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A new model for the dynamics of dispersions in a batch reactor: theory and numerical simulation

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Meccanica
37:
221–237, 2002.© 2002
Kluwer Academic Publishers. Printed in the Netherlands.
A New Model for the Dynamics of Dispersions in a Batch Reactor:Numerical Approach
⋆
ALBERTO MANCINI and FABIO ROSSO
University of Firenze, Department of Mathematics ‘Ulisse Dini’, Viale Morgagni, 67/a; 50134 Firenze, Italy
(Received: 23 April 2001; accepted in revised form: 16 January 2002)
Abstract.
In this paper, we develop some considerations concerning the structure of coalescence, breakage andvolume ‘scattering’ kernels appearing in the evolution equation related to a new model for the dynamics of liquid–liquid dispersions and show some numerical simulations. The mathematical model has been presented in [3, 4],whereaproof of theexistence anduniqueness for aclassical solutiontotheintegro–differential equation describingthe physical phenomenon is provided as well as a complete analysis of the general characteristics of the integralkernels. Numerical simulations agree with experimental data and with the expected asymptotical behavior of thesolution.
Key words:
Liquid–liquid dispersions, Integro–differential evolution equation, Coagulation–fragmentation mod-els, Numerical simulations.
1. Introduction
Dispersed phase systems and reactions among the component phases are of great importancein chemical, mining, food, and pharmaceutical industries.The literature on dispersions is considerably large and the list at the end of this paper is justa small part of it. Indeed the subject is challenging and still full of unanswered questions, mostof which related to the physics of the droplet mutual interactions. Essentially we distinguishtwomain interactions: coalescence and breakage. Both are complicated processes and the ﬁnaldistribution in a stirred vessel is governed by the dynamic equilibrium between coalescenceand breakage rates. It is generally believed that breakage is essentially a mechanical effect.However, turbulence adds considerable complication to a clear understanding of the wholeprocess. Coalescence is equally complex, appearing related not only to colliding trajectoriesof droplets but also to mechanics of the surrounding ﬁlm of the continuous phase betweenthem. This means that two colliding droplets may not coalescence if the energy involvedis not sufﬁcient to break the thin liquid ﬁlm of continuous phase. From the mathematicalpoint of view the system is described as a population of droplets characterized by its volume
⋆
This work was partially supported by the G.N.F.M. Strategic Project ‘Metodi Matematici in Fluidodinamica eDinamica Molecolare’.
222
Alberto Mancini and Fabio Rosso
distribution function. The standard procedure for modeling its evolution is to take into ac-count all local interactions and operating conditions via suitable forms of the integral kernelscharacterizing each kind of mutual interaction or decay. Thus, the complicated dynamicalprocess that determines the instantaneous conﬁguration of droplets is drastically simpliﬁed,all delicate matter concerning local interactions being transferred to an appropriate design of the relevant integral kernels. In principle the volume distribution function may depend uponthe space variables too, but this possibility is neglected in most industrial applications, both forthe high agitation speeds usually involved at regime and also for practical reasons: a perfecthomogeneous product is commonly the ﬁnal goal when preparing a dispersion.However, at least in principle, the mechanism is such that iterated coalescence events mayproduce bodies of arbitrarily large size. This is manifestly against both physical intuition andexperimental observation.Recently a new model for the dynamics of stirred liquid–liquid dispersions has been pro-posed. The main ideas are reported in [3, 4] and other developments of that approach havebeen worked out in [1]. In particular [1] is concerned with the possibility that the breakagekernel has a singularity when the droplet volume reaches a critical top level while in [3, 4]only bounded kernels are considered.Our approach to the problem of modeling the dynamics of stirred dispersions was quitedifferent from that proposed in most of the literature on this subject (we refer in particularto [5–7, 9, 10]): we assume – as in [11, 12] and consistently with experiments and most of the technical papers – that there exists a
ﬁnite
top size limit for allowable droplets, say
v
o
,which can be determined, at least in principle, when the operating conditions (stirring speed,characteristics of the two ﬂuids, vessel design, etc.) are ﬁxed. The problem is that, regardlessof whether the breakage kernel has or not a singularity, when the droplet size approaches
v
o
, coalescence would naturally lead to the occurrence in the system of droplets of arbitrarysize unless a speciﬁc ‘cut-off’ is imposed to the coalescence kernel. The question is then if it is possible to set up a model encompassing in some natural and physical consistent waythe decay of too large droplets. Indeed this instability is guaranteed by a new effect that wecalled
volume scattering
and which is just a suitable combination of the two main interactions(breakage and coalescence): scattering can be roughly described as the coalescence of twodroplets of sizes
u
and
w
which result in an unstable droplet with
u
+
w>v
o
followed by animmediate rupture into droplets both within the admissible range. Thus, the instability of largedroplets is naturally embedded into the dynamics without the necessity to make any artiﬁcialassumption about the form of the kernels. This implies agreat ﬂexibility and is the true noveltywith respect to other previous papers on this subject.Another feature introduced in [3, 4] is the presence, in the equation governing the dynam-ics, of a general
efﬁciency factor
, depending in a functional way on the unknown volumedistribution via the total number of droplets and the total interfacial area. Efﬁciency factorsare widely used in the technical literature, but they are always explicitly set as known func-tions of the relevant parameters and generally embedded in the structure of the two mainkernels.Mathematically speaking the model consists of an integro–differential equation for thevolume distribution, in which the scattering process is represented by terms resembling thecollisional operator in Boltzmann’s equation. In [1, 3, 4] it is proved that the correspondingCauchy problem is well posed and the solution exists for all times.The present paper is mainly devoted to a numerical investigation in order to present somegraphical simulations of the evolution of droplet size distribution with time. Simulations turn
A New Model for the Dynamics of Dispersions in a Batch Reactor
223out to be consistent with the physics of stirred dispersions and show clearly that, also fora coalescence kernel which does not vanish at
v
o
, droplets of arbitrary large volume cannotappear in the system
regardless of breakage efﬁciency
. Moreover, the asymptotic distributionturns out to be insensitive (consistently with experiments) to the shape of the initial one,depending only on its integral, that is, the percentage of dispersed phase.Another important fact which is emphasized by simulations is that the contributions of coalescence, breakage and volume scattering to the rate of change of the distribution functionare of comparable size, thus conﬁrming that volume scattering has the same relevance as theother two phenomena.
2. Mathematical Model
Let us summarize the mathematical models of [3, 4]: we ﬁrst deﬁne
f(v,t)
(
v
=
volume,
t
=
time) to be the distribution function of droplet size (per unit volume of dispersion). Weassume droplets to be uniformly distributed in the reactor so that
f
does not depend on spatialcoordinates. This is quite reasonable when the imposed shear rate is sufﬁciently high and theviscosity of the dispersion sufﬁciently low
1
. We also assume that the system is isolated, sothat there is no heat or mass exchange effects.Thus, it is possible to formulate the following evolution equation:
∂f ∂t
=
φ(t)(L
c
f
+
L
b
f
+
L
s
f ),
(2.1)where
L
c
,
L
b
,
L
s
denote the coalescence, breakage an scattering operator respectively and
φ
isthe efﬁciency factor. Equation (2.1) models the three interaction processes that make
f
evolvewith time. Let us describe their action in detail.(a)
Efﬁciency factor.
We set
φ(t)
=
[
N
(t),
S
(t)
], where
N
(t)
=
v
o
0
f(v,t)
d
v,
S
(t)
=
v
o
0
v
(
2
/
3
)
f(v,t)
d
v,
(2.2)represent respectively the number of droplets and the interfacial area per unit volume, and
(
N
,
S
)
is a given continuous function. The underlying idea is that all interactions are drivenby the internal power dissipation which in turn is related to both
N
and
S
.(b)
Coalescence operator.
We set
L
c
f(v,t)
=
v/
20
τ
c
(w,v
−
w)f(w,t)f(v
−
w,t)
d
w
−−
f(v,t)
v
o
−
v
0
τ
c
(v,w)f(w,t)
d
w,
(2.3)where
τ
c
(v,w)
represents the (binary)
coalescence kernel
. This is proportional to the prob-ability that two colliding droplets of respective volumes
v
and
w
coalesce to form a uniquedroplet of volume
v
+
w
. A natural choice is to assume that the probability of coalescence
1
When these conditions are not satisﬁed, droplets generally show observable convective motions due to buoy-ancy and gravity combined. However, most of research on dispersions focuses upon the ﬁnal droplet distribution
under perfect mixing condition
.
224
Alberto Mancini and Fabio Rosso
is proportional to the total cross-sectional area times a coalescence efﬁciency of exponentialtype (as in [2]), although other forms are possible (see [6]), that is
τ
c
≃
exp
−
v
1
/
3
+
w
1
/
3
v
1
/
3
w
1
/
3
4
(v
1
/
3
+
w
1
/
3
)
2
,
(2.4)where the proportionality factor depends on physical and rheological parameters.Clearly
τ
c
(v,w)
has to be symmetric:
τ
c
(v,w)
=
τ
c
(w,v).
(2.5)In the ﬁrst term, which represents the gain rate at level volume
v
, the arguments of
τ
c
havethe ordering
w<v
−
w
, otherwise coalescence events, giving rise to a droplet of volume
v
,would be counted twice. Instead, in the second term, representing loss rate from the same cellin the volume space, we must consider coalescence of droplets of volume
v
with droplets of any other size
w
such that
v
+
w
v
o
.(c)
Breakage kernel.
Let us set
L
b
f(v,t)
=
v
o
v
α(w)β(w,v)f(w,t)
d
w
−
α(v)f(v,t),
(2.6)where
τ
b
(w,v)
=
α(w)β(w,v), (w > v),
(2.7)represents the
breakage kernel
. Here
β(w,u)
d
u
is the probability that a droplet of volumein the interval
(u,u
+
d
u)
is generated by breakage of a droplet of volume
w
. We assume,for simplicity, that a single breakage event cannot produce more than two droplets (
binary
breakage). In this case
u
and
w
−
u
have the same probability, that is
β(w,u)
=
β(w,w
−
u),
(2.8)and
β(w,u)
=
0
,
if
w
u.
(2.9)Accordingly
β(w,u)
is normalized as follows
w/
20
β(w,v)
d
v
=
1
.
(2.10)(d)
Scattering kernel.
Let us set
L
s
f(v,t)
=
v
o
v
o
−
v
d
w
(v
+
w)/
2
v
+
w
−
v
o
τ
c
(u,v
+
w
−
u)β(v
+
w,v)f(u,t)
××
f(v
+
w
−
u,t)
d
u
−
f(v,t)
v
o
v
o
−
v
τ
c
(v,w)f(w,t)
d
w,
(2.11)where
τ
s
≡
βτ
c
represents the
scattering
kernel regulating interactions of pairs of dropletswhose cumulative volume exceeds
v
o
. This term is proportional to the frequency of collisionsresulting in the following volume re-distribution: a pair
(v,w)
such that
v
+
w>v
o
producesby coalescence a droplet which immediately decays into a pair
(u,v
+
w
−
u)
. Physically
A New Model for the Dynamics of Dispersions in a Batch Reactor
225this amounts to say that, due to
τ
s
, large droplets (with volume greater than
v
o
) produced bycoalescence are unstable. We called this process
volume scattering
for its analogy with thecollision term in Boltzmann equation. This interpretation is the reason of the choice
τ
s
(v,w
;
v
+
w,u)
=
τ
c
(v,w)β(v
+
w,u).
(2.12)The function
β(s,u)
for
s
∈
(v
o
,
2
v
o
)
preserves the symmetry property (2.8) and is nor-malized so that
s/
2
s
−
v
o
β(s,u)
d
u
=
1
.
(2.13)The last term is nothing but the continuation of the loss term in the coalescence operatorand is obtained by performing the integration with respect to the variable
u
, thanks to (2.13).The integration intervals are consistent with the range of
v,w
and the requirement that theoutcome of scattering is a pair in the admissible range.Equation (2.1) has to be completed with an initial condition
f(v,
0
)
=
f
o
(v)
on
[
0
,v
o
]
andsolved in the region
[
0
,v
o
]×[
0
,T
]
, where
f
is required to be continuous.Bearing in mind the introduced notation we have to ﬁnd a numerical method for thesolution of the Cauchy problem to the equation (2.1).Let us ﬁnally recall all the main properties of the kernels and state some of the
a priori
qualitative results (conservation of volume during the evolution and conservation of the num-ber of droplets under scattering only) that will be deﬁnitely of crucial relevance in the valida-tion of numerical results.The basic requirements on the data (
τ
c
, α, β, φ
) are:(i)
τ
c
(v,w)
is a positive continuous symmetric function on
[
0
,v
o
]×[
0
,v
o
]
; we put max
[
0
,v
o
]×[
0
,v
o
]
τ
c
=
τ
c
(<
+∞
)
.(ii)
β(w,v)
for
w
∈
(
0
,v
o
]
is a non-negative function continuous such that
β(w,v)
=
0
,
for
v
∈ [
w,v
o
]
and
v
∈ [
0
,v
b
]
,β(w,v)
=
β(w,w
−
v),
for
v
∈ [
0
,w
]
,
where
v
b
0 (
v
b
/v
o
≪
1) is the volume of the largest stable (i.e. unbreakable) drops inthe dispersion, usually proportional to We
−
1
.
8
, where We is the Weber number (see [11,13]). For
w
∈
(
0
,v
o
]
,
β(w,v)
is normalized as described by (2.10).(iii)
β(s,v)
for
s
∈
(v
o
,
2
v
o
]
has the same properties as above, with the normalization de-scribed by (2.13).(iv)
α(w)
is a continuous function in
[
0
,v
o
]
which vanishes in
[
0
,
2
v
b
]
and is increasing for
w>
2
v
b
; we put max
[
0
,v
o
]
α
=
α(<
+∞
)
(for an extension to unbounded breakage frequency,see [1]).(v)
φ
is a bounded and Lipschitz continuous function.We recall the basic properties proved in [3].PROPOSITION 1.
Let
f(v,t)
be a non-negative solution of equation
(2.1)
. Then
(a)
v
o
0
L
c
f(v,t)
d
v
0
,
that is coalescence tends to decrease the number of droplets per unit volume.

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