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8/3/2019 Simulation of Reacting Moving Droplet
1/18
Chemical Engineering Science 61 (2006) 6424 6441www.elsevier.com/locate/ces
Simulations of mass transfer limited reaction in a moving droplet to studytransport limited characteristics
Kiran B. Deshpande, William B. Zimmerman
Department of Chemical and Process Engineering, University of Sheffield, Newcastle Street, Sheffield S1 3JD, UK
Received 9 November 2004; received in revised form 7 June 2006; accepted 7 June 2006
Available online 13 June 2006
Abstract
Transport limited heterogeneous reactions with asymmetric transport rates in the non-reacting phase can exhibit an interesting switch in theconcentrations of the reactants in the reacting phase from one limiting reactant to the other. This switch, called cross-over [Mchedlov-Petrossyan
P.O., Khomenko G., Zimmerman W.B., 2003a. Nearly irreversible, fast heterogeneous reactions in premixed flow. Chemical Engineering Science
58, 30053023; Mchedlov-Petrossyan P.O., Zimmerman W.B., Khomenko G.A., 2003b. Fast binary reactions in a heterogeneous catalytic
batch reactor. Chemical Engineering Science 58, 26912703], relates to the optimum design of the tubular reactor as all the reactants in
the reacting phase are completely consumed at cross-over. The cross-over phenomenon, which has been studied by a number of researchers
using phenomenological modelling, is investigated here by developing a distributed model using level-set simulations, in order to explore the
possibility of the existence of cross-over in the frame of reference of a moving droplet. Cross-over occurs for a droplet moving due to buoyancy
with asymmetric transfer rates of the reactants in the non-reacting phase and an instantaneous reaction occurring inside the droplet (reacting
phase). The cross-over length obtained using the level-set simulation is found to be within 0.78% of that obtained using the phenomenological
model. Computational experiments are performed by varying the ratios of the initial concentrations of the reactants and the transfer rates of
the reactants, in order to obtain the parametric region for the existence of cross-over which is also compared with the theoretical prediction.
2006 Elsevier Ltd. All rights reserved.
Keywords: Simulation; Reaction engineering; Drop; Mass transfer
1. Introduction
Transport limited heterogeneous reactions are useful in the
chemical industry, e.g. propylene oxide manufacturing plants,
where propylene chlorohydrin is a precursor to propylene ox-
ide (Warnecke et al., 1999). In this system, gaseous chlorine
and propylene react in the aqueous phase, and due to fast re-action in the liquid phase, mass transfer of the two reactants
from the gas phase to the liquid phase is a controlling step. Si-
multaneous absorption of two gases reacting in the liquid phase
has been extensively studied theoretically by a number of re-
searchers (Roper et al., 1962; Ramachandran and Sharma, 1971;
Corresponding author. Tel.: +44114 2227517; fax: +44114 2227501. E-mail address: [email protected] (W.B. Zimmerman).
0009-2509/$- see front matter 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2006.06.013
Chaudhari and Doraiswamy, 1974; Juvekar, 1974; Hikita
et al., 1977; Zarzycki et al., 1981; Bhattacharya and Chaud-
hari, 1972). However, relatively little attention has been paid
to simulating mass transfer across moving drops (Petera and
Weatherley, 2001; Lakehal et al., 2002).
In the present work, two reactants are dissolved in a contin-
uous phase but do not react with each other due to possible sol-vation effects opposing catalysis. Instead, the reactants diffuse
into a dispersed phase whose solvent is catalytic. The possible
reactions following the above scheme are discussed in detail in
Deshpande and Zimmerman (2005a) and Deshpande (2004).
Transport limited heterogeneous reactions with asymmetric
transfer rates in the non-reacting phase can exhibit an interest-
ing switch, called cross-over, in the excess concentrations of
the reactants in the reacting phase from one limiting reactant
to the other. The origin of cross-over is attributed to the asym-
metric transfer rates of the two reactants from the continuous
http://www.elsevier.com/locate/ceshttp://-/?-http://-/?-mailto:[email protected]://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-mailto:[email protected]://-/?-http://www.elsevier.com/locate/ces8/3/2019 Simulation of Reacting Moving Droplet
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K.B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 61 (2006) 6424 6441 6425
U
CA + B
A
BC
A
B
Fig. 1. Schematic of a tubular reactor with premixed reactants A and B entering with superficial velocity U. The outlet stream contains product C and unreacted
A and B.
phase to the dispersed phase and is experimentally studied
in Deshpande and Zimmerman (2005a, b). The existence of
cross-over relates to the optimum length of the tubular reactor,
since localization of the reaction occurs near cross-over and the
molecular efficiency is also higher in the vicinity of cross-over.
The cross-over phenomenon is a good indicator for optimizing
the reactor length as the conversion level was found to be 99%
at a distance five times longer than that is required for cross-
over to occur for a tubular reactor (Mchedlov-Petrossyan et al.,
2003a). A similar result is reported for a batch reactor in terms
of modified Thiele modulus (mTm) by Mchedlov-Petrossyan
et al. (2003b). The greatest molecular efficiency at the point
of cross-over occurs in a broader sense as all the molecules of
both reacting species that arrive at the dispersed phase react si-
multaneously. In general, Mchedlov-Petrossyan et al. (2003a)
show that product yields are greater per unit reactor length
when cross-over occurs than when it does not. This feature is a
generalization of using an excess reagent, suitably tailored for
unequal mass transfer coefficients. Mchedlov-Petrossyan et al.
(2003a) demonstrate cross-over with a bulk parameter model
for a dispersed phase of droplets, as shown in Fig. 1. Our
premise is that if the cross-over phenomenon originates fromasymmetric transport rates and occurs in the bulk, it should
occur in an isolated droplet, with a distributed model. The pur-
pose of this paper is to demonstrate the occurrence of cross-
over in the frame of reference of a single droplet with detailed
simulations of diffusion and reaction.
The paper is organised as follows: First the phenomenolog-
ical model, proposed by Mchedlov-Petrossyan et al. (2003a),
is implemented in order to demonstrate localization of the
reaction at cross-over and then modified by incorporating the
effect of accumulation of excess reactant in the dispersed phase
in Section 2. The present article mainly aims to investigate if
cross-over occurs for a single buoyancy driven droplet. Hence,cross-over is further investigated in Section 3 by developing a
distributed model for mass transfer with a strictly irreversible
reaction inside a moving droplet using the level-set simulations,
which are described in detail in Deshpande and Zimmerman
(2005c) and Deshpande (2004). The cross-over length pre-
dicted using the phenomenological model and the distributed
model is compared and the theoretical criterion (proposed by
Mchedlov-Petrossyan et al., 2003a) for the phenomenologi-
cal model to estimate the parameter space for existence of
cross-over is tested for the distributed model in Section 4. A
distributed model is further applied for a nearly irreversible
reaction in Section 5, in order to bring out the salient feature
of cross-over highlighting localization of the reaction.
2. Theory
The transport of reactants in a heterogeneous system in-
cludes physical processes such as bulk convection, bulk dif-
fusion, mass transfer of reactants from one phase to the other
and chemical reaction. The transport mechanism of the reac-
tants capturing all these processes for a binary reaction can be
represented as a six variable system which includes species
conservation equations (PDEs) for the reactants and prod-
uct, and reaction constraints (algebraic equations). A theo-
retical approach proposed to simplify a six variable system
in terms of intermediate variables can be found in detail in
Mchedlov-Petrossyan et al. (2003a). Here, the governing equa-
tions for modelling of axial transport of reactants and product
and their numerical implementation are discussed. The compu-
tations of Mchedlov-Petrossyan et al. (2003a) are reproduced
here and then compared with the 2-D level-set simulations.
2.1. Numerical implementation
The system of PDEs and algebraic equations is written for a
steady state as
UjCA
jz= DA
j2CA
jz2 Aa(CA CA,s ), (1)
UjCB
jz= DB
j2CB
jz2 B a(CB CB,s ), (2)
UjCC
jz= DC
j2CC
jz2 C a(CC CC,s ), (3)
Aa(CA CA,s ) = B a(CB CB,s ), (4)
Aa(C
A C
A,s)=
C
a(CC
CC,s
) (5)
and
CA,s CB,s = KC C,s . (6)The above set of three PDEs and three algebraic equations can
be numerically solved using a FEM based commercial software,
FEMLAB. Eq. (6) is a consequence of an assumption that the
reaction is very fast, as compared to local mass transfer and this
assumption for catalytic reaction is relaxed in Zimmerman et al.
(2005). Since the concentration of the reactants and the product
varies in the z-direction only, the system is one-dimensional.
The parameter a is the active surface area per unit volume of
the reactor. There are two reactants, A and B, and a product, C,
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6426 K.B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 61 (2006) 6424 6441
Fig. 2. The change in bulk concentration of reactants A and B along the
length of a 1-D tubular reactor.
and hence six variables: bulk concentration ofA, B and Cwhich
is represented as CA, CB and CC , respectively, and surface
concentration ofA, B and Cwhich is represented as CA,s ,CB,sand CC,s , respectively.
In FEMLAB, the PDEs (Eqs. (1)(3)) and the algebraic equa-
tions (Eqs. (4)(6)) are solved using the convectiondiffusion
application mode. Both the reactants and the product are con-
vected by a velocity, U, equal to 0.5 c m s1, and the diffusioncoefficient of two reactants in the bulk phase (DA and DB ), av-
eraged over cross-section of the tubular reactor, is considered
to be the same and is equal to 1cm2 s1. Kinetic asymmetryis introduced through different mass transfer coefficients (A,
B and C ). The Aa of the reactant A is considered to be
0.2 s1 and that of the reactant B (B a) is considered to be1s1. The reaction is considered to be nearly irreversible withequilibrium constant, K, equal to 105. The initial concentra-tion of the reactants A and B is taken as 1 and 0.4 M, respec-
tively. The concentration of the reactants A and B at the inlet
boundary is assumed to be equal to their initial concentration,
and the boundary condition, convective outflow, is used at the
outlet boundary. Although the results here duplicate those of
Mchedlov-Petrossyan et al. (2003a), the use of finite element
methods to treat the boundary value problem is inherently morestable than the shooting method used by those authors, which
required a stiff solver.
The bulk concentration of the reactants A and B smoothly
decreases along the length of the reactor, as shown in Fig. 2.
The decrease in the bulk concentration of both the reactants is
due to the reaction occurring in the dispersed phase, which is
assumed to be in a pseudo-steady state with mass transfer. A
more interesting feature is observed when the surface concen-
tration of the reactants, A and B, is plotted along the length of
the reactor, as shown in Fig. 3. The surface concentration of
the reactant B is found to be in excess of reactant A during the
initial stages, whereas the surface concentration of reactant A
is found to be in excess of reactant B at later stages. The shift
Fig. 3. The change in surface concentration of reactants A and B along the
length of a 1-D tubular reactor.
in the excess surface concentration of the reactants is termed as
cross-over phenomenon, which is studied in detail and rep-
resented later in this article.
2.2. Asymptotic theory
The model equations represented in the previous section with
suitable initial and boundary conditions form a six variable
system which includes three PDEs and three nonlinear algebraic
constraints. Mchedlov-Petrossyan et al. (2003a) proposed an
asymptotic theory in order to simplify the six variable system byrecasting the system with different dependent variables in terms
of bulk and surface concentrations that reflect key physical
processes such as bulk diffusion and mass transfer.
Mass transfer phenomenon is always driven by difference
in chemical activity, which is usually equivalent to concen-
tration difference. Since mass transfer transients die quickly,
steady state attains for the system. Bulk concentration and sur-
face concentration can be expressed in terms of supersaturation,
, as
CA = CA,s +m
n1, (7)
CB = CB,s + (8)and
CC = CC,s +m
n2. (9)
1 and 2 are the ratios of mass transfer coefficients, A/Band C /B , respectively, and m and n are stoichiometric coeffi-
cients of the two reactants, A and B, respectively. Supersatura-
tion concentration, , calculated using the numerical approach
discussed in the previous section, is plotted along the length of
the reactor as shown in Fig. 4. Supersaturation concentration
is calculated using the same bulk and surface concentrations
shown in Figs. 2 and 3 and for the same operating conditions.
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K.B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 61 (2006) 6424 6441 6427
Fig. 4. The change in supersaturation concentration which is defined in terms
of bulk concentration of reactant A and mass transfer coefficients of reactantsA and B (Eq. (7)), along the length of a 1-D tubular reactor.
The decay rate of supersaturation is found to be discontinu-
ous at the point of cross-over and is also found to be relatively
smaller just before cross-over than that after cross-over, indi-
cating more product formation before cross-over. We will apply
similar definitions of supersaturation to study its behaviour for
simulations of mass transfer across a moving droplet later in
this article.
In order to build the algebraic constraints into the differential
equations, auxiliary variables were introduced. The concept of
auxiliary dependent variables is adopted to study a diffusionlimited heterogeneous reaction where the reactants are initially
separated (Zimmerman et al., 1999). A six variable system was
thus reduced to three PDEs in terms of auxiliary variables and
supersaturation concentration, but still contained the surface
concentration terms. In order to present the surface concen-
tration in terms of auxiliary variables, modified Thiele-moduli
(mTm) as proposed by Mchedlov-Petrossyan et al. (2003a, b),
were introduced as
F1 =m2/1CA,s
n2/CB,s + m2/1CA,s + 1/2CC,s, (10)
and
F2 =1/2CC,s
n2/CB,s + m2/1CA,s + 1/2CC,s. (11)
The mTm are obtained by differentiating the auxiliary variables
and then solving the derivatives with a nonlinear algebraic con-
straint (Eq. (6)). The intermediate variables obtained are termed
modified Thiele moduli as they are ratios of rates of different
physical mechanisms analogous to the classical Thiele mod-
ulus. The novel theoretical technique discussed here was first
applied by Mchedlov-Petrossyan (1998) in a solid state mix-
ture for the precipitation reaction. The mTms are analysed here
to illustrate the abrupt change which occurs at cross-over. The
system of the equations transformed in terms of the saturation
Fig. 5. The change in modified Thiele modulus, F1, which is defined in
terms of the surface concentration of reactants A and B and product C and
the mass transfer coefficients of reactants A and B and product C, along the
length of a 1-D tubular reactor. The abrupt level change of F1 corresponds
with cross-over.
Fig. 6. The change in modified Thiele modulus, F2, which is defined in
terms of the surface concentration of reactants A and B and product C and
the mass transfer coefficients of reactants A and B and product C, along the
length of a 1-D tubular reactor.
variable, , and mTm, F1 and F2, eliminating all the concen-
tration variables is comparatively easier to visualize than a six
variable system.
Modified Thiele moduli, F1 and F2, are calculated using the
numerical scheme discussed in the previous section for the same
operating conditions, and is shown in Figs. 5 and 6, respectively.
The mTm undergo a sudden change at the cross-over length
which is equal to 1.5 cm, as shown in Fig. 3. Modified Thiele
modulus, F1, shifts from its value equal to 10 at the cross-over
length approximating a Heaviside step function. The modified
Thiele modulus, F2, also has an interesting feature: a local
maximum at the point of cross-over. Since F2 is defined in
terms of the surface concentration of product C, the maximum
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6428 K.B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 61 (2006) 6424 6441
in F2 indicates that there is greatest molecular efficiency near
the cross-over region where reactants A and B and product C
co-exist to the greatest extent.
It should be noted that the results we have discussed here
are obtained for a nearly irreversible reaction. The mTm, F2, is
always zero for a strictly irreversible reaction, which is less in-
formative. We will extend our analysis of mTm to simulationsof mass transfer across a moving droplet to explore the pos-
sibility of cross-over and to study the behaviour of mTm near
the cross-over length.
2.3. Modified approach
In the numerical implementation of the theory (Eqs. (1)(6)),
which is proposed to study transport limited heterogeneous re-
action, equal mass transfer flux of the two reactants and the
product was a necessary consequence of quasi-stationary state
of local mass transfer and surface character of instantaneous
reaction which led to non-accumulation of reactants in the re-active domain. The phenomenological model, as discussed in
Section 2.1, results in the switch in the surface concentration
of the reactants as shown in Fig. 3. In the above mentioned fig-
ure, we can see that the surface concentration of the reactant
with higher transfer rate is finite at the entrance of the reactor
indicating that the dispersed phase is occupied by the reactant
with higher transfer rate. Since the phenomenological model is
for a quasi-stationary state, transients die out quickly. In order
to investigate the transient behaviour of the system, we replace
the constraints (Eqs. (4)(6)) considered in the theory discussed
in the previous section by incorporating accumulation of the
reactants and the product.
Convection, diffusion and mass transfer of the reactants in thebulk phase are captured by using the same system of PDEs (Eqs.
(1)(3)) as used in the previous section. However, the algebraic
constraints used in the theory (Eqs. (4)(6)) are replaced by a
set of PDEs as follows:
UjCA
jz= DA
j2CA
jz2 Aa(CA CA,s ), (12)
UjCB
jz= DB
j2CB
jz2 B a(CB CB,s ), (13)
U
jCC
jz = DCj2CC
jz2 C a(CC CC,s ), (14)
UjCA,s
jz= Aa(CA CA,s ) k(CA,s CB,s KC C,s ), (15)
UjCB,s
jz= B a(CB CB,s ) k(CA,s CB,s KC C,s ) (16)
and
UjCC,s
jz= k(CA,s CB,s KCC,s ) Ca(CC,s CC). (17)
The constraints we have used in this modified approach in
terms of PDEs capture the accumulation of the reactants and
Fig. 7. The change in surface concentration of reactants A and B along the
length of a 1-D tubular reactor.
the product in the dispersed phase as they comprise the mass
transfer flux of the reactant from the continuous phase to the
dispersed phase and the disappearance of the reactants due to
chemical reaction in the dispersed phase. The emergence of
Eqs. (15)(17) is provided in the Appendix section.
The above set of equations is solved in FEMLAB using the
convectiondiffusion application mode. Both the reactants and
the product are convected by a velocity equal to 0.5 c m s1,and the diffusion coefficient of two reactants is considered to
be the same and is equal to 1 cm2 s1. Kinetic asymmetry is in-
troduced through different mass transfer coefficients. The Aaof the reactant A is considered to be 0.2 s1 and that of thereactant B (B a) is considered to be 1 s
1. The reaction is con-sidered to be very fast and almost irreversible with equilibrium
constant, K, equal to 105. The reaction rate constant, k, for theforward reaction is considered to be 106 M1 s1. The initialconcentration of the reactants A and B is considered to be 1 and
0.225 M, respectively. The above values are chosen in order to
induce kinetic asymmetry, as reported in Mchedlov-Petrossyan
et al. (2003a, b). The initial surface concentrations of the reac-
tants are taken as zero and there are no traces of the product
at the initial stage. The concentrations of the reactants and the
product at the inlet boundary are assumed to be equal to theirrespective initial concentrations, and the boundary condition,
convective outflow, is used at the outlet boundary. It should
be noted that Eqs. (15)(17) are essentially more complicated
than their mTm counterparts, and the mTm theory is not par-
ticularly simpler for the system of equations (12)(17) than the
original transformed variables. One important question is the
extent to which the mTm theory for Eqs. (1)(6) leads to ro-
bust predictions of cross-over length even in systems with accu-
mulation (Eqs. (15)(17)) rather than steady interphase fluxes
(Eqs. (4)(6)).
The surface concentrations of the reactants A and B are plot-
ted along the length of the reactor, as shown in Fig. 7, where
the interesting effect of incorporating the accumulation of the
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K.B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 61 (2006) 6424 6441 6429
reactants and the product can be seen. At the inlet of the re-
actor, the surface concentrations of both the reactants vanish,
which is more appropriate to this application than that obtained
using the previous theory (Mchedlov-Petrossyan et al., 2003a).
Thus, we capture the presence of a boundary layer for accumu-
lation using this modified approach, where the surface concen-
tration of the faster reactant first increases, attains a peak andthen decreases due to chemical reaction. The distance required
for the faster reactant to attain the peak value corresponds to
the boundary layer thickness. The mTm theory proposed by
Mchedlov-Petrossyan et al. (2003a) does not seem to capture
the existence of this accumulations boundary layer. The cross-
over length obtained using the previous theory (Fig. 3) and
that obtained after incorporating accumulation (Fig. 7) are dif-
ferent because these simulations are performed under different
parametric conditions (the ratio of the initial concentrations of
the reactants is different). One should note that since the en-
trance boundary layer is not short, it requires different para-
metric conditions for the existence of cross-over and hence,
the predicted cross-over length (0.48cm) differs from mTm
theory (1.5 cm).
3. Numerical simulations
The phenomenon of cross-over has been theoretically studied
in a 1-D tubular reactor for cases where the reactants are initially
segregated (Zimmerman et al., 1999), for premixed reactants in
a tubular reactor (Mchedlov-Petrossyan et al., 2003a) and for
premixed reactants in a batch reactor (Mchedlov-Petrossyan et
al., 2003b). The existence of a cross-over region for initially
segregated reactants is due to the fact that surface concentra-
tions of reactants are trivially separated and hence the cross-over region is limited to diffusion of reactants. The existence
of a cross-over region for a premixed case is not obvious a pri-
ori, but is attributed to kinetic asymmetry in the transfer rate
of reactants. It should be noted that since the above mentioned
theoretical study for a premixed flow was performed over a
1-D tubular reactor for multiple droplets, bulk concentrations
of reactants and surface concentrations of reactants were aver-
aged over the respective cross-sectional area. If the existence of
a cross-over region is indeed due to kinetic asymmetry, cross-
over would probably exist for a single moving droplet too, since
this is the dilute limit of small phase fraction.
We discuss a numerical scheme to simulate mass transfer of
two reactants from a continuous phase to a single droplet rising
due to buoyancy, using the level-set methodology and extend
the same further to explore the possibility of cross-over for a
single moving droplet in this section, by introducing kinetic
asymmetry in the transfer rate of the two reactants in the
continuous phase.
3.1. Level-set method
The level-set approach was originally introduced by Osher
and Sethian (1988) to simulate the motion of an incompressible
two phase flow and is discussed in detail in Deshpande and
Zimmerman (2005c). Here, we discuss the governing equations
of the level-set method that are used to compute velocity vector
for a buoyancy driven droplet.
3.1.1. Governing equations
The motion of the interface is captured by solving the ad-
vection equation of the level-set function, represented as
j
jt+ u = 0. (18)
The velocity vector, u, required for the advection of the
level-set function, is obtained by solving the incompressible
NavierStokes equations,
ju
jt (u + (u)T) + (u )u + p = F (19)
and
u = 0, (20)
where is density, is viscosity, F is a body force term and pis pressure.
The above equations are discussed in detail in Deshpande and
Zimmerman (2005c) along with re-initialization of the level-set
function which is required to conserve the mass. The velocity
vectors obtained using the level-set method can be used to
simulate mass transfer across a moving droplet. Since the effect
of change in concentration on momentum equations is assumed
to be negligible, we can solve the governing equations of the
level-set method and mass transfer equations separately.
3.2. Mass transfer across a moving droplet
Mass transfer with chemical reaction occurs frequently in
the chemical and processing industry. Mass transfer phenomena
are generally quantified in terms of a mass transfer coefficient,
which is obtained using empirical correlations. One of the major
objectives of the present work is to simulate numerically mass
transfer phenomena across a moving droplet that can be used to
evaluate mass transfer coefficients without using any empirical
correlations, as discussed in our previous communication.
3.2.1. Governing equations
Transport of reactants from the continuous phase to the
dispersed phase can be captured by solving the convection
diffusionreaction equation which is written for reactants A
and B, respectively, as follows:
jCA
jt+ u CA = DA2CA r (21)
and
jCB
jt+ u CB = DB2CB r , (22)
where CA and CB are the concentrations of reactants A and
B, respectively, DA and DB are the diffusion coefficients of
reactants A and B, respectively, and r is the reaction rate con-
stant. Reactants A and B are convected by the velocity vector,
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6430 K.B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 61 (2006) 6424 6441
Fig. 8. The evolution of the concentration profile of reactant A across the droplet, rising due to buoyancy, at various time steps: (a) t= 0.025s; (b) t= 0.25s;(c) t = 0.6 s.
u, which is obtained by solving the governing equations of the
level-set methodology. The mass transfer equation contains a
time dependent term to capture the change in the concentration
of reactants with time, a flux term due to convection of the re-
actants, a flux term due to diffusion of the reactants, and a re-
action term. A second order irreversible reaction is considered
in the present simulations. Since the reaction occurs only in the
dispersed phase, the reaction term should be applied only inthe dispersed phase. The level-set method is again very useful
to define the reaction rate term only in the dispersed phase, as
follows:
r = kC ACB (1 H ()). (23)
This equation ensures that reaction term is valid when the level-
set function is less than or equal to zero, which indicates the
dispersed phase. His the Heaviside function with H (0)=0and H (> 0) = 1. Numerical implementation uses smoothedapproximation to H.
3.3. Existence of cross-over for a single moving droplet
The system we consider can be physically described as fol-
lows: There is a two-phase system, where the dispersed phase
(a droplet) is rising due to density difference between the
dispersed phase and the continuous phase. There are two reac-
tants, A and B, dissolved in the continuous phase but not re-
acting with each other. Instead, the two reactants diffuse into
the aqueous droplet and react therein. Since the rate of reaction
that is occurring in a dispersed phase is very fast, transfer of
reactants from the continuous phase to the dispersed phase is a
controlling parameter. Hence, we are interested in quantifying
the mass transfer from one phase to the other.
Diffusivity of reactants plays a crucial role in mass transfer
phenomena. In the present case, we assume that the diffusion
coefficients of the two reactants in the continuous phase are
different. The diffusivity of the reactant A is considered to be
1 cm2 s1 and that ofB is 5cm2 s1. We also assume that dif-fusion coefficients of two reactants are relatively higher in the
dispersed phase than those in the continuous phase, to ensure
uniform concentration of reactants inside the droplet. Diffu-sivity of both the reactants in the dispersed phase is consid-
ered to be 50 cm2 s1. Since the reaction is occurring only inthe dispersed phase and is very fast in nature, mass transfer
of reactants is a controlling step and reaction kinetics is not
significant.
Since the transfer rate of reactant B is considered to be higher
than that of reactant A, initial concentration of reactant B is as-
sumed to be lower than that of reactant A. The initial concen-
trations of reactants A and B are considered to be 1 and 0.4 M,
respectively. These conditions ensure kinetic asymmetry in the
transfer rates of reactants. The simulations will be analysed for
the effect of kinetic asymmetry on transfer rates.Simulation results of mass transfer across a moving droplet
are represented in Figs. 8 and 9 to capture the evolution of con-
centration profiles of reactants A and B, respectively. The sur-
face plots of the concentration profiles of reactants A and B, for
time steps of 0.025, 0.25 and 0.6 s, indicate that mass transfer is
indeed taking place from the continuous phase to the dispersed
phase. The concentration of both reactants is disappearing in-
side the droplet which indicates that reaction is occurring in
the dispersed phase at all time steps. It should be noted that al-
though there was a negligible wall effect on velocity profile of
the droplet (see Fig. 5 in Deshpande and Zimmerman, 2005c),
concentration profiles of the two reactants seem to have been
affected markedly due to the instantaneous reaction occurring
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Fig. 9. The evolution of the concentration profile of reactant B across the droplet, rising due to buoyancy, at various time steps: (a) t= 0.025s; (b) t= 0.25s;(c) t = 0.6 s.
inside the moving droplet. The concentration profiles of the
two reactants (concentration boundary layer) which seem to be
circular surrounding the droplet during the initial stages were
found to be almost linear in the wake of the droplet at the later
time steps. This linear profile at the later stages is attributed
to an infinitely fast reaction, since the concentration profiles
of the species in the absence of the reaction do not show sig-
nificant wall effects (see Figs. 10 and 11 in Deshpande and
Zimmerman, 2005c). One would have to use a square domain
to account for the possible wall effect and at the expense of
computational time. Since only the average concentration of
the reactants in the continuous phase is used for analysis, we
persisted with a rectangular domain.
The analysis using the level-set methodology is further ex-
tended to bring out the change in the bulk concentration of
reactants and the concentration of reactants in the dispersed
phase with time. The average concentration of reactants A and
B in the continuous phase, also termed as bulk concentration,
can be represented in terms of the level-set function as
CA,b =>0 CA d>0 d
(24)
and
CB,b =>0 CB d>0 d
. (25)
The average concentration of reactants A and B in the dispersed
phase, also termed as surface concentration, can be represented
in terms of the level-set function as
CA,s=
0 CA d0 d
(26)
0 0.2 0.4 0.60.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
Bulkconcentra
tionofreactants
CA,b
CB,b
Fig. 10. Change in average bulk concentration of reactants A and B with time.
and
CB,s =0 CB d0 d
. (27)
Concentrations of reactants A and B are averaged over the
cross-sectional area of the dispersed phase and the continuous
phase to evaluate CA,s , CB,s and CA,b, CB,b , respectively.
The change in the average bulk concentration of reactants
A (Eq. (24)) and B (Eq. (25)) with time is shown in Fig. 10.
The average bulk concentration of both reactants is found to
decrease with time and attain a steady concentration at later
stages, indicating mass transfer from the continuous phase to
the dispersed phase with reaction occurring inside the droplet.
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0 0.2 0.4 0.60
0.05
0.1
0.15
0.2
0.25CB,s
CA,s
Surfaceconcentratio
nofreactants
Time
Fig. 11. Change in average surface concentration of reactants A and B with
time.
A more interesting result is found when change in the average
surface concentration of reactants A and B is plotted against
time, as shown in Fig. 11. The surface concentration of reactant
B is found to be in excess at the initial stages, which eventually
decreases with time attaining a constant value close to zero.
At this point, there is a switch in the excess surface concentra-
tion of the reactants and thereafter, the excess concentration of
reactant A increases with time.
If we look at the formulation of our problem, reactant B is
considered to have a higher transfer rate and a lower initial
concentration than that of reactant A. At initial stages, surfaceconcentration of reactant B is in excess because of its higher
transfer rate. The transfer rate of reactant B to the droplet is
higher than consumption rate of reactant B inside a droplet,
whereas transfer rate of reactant A and its consumption rate are
in equilibrium at the initial stages. At later stages, the rate of
consumption of reactant B increases and eventually attains an
equilibrium with the rate of transfer of reactant B, leading to
zero surface concentration of reactant B. At the same time, the
rate of transfer of reactant A exceeds the rate of consumption
A and ultimately results in the excess surface concentration of
reactant A. Thus, the phenomenon with a switch in the excess
surface concentration of the reactants, also termed as cross-over, is observed for simulations of mass transfer across a
moving droplet.
Now, since it has been shown that cross-over phenomenon
exists for a single moving droplet, cross-over length will be cal-
culated using simulations and then will be compared with those
obtained using the 1-D theoretical approach. The behaviour
of the cross-over length for different hydrodynamic conditions
will be studied in detail in the next section. Before that, it is first
ensured that the reaction we have considered is indeed mass
transfer limited, by varying the reaction kinetics.
To explore the limit of the extent of reaction, simulations
were performed varying the reaction rate constant in an ex-
treme limit without changing hydrodynamic conditions. The
cross-over time is found to change for the different reaction
rate constant, k, ranging from 104 to 5 106 M1 s1 (resultsnot shown). It is observed that the cross-over time increases
slightly with increase in the reaction rate constant for lower
values of the reaction rate constant, eventually attaining a con-
stant value. This indicates the limit in which reaction is mass
transfer limited and is independent of reaction kinetics. Sincewe are dealing with fast/instantaneous reaction, a considerably
higher reaction rate constant (k = 106 M1 s1) is used in thecomputations performed hereafter to ensure that there is a shift
in CA,s = 0 to CB,s = 0 and that the reaction is mass transferlimited.
4. Comparison of simulation results with theory
The interesting phenomenon of cross-over, which has been
theoretically well studied for an irreversible reaction in a sim-
plified 1-D domain, also exists in the frame of reference of a
single moving droplet. We first discuss the calculations to eval-uate the cross-over length and then study the behaviour of the
cross-over length for various hydrodynamic conditions.
4.1. Evaluation of cross-over length
We are interested in simulations of mass transfer across a
moving droplet, where the droplet is rising due to density dif-
ference between the continuous phase and the dispersed phase.
Hence, we are dealing with time dependent simulations. After
the analysis of the simulation results of mass transfer across a
moving droplet, as discussed in the previous section, we ob-
tain the cross-over time which is defined as the time at which
the excess surface concentration of reactants is switched. To
evaluate the cross-over distance or length from the cross-over
time, we have to multiply the cross-over time by an appropri-
ate velocity field. We have used various velocities such as inlet
velocity, relative velocity and drop velocity in our simulations.
Since the cross-over phenomenon is concerned with the excess
surface concentration of reactants which is defined as the av-
erage concentration of reactants in the dispersed phase, we use
the droplet velocity in all the calculations related to cross-over.
The average droplet velocity is defined in terms of the level-set
function as
Udrop = 0 v d0 d
, (28)
where v is the y-component (vertical direction) of the veloc-
ity field. The average droplet velocity is changing with time,
since the droplet is deforming as it rises due to buoyancy. The
change in the average droplet velocity, Udrop, with time is shown
in Fig. 12, for various hydrodynamic conditions. The average
droplet velocity is increasing at the initial stages, attaining a
constant velocity which is equivalent to the terminal velocity,
for all the simulations performed by varying the hydrodynamic
conditions.
Now, the cross-over length can be evaluated by multiplying
the cross-over time by the average droplet velocity. But, since
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0 0.05 0.1 0.15 0.2 0.25
2
4
6
8
10
12
14
16
Time, sec
Averagedropletve
locity,cm/s
Re=2.88Re=2.75Re=2.61Re=2.49Re=2.36Re=2.22Re=2.1Re=1.97Re=1.83Re=1.56
Fig. 12. Transient behaviour of the average droplet velocity of a moving
droplet.
Table 1
The cross-over lengths obtained using the level-set methodology, for modified
Peclet number, P, obtained using the empirical correlation
Inlet Drop Bulk Cross-over Cross-over P
velocity velocity velocity time length
0.5 3.2 0.575 0.3852 1.224 0.2376
1 3.95 0.9 0.352 1.3904 0.358
1.5 4.6 1.325 0.3338 1.5355 0.5178
2 5.3 1.8 0.31725 1.6814 0.6918
2.5 6.0 2.3 0.3065 1.839 0.8718
3 6.8 2.875 0.3078 2.093 1.0738
3.5 7.45 3.3 0.2958 2.2037 1.21554 8.15 3.8 0.2852 2.3244 1.3832
4.5 8.85 4.275 0.2765 2.447 1.5366
5 9.6 4.8 0.2642 2.5363 1.7048
the average droplet velocity is changing with time (Fig. 12), it
becomes more difficult to choose the average droplet velocity
for the evaluation of the cross-over length. The average droplet
velocity at the time of cross-over is the most appropriate veloc-
ity field to evaluate the cross-over length. Thus, the cross-over
length can be evaluated as
cross-over length = cross-over time average droplet velocityat the time of cross-over.
The cross-over time for different hydrodynamic conditions and
the corresponding cross-over lengths are represented in Table 1,
which will later be compared with the theoretical results.
4.2. Effect of hydrodynamic conditions on cross-over
Numerical simulations are performed for different hydrody-
namic conditions to study its effect on cross-over phenomenon.
The hydrodynamic conditions are changed by varying the in-
let boundary condition. The velocity at inlet stream is varied to
change the hydrodynamic conditions in the present case. The
simulation results of the average surface concentrations of the
two reactants are shown in Fig. 13. It can be clearly seen that
the cross-over time decreases with increase in the inlet velocity.
The cross-over length, evaluated using the procedure discussed
in the previous section, increases with increase in the inlet ve-
locity. The convection of reactants increases with the increase
in the inlet velocity and hence, the average concentration ofreactants in the bulk phase increases, ultimately delaying the
phenomenon of cross-over.
To compare the simulation results with the theory, we need
to use the same scaling parameters that have been used in the
theory. An important parameter is the modified Peclet number,
P, which is defined as
P = U lDB
, (29)
where P is not the exact Peclet number, since the parameter, l,
used to define P is not a geometrical parameter, but a diffusive
length scale and is written in terms of the diffusion coefficient
and the mass transfer coefficient as l=DB /aB . In the present2-D simulations, the active surface area per unit volume, a, is
considered to be ratio of the circumference of the droplet to the
cross-sectional area of the computational domain. In the theory,
kinetic asymmetry was introduced in terms of the ratio of the
mass transfer coefficients of the two reactants. However, kinetic
asymmetry is introduced in simulations in terms of asymmetric
transfer rates (diffusion coefficients) of the two reactants. We
use mass transfer coefficients evaluated using the empirical cor-
relation, discussed in our earlier communication (Deshpande
and Zimmerman, 2005c), to calculate the diffusive length
scale, l.
Another important feature in the evaluation of the modifiedPeclet number is the velocity field, U. In theory, the reactants
are convected in a tubular reactor by a constant velocity, U.
The change in hydrodynamics due to moving droplets has
not been considered in the theory. In simulations, the droplet
is moving along with the reactants in the continuous phase
affecting the average velocity of the dispersed phase and the
continuous phase. Hence, the average velocity of the contin-
uous phase is used to evaluate the modified Peclet number.
The simulation results of cross-over time, cross-over length
and modified Peclet number evaluated using the empirical
correlation are tabulated in Table 1, for various operating
conditions.The cross-over length obtained using the level-set method-
ology is plotted against the modified Peclet number evaluated
using the different empirical correlations for mass transfer co-
efficient, and is compared with the theoretical results, as shown
in Fig. 14. The theoretical results are obtained by solving a set
of equations (Eqs. (1)(6)) discussed in the asymptotic theory
section, for various modified Peclet numbers. The simulation
results match quite well with the theoretical results, indicat-
ing that an increase in the modified Peclet number delays the
cross-over phenomenon. Although the theory and the simula-
tions address different configurations (hydrodynamics is not
considered in the theory), they follow the same trend to predict
cross-over length for various modified Peclet numbers, which
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0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5CA,sCB,s
CA,s
CB,s
CA,s
CB,s
CA,sCB,s
CA,sCB,s
Surfaceconcentrationofreactants
Time
0 0.2 0.4 0.60
0.05
0.1
0.15
0.2
0.25
Surfaceconcentratio
nofreactants
Time
0 0.1 0.2 0.3 0.4 0.50
0.05
0.1
0.15
0.2
0.25
Surfaceconcentrationofreactants
Time
0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
Surfaceconcentrationofreactants
Time
0 0.1 0.2 0.3 0.40
0.02
0.04
0.06
Time
Surfaceconcentrationofreactants
0 0.1 0.2 0.30
0.02
0.04
0.06
0.08
0.1
0.12
Surfacec
oncentrationofreactants
Time
(a)
(c)
(e) (f )
(d)
(b)
CA,s
CB,s
Fig. 13. The change in the average surface concentrations of the two reactants for various operating conditions, to bring out the effect of hydrodynamic
conditions on cross-over. Inlet velocity: (a) 0.5 cm s1; (b) 1cms1; (c) 2cms1; (d) 3cms1; (e) 4cms1; (f) 5cms1.
indicates that cross-over indeed occurs due to kinetic asymme-
try in transfer rates of the reactants. The change in hydrody-
namic conditions does not seem to have any significant effect
on cross-over length as hydrodynamic conditions affect both
the reactants present in the continuous phase in the same way.
Thus, the theory proposed for a stationary premixed tubular
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10-1 10010-1
100
101
Modified Peclet number, P
Cross-overl
ength,
X
TheorySimulation
Fig. 14. Comparison between the theoretical and the simulation results for
the change in the cross-over length for different modified Peclet numbers.
reactor can be successfully applied for the case of a single
droplet in the frame of reference of a moving droplet.
The error in cross-over length estimated using the level-
set simulations for various modified Peclet numbers and that
obtained using the mTm theory is calculated on the basis
of theoretically predicted cross-over length. The discrep-
ancy in cross-over length is estimated to be within 0.78%
of that obtained using mTm theory for modified Peclet
number.
4.3. Parametric space for cross-over
We have observed the existence of cross-over phenomenon
for a moving droplet in the previous section. The effect of hy-
drodynamic conditions on cross-over is studied and compared
with the proposed 1-D theoretical approach. Since intensifi-
cation of reaction was observed near cross-over, the study of
the parametric space for the existence of the cross-over phe-
nomenon is crucial, in order to optimize the length of a tubular
reactor.
Mchedlov-Petrossyan et al. (2003a) theoretically analysed
various operating conditions in order to evaluate the paramet-
ric space for the existence of cross-over phenomenon for an ir-
reversible binary reaction. They considered the following four
possible cases:
(1) ReactantA is in excess of reactantB over the entire domain.
(2) ReactantB is in excess of reactantA over the entire domain.
(3) Both the reactants are depleted due to chemical reaction,
but the surface concentration of reactantA is more depleted
at the entrance of the reactor, whereas reactant B is more
depleted at the other end.
(4) Both the reactants are depleted due to chemical reaction,
but the surface concentration of reactant B is more depleted
at the entrance of the reactor, whereas reactant A is more
depleted at the other end.
Table 2
The various operating conditions considered to test the validity of proposed
criterion (Eq. (30)) for possible existence of the cross-over phenomenon
CA0 CB0 DA DB 11
2 1A2
1 0.2 1 5 5 5 0.0826 0.1819
1 0.3 1 5 3.33 5 0.0826 0.1819
1 0.35 1 5 2.86 5 0.0826 0.18191 0.375 1 5 2.67 5 0.0826 0.1819
1 0.4 1 5 2.5 5 0.0826 0.1819
1 0.45 1 5 2.22 5 0.0826 0.1819
1 0.5 1 5 2 5 0.0826 0.1819
1 0.6 1 5 1.67 5 0.0826 0.1819
It can be clearly seen that cross-over phenomenon is not pos-
sible for the first two cases, since either of the reactants is in
excess over the entire domain of the reactor. Cross-over phe-
nomenon is possible for cases (3) and (4). We focus on case (3)
in the present analysis, since cases (3) and (4) are exactly thesame except for the fact that the concentrations of two reactants
are interchanged.
The theoretical criterion for the existence of cross-over phe-
nomenon for case (3) can be written as
1
A2
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0 0.2 0.4 0.60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
CA,sCB,s
CA,sCB,s
CA,sCB,s
CA,sCB,s
CA,sCB,s
CA,sCB,s
Time
Surfaceconcentrationofreactants
0 0.2 0.4 0.60
0.1
0.2
0.3
Time
Surfaceconcentrationofreactants
0 0.2 0.4 0.60
0.05
0.1
0.15
0.2
0.25
Surfaceconcentrationofreactants
Time
0 0.2 0.4 0.60
0.02
0.04
0.06
0.08
0.1
0.12
Surfaceconcentrationofreactants
Time
0 0.2 0.4 0.60
0.05
0.1
0.15
Surfaceconcentrationofreactants
Time
0 0.2 0.4 0.60
0.05
0.1
0.15
0.2
0.25
Surfaceconcentrationofreactants
Time
(a)
(c)
(e)
(d)
(f)
(b)
Fig. 15. The change in the average surface concentrations of the two reactants for various operating conditions, to evaluate the parametric space for the existence
of cross-over phenomenon: (a) = 5, 1 = 0.2; (b) = 2.67, 1 = 0.2; (c) = 2.5, 1 = 0.2; (d) = 2.22, 1 = 0.2; (e) = 2, 1 = 0.2; (f) = 1.67, 1 = 0.2.
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in Fig. 15 for representative data, for the various operating
conditions discussed in Table 2.
The left hand side inequality of Eq. (30) is not very significant
in the present case, since is greater than 1/A2 for all the
conditions considered. Even in the limit of equal to 1/A2,
we need to have the initial concentration of the reactant with a
lower transfer rate (reactant A) to be much smaller than that ofthe reactant with a higher transfer rate (reactant B). This limit
will never result in cross-over, since the kinetic asymmetry
required for the cross-over is not satisfied.
The right hand side inequality of Eq. (30) is important in or-
der to explore the possibility of cross-over. Simulations chang-
ing the ratio of the initial concentration of reactants, , were
conducted for a wide range, maintaining a constant ratio of the
diffusion coefficients of the two reactants. In Fig. 15(a), we
explore the limit where is equal to 1/1. It is apparent that
the surface concentration of reactant A is in excess of reactant
B throughout the length of the reactor. Although the transfer
rate of reactant B is higher than that of reactant A, the initial
concentration of reactant A is large enough to occupy the dis-
persed phase. This case is very much similar to case 1 studied
by Mchedlov-Petrossyan et al. (2003a).
In Fig. 15(b), is smaller than 1/1 and it is observed that
cross-over occurs almost at the entrance of the reactor. With
further increase in the initial concentration of reactant B, which
means that is considerably smaller than 1/1, there is a delay
in the occurrence of the cross-over phenomenon, as shown in
Figs. 15(c) and (d).
With a further increase in the initial concentration of reactant
B, attains a value which is very much smaller than 1/1, but
still larger than 1/A2. The average surface concentration plots
of reactantsA andB for this case, as shown in Figs. 15(e) and (f),indicate that cross-over does not occur within a definite length
of the reactor. The average surface concentration of reactant B
is still decreasing with time, indicating that cross-over could
occur further down the length of the reactor. Thus, using the
level-set methodology it is found that the proposed criterion
for the existence of cross-over phenomenon (Eq. (30)) indeed
holds true for a transport limited heterogeneous reaction in the
frame of reference of a moving droplet.
5. Nearly irreversible reactions
In Section 2 on theory, it was commented that a six variablesystem of PDEs and algebraic equations can be simplified in
terms of supersaturation concentration, , and modified Thiele
moduli, F1 and F2. The change in the supersaturation concen-
tration along the length of the reactor is a meaningful way of
representing the cross-over phenomenon, since there is a dis-
continuity observed in at the point of cross-over. The mTm
F2 has the significance of representing the intensification of
the reaction at the time of cross-over. These important features
have been captured theoretically in the previous section for a
nearly irreversible reaction. The computations performed so far
using the level-set methodology are strictly for an irreversible
reaction where the effect of concentration of the product is de-
coupled from the dynamics of the system. In order to bring out
0 0.2 0.4 0.60
0.1
0.2
0.3
0.4
0.5
Time
Surfaceconcentrationofreactants
CA,s
CB,s
Fig. 16. The change in the average surface concentration of the reactants with
time, to represent cross-over phenomenon for a nearly irreversible reaction.
the above mentioned features of cross-over, we now apply the
level-set methodology for a nearly irreversible reaction, incor-
porating contribution of the product.
The analysis of the asymptotic theory proposed for trans-
port limited reactions indicates that a strictly irreversible reac-
tion case is not very informative. Since the concentration of
the product is decoupled for a strictly irreversible reaction, the
mTm F2, which is represented in terms of the surface concen-
tration of product, is always zero and mTm F1 plays a signifi-
cant role. Hence, the level-set methodology is further extended
to simulate mass transfer across a moving droplet for a nearlyirreversible reaction.
For a nearly irreversible reaction, the convection
diffusionreaction equation for the reactants (similar to Eqs.
(21) and (22)) as well as the product can be written as
jCA
jt+ u CA = DA2CA k(CACB KC C ), (31)
jCB
jt+ u CB = DB2CB k(CACB KCC ) (32)
and
jCCjt
+ u CC = DC2CC + k(CACB KCC ). (33)
The above set of equations can be solved using the same numer-
ical scheme as discussed for a strictly irreversible reaction. We
use the same physical properties, initial conditions and bound-
ary conditions for reactants A and B, as that used for a strictly
irreversible reaction. We need to introduce one more variable
CC to incorporate the effect of the concentration of product on
the dynamics of the system. Diffusivity of the product is con-
sidered to be 1 cm2 s1 in the continuous phase and 50 cm2 s1
in the dispersed phase, which rapidly leads to a uniform con-
centration of the product in the dispersed phase. The initial
concentration of the product is considered to be 0 M, as there
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0 0.2 0.4 0.60.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
Time
Supersatur
ation
Fig. 17. The change in supersaturation concentration, defined in terms of the
level-set function, with time, for a nearly irreversible reaction.
is no product formation at the early stage. The same boundary
conditions are applied for the product as those used for the re-
actants. Since the reaction is nearly irreversible, the equilibrium
constant, K, is considered to be 105. The major change in thepresent formulation for a nearly irreversible reaction is that the
reaction rate term is rewritten for a slightly reversible reaction
incorporating the concentration of a product term. The reaction
rate constant, k, is considered to be 106 M1 s1. The reactionis still occurring in the dispersed phase which is incorporated
using the characteristics of the level-set function.
The average surface concentration of reactantsA andB, as de-fined by Eqs. (26) and (27), respectively, is plotted against time
to represent cross-over phenomenon for a nearly irreversible
reaction, and is shown in Fig. 16. The cross-over phenomenon
is found to occur at a relatively earlier stage for a nearly ir-
reversible reaction than that for a strictly irreversible reaction.
Since the level-set simulations for a nearly irreversible reac-
tion and a strictly irreversible reaction are performed for the
same hydrodynamic conditions, cross-over time can be directly
compared. It can be seen that cross-over occurs at 0.352 s for
a strictly irreversible reaction, whereas it is found to occur at
0.128s for a nearly irreversible reaction. The change in cross-
over time can be attributed to different reaction kinetics. Aslight coupling of concentration of the product introduced by
the equilibrium constant, K = 105, changes the dynamics ofthe system drastically, as indicated by the cross-over times for
a nearly irreversible reaction and a strictly irreversible reaction.
It seems that reversibility effects are important along with ac-
cumulation of reactants in the dispersed phase. This complex-
ity is attributed to non-equilibrium effects studied in a batch
reactor (Zimmerman et al., 2005).
The major advantage of the present level-set methodology,
as discussed earlier, is its convenience in easily describing the
continuous phase and the dispersed phase. The six variable
system can be expressed in terms of just three concentrations,
two of the reactants and one of the product, using the level-set
formulation, as described by Eqs. (31)(33). The concentration
of the reactants and the product in the dispersed phase and the
continuous phase can be easily defined using the characteristics
of the level-set function.
Using the present formulation, supersaturation can also be
defined in terms of the concentration of either the reactants
or the product. Supersaturation concentration, , which wasdefined in the proposed asymptotic theory by Eqs. (7)(9), can
be rewritten as
= (CA,b CA,s )DA
DB, (34)
= CB,b CB,s , (35)
and
= (CC,s CC,b)DC
DB, (36)
where CA,b and CB,b are the average bulk concentrations of
the reactants, as defined by Eqs. (24) and (25). The average
bulk concentration of the product, CC,b , can be similarly de-
fined. CA,s and CB,s are the average surface concentrations of
the reactants, as defined by Eqs. (26) and (27). The average
surface concentration of the product, CC,b , can be similarly de-
fined. Supersaturation concentration, , is plotted against time
as shown in Fig. 17.
The discontinuity in supersaturation concentration at the
point of cross-over, which was apparent for the 1-D theoret-
ical results as shown in Fig. 4, is not so obvious but can be
observed for the simulation of mass transfer across a moving
droplet for a nearly irreversible reaction. The rate of change in
supersaturation before cross-over is found to be slightly lowerthan that after cross-over.
A major difference between the numerical implementation
of the asymptotic theory and the simulations using the level-set
methodology is the parameter responsible for the transport of
the reactants, and the kinetic asymmetry in the transfer rates. In
the proposed asymptotic theory, diffusivity of the two reactants
in the continuous phase is assumed to be the same and kinetic
asymmetry was introduced through asymmetric mass transfer
coefficients of the reactants, which are often evaluated using
empirical correlations. In the simulations using the level-set
methodology, kinetic asymmetry is incorporated by consider-
ing two reactants of different diffusion coefficients. For a trans-port limited reaction, the rate of mass transfer of reactants and
the rate of chemical reaction are assumed to be in equilibrium,
since the reaction is almost instantaneous. We do not use mass
transfer coefficients in simulations since the disappearance of
reactants is defined in terms of the reaction rate. Instead, we
have proposed a numerical scheme to infer mass transfer coef-
ficients using the level-set methodology (Deshpande and Zim-
merman, 2005c).
The mTm, F1 and F2, as defined by Eqs. (10) and (11), bring
out the interesting feature of cross-over phenomenon as dis-
cussed in the asymptotic theory. To explore the possibility of
such an interesting feature to be observed in the simulation of
mass transfer across a moving droplet for a nearly irreversible
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0 0.2 0.4 0.60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
ModifiedThielem
odulus,
F1
Fig. 18. The change in modified Thiele modulus, F1, with time, for a nearly
irreversible reaction.
0 0.2 0.4 0.60
1
2
3
4
5
6
7x 10-3
Time
ModifiedThielemodulus,
F2
Fig. 19. The change in modified Thiele modulus, F2, with time, for a nearly
irreversible reaction.
reaction, we modify the definition of mTm used earlier (Eqs.
(10) and (11)) and represent mTm in terms of the surface con-
centration using the level-set function as
F1 =m2/1CA,s
n2/CB,s + m2/1CA,s + 1/2CC,s(37)
and
F2 =1/2CC,s
n2/CB,s + m2/1CA,s + 1/2CC,s, (38)
where CA,s , CB,s and CC,s are the average surface concentra-
tions of the reactants, A and B, and the product, C, respectively.
It should be noted that 1 and 2 used in the asymptotic the-
ory are the ratios of the mass transfer coefficients of the reac-
tants and the product, whereas 1 and 2 used in the present
simulations are the ratios of the diffusion coefficients of the
reactants and the product. The change in mTm, F1 and F2, is
plotted against time as shown in Figs. 18 and 19, respectively.
The modified Thiele modulus, F1, changes smoothly follow-
ing a smoothed Heaviside step function behaviour from 1 to
0 at the cross-over time which is 0.128 s in the present case
(Fig. 18). The modified Thiele modulus, F2, which is repre-sented in terms of the surface concentration of the product,
represents a more interesting feature indicating a maximum
value at the cross-over time (Fig. 19). The mTm, F2, represents
the intensification and localization of reaction near cross-over.
Thus, representation of the surface concentration of the reac-
tants and the product in terms of mTm is a more meaningful
way to understand the cross-over phenomenon.
6. Conclusion
The phenomenon of cross-over, which occurs due to kinetic
asymmetry in the transfer rate of reactants for a transport lim-
ited reaction, is first captured by solving a six variable system
of PDEs and algebraic numerically, in order to optimize the
design of a tubular reactor. An asymptotic theory proposed by
Mchedlov-Petrossyan et al. (2003a), in order to simplify the six
variable system in terms of supersaturation concentration and
modified Thiele modulus (mTm), is numerically implemented
to bring out the salient features of cross-over phenomenon. A
modified approach has been represented in this work by incor-
porating the accumulation of the reactants and the product in
the dispersed phase, in order to capture the existence of the
concentration boundary layer, which was not captured in the
theory proposed by Mchedlov-Petrossyan et al. (2003a).Since the study of transport limited heterogeneous reactions
leading to a cross-over phenomenon requires knowledge of
concentration of the reactants and the product in both the dis-
persed phase and the continuous phase, a numerical scheme has
been developed using the level-set methodology that reduces a
six variable system to just three convectiondiffusionreaction
equations. The level-set function is efficiently used to describe
the concentration of the reactants and the product in the dis-
persed phase and in the continuous phase. The reaction, which
occurs only in the dispersed phase, is very easily incorporated
in the present formulation using the unique characteristics of
the level-set function. The level-set methodology, which wasused earlier to simulate mass transfer across a moving droplet,
has been further extended to explore the possibility of cross-
over phenomenon in the frame of reference of a single moving
droplet. The cross-over phenomenon indeed exists for a single
moving droplet for a strictly irreversible reaction. The cross-
over length obtained using the asymptotic theory is then com-
pared with that obtained using the level-set methodology for
a single moving droplet and the results match quite well for
various hydrodynamic conditions. Cross-over length obtained
using the level-set simulation is found to be within 0.78% of
that obtained using mTm theory for modified Peclet number.
The study of evaluation of the parametric space followed the
theoretically proposed criterion for the existence of cross-over.
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6440 K.B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 61 (2006) 6424 6441
The analysis of the asymptotic theory is then extended to
the simulation of mass transfer across a moving droplet for a
nearly irreversible reaction. The intensification of the reaction
at the time of cross-over is captured and is represented in terms
of mTm, F2, which is a good measure to evaluate the optimum
length of the reactor. The theoretical criterion is numerically
validated to obtain the parametric space for the existence ofcross-over in the frame of reference of a moving droplet, and
can be used for the design of a tubular reactor since greater
molecular efficiency can be achieved than when operating in a
regime without cross-over.
Acknowledgements
K.B.D. would like to thank Prof. Mchedlov-Petrossyan for
very useful discussions. W.B.Z. would like to thank the EPSRC
(GR/A01435, GR/S67845, GR/R72754 and GR/N20676) for
financial support for this work.
Appendix
Let us consider a layer Q perpendicular to the direction of
the overall fluxU, thick enough to comprise many droplets
(that is, having thickness much larger than interdroplet
distance), but much smaller than the scale of macroscopic
gradients. The full flux of any reagent, say A, inside the
layer is
U CA DCA. (39)
Now, let us integrate this expression over all surfaces = 0inside Q, taking outward direction from < 0 to > 0 do-mains as positive directions of the surface elements. We de-
note the sum of all such surfaces inside Q by . Then the
total inflow (per unit time) of A into < 0 domain inside
Q is
(U CA DCA) d s. (40)
For the stationary state it should be equal to total consumption
of A in all < 0domains inside Q; denoting by the total
volume of domains with negative ,
(U CA DCA) d s =
k(CACB KCC ) d. (41)
Because the layer is thin enough, the surface concentrations as
defined in Deshpande and Zimmerman (2005c) (Eq. (21))
CA,s =
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