simulation of reacting moving droplet

8/3/2019 Simulation of Reacting Moving Droplet

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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
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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


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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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


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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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


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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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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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.


9/18


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


10/18


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


11/18


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


Time

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2


Time

0 0.1 0.2 0.3 0.40

0.02

0.04

0.06

Time


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


12/18


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


13/18


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


0 0.2 0.4 0.60

0.1

0.2

0.3

Time


0 0.2 0.4 0.60

0.05

0.1

0.15

0.2

0.25


Time

0 0.2 0.4 0.60

0.02

0.04

0.06

0.08

0.1

0.12


Time

0 0.2 0.4 0.60

0.05

0.1

0.15


Time

0 0.2 0.4 0.60

0.05

0.1

0.15

0.2

0.25


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.


14/18


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


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


15/18


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


16/18


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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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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Hikita, H.S., Asai, S., Ishikawa, H., 1977. Simultaneous absorption of two

gases which react between themselves in a liquid. Industrial Engineering

Chemistry Fundamentals 16, 215219.

Juvekar, V.A., 1974. Simultaneous absorption of two gases which react

between themselves in a liquid. Chemical Engineering Science 29,

18421843.

Lakehal, D., Meier, M., Fulgosi, M., 2002. Interface tracking towards the direct

simulation of heat and mass transfer in multiphase flows. InternationalJournal of Heat and Fluid Flow 23, 242257.

Mchedlov-Petrossyan, P.O., 1998. Heterogeneous diffusion-limited chemical

reactions of initially separated species. Kharkov University Bulletin,

Chemistry Series 2, 97101.

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


Osher, S., Sethian, J.A., 1988. Front propagating with curvature-dependent

speed: algorithm based on HamiltonJacobi formulations. Journal of

Computational Physics 79, 1249.

Petera, J., Weatherley, L.R., 2001. Modelling of mass transfer from falling

drops. Chemical Engineering Science 56, 49294947.

Ramachandran, P.A., Sharma, M.M., 1971. Simultaneous absorption of

two gases. Transactions of the Institution of Chemical Engineers 29,

18421843.

Roper, G.H., Hatch, T.F., Pigford, R.L., 1962. Theory of absorption and

reaction of two gases in a liquid. Industrial & Engineering Chemistry

Fundamentals 1, 144147.Warnecke, H.J., Schafer, M., Pruss, J., Weidenbach, M., 1999. A concept

to simulate an industrial size tube reactor with fast complex kinetics

and absorption of two gases on the basis of CFD modelling. Chemical


Zarzycki, R., Ledakowicz, S., Starzak, M., 1981. Simultaneous absorption of

two gases reacting between themselves in a liquid. Chemical Engineering

Science 36, 105111.

Zimmerman, W.B., Mchedlov-Petrossyan, P., Khomenko, G.A., 1999.

Diffusion limited mixing and reaction in heterogeneous catalysis of initially

segregated species. IChemE Symposium Series 146, 317324.

Zimmerman, W.B., Mchedlov-Petrossyan, P., Khomenko, G.A., 2005.

Nonequilibrium effects on fast binary reactions in a heterogeneous catalytic

batch reactor. Chemical Engineering Science 60, 30133028.

simulation of reacting moving droplet

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