(
)
1
3
μ
ecbb
cckB =
,
()
()
'
0',
vvb
vv
+=
π
and apply-
ing the cumulant-neglect closure (Lakatos, 2010) a
set of 13 ODE’s was generated for the joint mo-
ments of concentrations
c
a
, c
b
and c
c
and for the raw
moments of crystal volume
v up to the second order.
Figure 1 presents the time evolution of axial dis-
tribution of the number of crystals for the case of
small seeding and micromixing intensity 0.15. In
this case the feed of species
A and B was segregated.
Figure 1 shows well the effects of increasing nuclea-
SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and
Applications
190 Figure 1: Time evolution of the axial distribution of popu-
lation density function of crystals for Pe=10 and
1
=
t .
tion rate as the state of mixing was increased along
the axial coordinate. At the same time, the number
of crystals is decreased due to crystal agglomeration.
Figure 2 shows the effects of micromixing in-
tensity on the steady state axial distribution of crys-
tals illustrating the strong influence of micromixing
on the number of crystals produced in the process.
Figure 2: Effects of micromixing intensity on the steady
state axial distribution of population density function of
crystals for Pe=10 and
1=t .
6 CONCLUSIONS
The population balance approach was applied to de-
velop a three-scale model for disperse systems using
the multi-scale methodology of complex systems. It
was shown that marking out the individual disperse
objects for microlevel of the system the population
balance equation, containing also terms describing
collision interchange of extensive quantities between
the disperse elements and motion in the physical
space is a meso-scale model. In this interpretation,
the macro-scale model is formulated by means of the
moments of internal quantities of disperse elements.
As an example, a two-population model, governing
the coupled behaviour of crystal and fluid element
populations is presented for describing micromixing
in reaction crystallization.
ACKNOWLEDGEMENTS
This work was supported by the Hungarian Scien-
tific Research Fund under Grant K 77955 which is
gratefully acknowledged.
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