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International Journal of Heat and Mass Transfer xxx (2010) xxx–xxx

HMT 7935 No. of Pages 12, Model 5G

15 October 2010

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Effects of nonlinear chemical reactions on the transport coefficients associatedwith steady and oscillatory flows through a tube

Suvadip Paul, B.S. Mazumder ⇑Fluvial Mechanics Laboratory, Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700 108, India

a r t i c l e i n f o a b s t r a c t

222324252627282930313233

Article history:Received 5 June 2009Received in revised form 8 June 2010Accepted 25 August 2010Available online xxxx

Keywords:Phase exchangeHomogenizationTransport coefficientNonlinear reactionPhase partition coefficient

3435

0017-9310/$ - see front matter � 2010 Published bydoi:10.1016/j.ijheatmasstransfer.2010.08.028

⇑ Corresponding author. Tel.: +91 3325753033.E-mail address: [email protected] (B.S. Mazumder)

Please cite this article in press as: S. Paul, B.S. Moscillatory flows through a tube, Int. J. Heat Ma

The paper concerns with the determination of effective transport coefficients associated with theoscillatory flow through a tube where a solute undergoes nonlinear chemical reactions both within thefluid and at the boundary. Method of homogenization, a multiple-scale method of averaging, is adoptedto derive the transport equation that contains advection, diffusion and reaction. The resultant equationshows how the transport coefficients are influenced by the rate and degree of the nonlinear chemicalreaction. Two different nonlinear reactions are considered at the bulk flow and the boundary. The reac-tions at the boundary may be reversible and irreversible in nature. Several facts are established from themodel by fixing the rate or degree of the nonlinear reactions. Results demonstrate that the reaction at theboundary is more influential than the bulk-flow reaction in determining the transport coefficients. Alsofluid-phase reaction coefficient diminishes as the nonlinearity increases, whereas the trend is oppositefor the nonlinear wall-phase reaction coefficient. Different controlling parameters are found to play sig-nificant role on the transport coefficients when the ratio of wall-phase concentration to the fluid-phaseconcentration is low.

� 2010 Published by Elsevier Ltd.

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1. Introduction

Transport of solute through a tube has always been an impor-tant topic in fluid mechanics. Due to its applications in the fieldof chemical, environmental and biomedical engineering, masstransport phenomena have been extensively studied during thepast fifty years. Investigation was initiated by Taylor [1] when hepublished the landmark paper concerning the transport of a con-taminant dissolved in a fluid flowing through a pipe of narrowdiameter. He demonstrated that an interaction between the trans-verse variations in the fluid’s velocity field and the transverse dif-fusion of the solute yielded an effective downstream mixingmechanism for the solute. This mechanism has since been dubbed‘Taylor Dispersion’. He found an approximate analytical formulafor the transport coefficient, more particularly for the dispersioncoefficient which is a measure of the rate at which a solute spreadsalong the flow. Since then many related studies have been sur-faced. Incorporating the effect due to molecular diffusion, Aris [2]modified Taylor’s expression for the dispersion coefficient usinghis method of moments. Later on Chatwin [3], Gill and Sankara-subramanian [4], Smith [5], Sullivan [6], Pedley [7] and others con-tribute significantly on this complex field.

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azumder, Effects of nonlinear css Transfer (2010), doi:10.1016

The phenomena of mass transport becomes more complexwhen the transported material is chemically reactive. Knowledgeof the transport coefficients associated with the flow of a reactivespecies has applications as diverse as chromatographic separationsin chemical engineering, the spreading of pollutants in a stream,the mixing and transport of drug and toxins in physiological sys-tems and so on. Reactions originated from chemical or biologicalprocesses are very common in aqueous environment. The fluid dy-namic behaviour of the transport coefficients can be greatly influ-enced by the presence of chemical processes like phase exchange,partitioning, boundary uptake or chemical reactions. Transportthrough these processes has widely been extended to applicationsin biological or physiological transport (Davidson and Schroter [8],Grotberg et al. [9], Phillips and Kaye [10]). Reactions may occur inthe bulk flow or at the boundary. It may be reversible or irrevers-ible in nature and linear or nonlinear in degree. Linear reversible orirreversible reactions at the boundary has received a considerableattention by the researchers such as Gupta and Gupta [11], Purn-ama [12], Mazumder and Das [13], Jiang and Grotberg [14], Sarkarand Jayaraman [15], Mazumder and Mondal [16], Ng [17,18], Pauland Mazumder [19] and others.

Thus prosperous literature exists on mass transport when thetransported substance is chemically inert or it undergoes a first or-der chemical reaction either within the bulk flow or at the bound-ary. But the case is different when the reactions are not linear. Inspite of its biological and industrial applications, less attention

hemical reactions on the transport coefficients associated with steady and/j.ijheatmasstransfer.2010.08.028

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2010.08.028

mailto:[email protected]


http://www.sciencedirect.com/science/journal/00179310

http://www.elsevier.com/locate/ijhmt


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Nomenclature

a radius of pipeC concentration of mobile phaseCs concentration of immobile phaseD molecular diffusion coefficientDa Damkohler numberDeff effective dispersion coefficientDTs steady component of dispersion coefficientDTw oscillatory component of dispersion coefficientk reversible reaction rate constantL characteristic longitudinal distancem degree of nonlinear reaction at the boundaryn degree of nonlinear reaction in the bulk flowP pressure gradientPe Peclet numberR nonlinear reaction term occurring in the bulk flowRs nonlinear reaction term occurring at the boundaryR retardation factorr radial coordinateSc Schmidt numberT0 time scale for radial diffusionT1 time scale for advection and reaction

T2 time scale for axial diffusion/dispersionu axial component of velocityus steady part of velocityuw oscillatory part of velocityt timex axial coordinate

Greek lettersa partition coefficientC rate of irreversible reaction at the boundary� perturbation parameterk rate of nonlinear reaction in the bulk flowks rate of nonlinear reaction at the boundarym kinematic viscosityq densityw amplitude of pressure pulsationx frequency of pressure pulsationd oscillation parameterK Womersley number

2 S. Paul, B.S. Mazumder / International Journal of Heat and Mass Transfer xxx (2010) xxx–xxx


15 October 2010

has been made to the effects of nonlinear reactions upon a systemof solutes undergoing Taylor Dispersion. Established literature pro-vides no insight for the effective transport coefficients when thenonlinearity is of the order of two or more. Though Barton [20]and Smith [21] considered the effect of second order reaction ondispersion phenomena, but such limited attempts are too deficientconsidering its vast environmental and biological importance. Intwo successive papers, Revelli and Ridolfi [22,23] have shown theeffect of nonlinear chemical reactions on suspended material insediment-laden turbulent streams. Still there remains a lot toknow about the effect of higher order chemical reactions on solutetransport.

The mathematical models that describe chemical reactionkinetics provide chemists and chemical engineers with tools tobetter understand and describe chemical processes such as fooddecomposition, microorganism growth, stratospheric ozonedecomposition, and the complex chemistry of biological systems.These models can also be used in the design or modification ofchemical reactors to optimize production, more efficiently segrega-tion of products, and elimination of harmful by-products. Whenperforming catalytic cracking of heavy hydrocarbons into gasolineand light gas, for example, kinetic models can be used to find thetemperature and pressure at which the highest yield of heavyhydrocarbons into gasoline will occur. The kinetics of chemicalreactions, which determines the order of nonlinear reactions, illu-minate the process by which a reaction takes place. The principalfactor in kinetics is the rate of the reaction. Many factors, such asconcentration and temperature, influence the rate of a reaction. Gi-ven the ubiquity of chemical reactions, kinetics can be applied to awide variety of practical situations. An understanding of kineticsenables the precise dealing of important reaction rates in biologicaland industrial fields. Chemical kinetics helps to reveal the nuancesof important rate-based processes such as metabolism. In turn, thisknowledge helps scientists and other professionals to determinethe best course of action for one’s personal nutrition. Nonlinearchemical reactions have a wide variety of application in the worldof medicine. In addition to the ways in which the human bodyundergoes processes of respiration and metabolism, nonlinearreactions also play a large part in the supervision of drugs. Forexample, the mechanisms for the sustained release of drugs are

Please cite this article in press as: S. Paul, B.S. Mazumder, Effects of nonlinear coscillatory flows through a tube, Int. J. Heat Mass Transfer (2010), doi:10.1016

based on the half-life of the substances used and sometimes thepH of the body as well. Practically, this affects the way in whichdosages are determined and prescribed. The rate of reactions andthe various conditions under which they occur are crucial fordetermining certain aspects of environmental protection.

The main objective of the present paper is to examine the influ-ence of nonlinear reactions on the transport coefficients, when theflow is driven by a pressure gradient comprising steady and peri-odic components. We present here a rigorous mathematical modelfor the evolution of the concentration of reacting solutes that travelwithin a fluid flowing down a pipe of circular cross-section. Theintention is to provide analytical expressions for the transportcoefficients and to look into the effect of nonlinear chemical reac-tions on those coefficients. With this in view, a two dimensionalmathematical model is formulated to take into account advection,diffusion and reaction. The mathematical homogenization theory(Mei et al. [24]) is adopted to take cross-sectional and time aver-ages of the two dimensional model, and the most general case isdealt with in which the reactions are nonlinear both in the bulkflow and boundary.

2. Velocity distribution

We consider a fully developed, axi-symmetric laminar flow of ahomogeneous, incompressible viscous fluid through a pipe of ra-dius a. We have used a cylindrical coordinate system in whichthe radial and axial co-ordinates are r and x respectively. The flowis assumed to be unidirectional and so the velocity has only axialcomponent u(r, t) which satisfies the Navier–Stokes equation as:

@u@t¼ � 1

q@p@xþ m

1r@

@rr@u@r

� �;

where @p@x is the axial pressure gradient, q is the fluid density and m is

the kinematic viscosity.The horizontal pressure gradient, which drives the flow, con-

sists of steady and harmonically fluctuating components,

� 1q@p@x¼ P½1þ wReðeixtÞ�;



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15 October 2010

where P > 0 is the steady part of the pressure gradient, w is a factorsuch that Pw is the amplitude of the oscillatory part of the pressuregradient, and x is the frequency of the pressure pulsation.

The no-slip conditions on the boundary of the pipe produces thefollowing velocity profile,

uðr; tÞ ¼ usðrÞ þ Re½uwðrÞeixt �; ð1Þ

where the steady component (us(r)) of the velocity is,

usðrÞ ¼ husi 1� ra

� �2� �

and its unsteady component (uw(r)) is given by

uwðrÞ ¼ �iPwx

1� J0ðrrÞJ0ðraÞ

� �:

Here husi ¼ Pa2

8m

� �is the velocity averaged over the cross-section of

the pipe (the angle brackets denote averaging across the tube sec-tion). J0 is the zeroth-order Bessel function of the first kind, andr2 =�ix/m or r = (1�i)/d where d ¼

ffiffiffiffiffiffiffiffiffiffiffiffi2m=x

p3. Governing equation and boundary conditions

We consider the transport of a chemically reactive solute along atube whose walls are made of retentive and reactive materials. It isassumed that the solute is completely miscible in the fluid andundergoes nonlinear reactions with it in the bulk flow. Also at theboundary of the tube, the solute is subject to linear and nonlinearreactions with the wall material. As a result a portion of the soluteis absorbed irreversibly by the wall material and some portion takespart in a reversible phase exchange process with the flowing fluid.The phase of the solute that flows with the fluid and that retains atthe boundary are termed as mobile and immobile phasesrespectively.

If C is the concentration (mass of solute per bulk volume of theflowing fluid) of the mobile phase and Cs is the concentration (massof solute per surface area of the boundary wall) of the immobilephase, then at equilibrium

Cs

C¼ a;

where a is a partition coefficient having the dimension of length.When, in general, equilibrium is not attained, the phase ex-

change will take place in either forward or backward directionaccording to

@Cs

@t¼ kðaC � CsÞ þRs;

where k is the reversible reaction rate constant and Rs is the so callednonlinear reaction occurring at the boundary to be defined later.

The problem for the transport of the reactive solute can now beformulated as follows:

@C@tþ uðr; tÞ @C

@x¼ D

@2C@x2 þ

Dr@

@rr@C@r

� �þR; 0 < r < a; ð2Þ

where D is the molecular diffusion coefficient assumed to be con-stant. The term R on the right hand side of (2) represents the effectof nonlinear chemical reaction occurring in the bulk flow.

Two different nonlinear reactions (represented by R and Rs) areassumed to affect the mobile and immobile phases. The corre-sponding reaction rates, R and Rs, are therefore prescribed accord-ing to the power laws,

R ¼ �kCn; Rs ¼ �ksCms ;

where coefficients k and ks are the nonlinear reaction rate constantsfor the mobile and immobile phases respectively (their units are


(kg m�3)1�ns�1 and (kg m�2)1�ms�1) and the corresponding degreesof nonlinearity are given by the exponents n and m. When n and mare zero, k and ks simply play the role of a source term. It should beremarked that if k = ks = 0, a great deal is known from the work ofNg [17].

The boundary conditions for (2) can now be formulated as:

@C@r¼ 0; r ¼ 0 ð3Þ

� D@C@r� CC ¼ @Cs

@t¼ kðaC � CsÞ � ksC

ms ; r ¼ a; ð4Þ

where C is the irreversible absorption rate.

4. Assumptions

The following assumptions are made for carrying out the per-turbation analysis:

1. A sufficiently long time has passed since the discharge of thesolute into the flow so that the length scale for the longitudinalspreading of the reactive species is much greater than the tuberadius. By this we mean that x = O(L) and r = O(a), where L is acharacteristic longitudinal distance for the solute transport.The ratio

� ¼ a=L << 1

is small enough to use as ordering parameter.2. The time period of flow oscillation is so short that within this

period there can be no appreciable transport effects down thetube. In other words, flow oscillations are sufficiently fast anda large number of flow oscillations are required for the spread-ing of solute over the length O(L) along the tube. However, thetube is so fine in bore that diffusion across the entire cross-sec-tion may be accomplished within this short time scale.

3. The two reactions occurring at the boundary are of differentorders. The reversible phase exchange is much faster than theirreversible reaction. This ensures that local equilibrium canbe largely achieved over a finite number of oscillations. The rateof irreversible absorption is much slower and it is comparablewith the advection speed down the tube.

4. The Peclet number is equal to or greater than order of unity:

Pe � ahusi=D P Oð1Þ:

Under these assumptions, three distinct time scales, which gov-erns the total transport process, may be defined as (Ng [17])

Time scale for the radial diffusion : T0 ¼2p=x¼Oða2=DÞ¼Oðk�1Þ;Time scale for advection and reactions : T1 ¼ L=husi¼OðaC�1Þ¼ T0=�;

Time scale for axial diffusion=dispersion : T2 ¼ L2=D¼ T0=�2:

Thus T1 and T2 are respectively one and two order of magnitudelonger than T0. Based on these time scales, we may introduceaccordingly

t0 ¼ t; t1 ¼ �t; t2 ¼ �2t;

which are, respectively, the fast, medium and slow timevariables.

5. Asymptotic analysis

Let us assume that the radial diffusion is two order of magni-tude greater than the longitudinal dispersion and one order ofmagnitude greater than the advection which is again comparablewith the rate of irreversible reaction at the boundary as mentionedearlier. The rate of nonlinear reactions occurring in the bulk flowand at the boundary are also assumed to be comparable with the



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15 October 2010

advection. The reactions play a significant role in the whole trans-port process if the time scales of the reactions and advection arecomparable. If the reactions are fast enough, they dominate andthat lead to a considerable reduction of the influences of longitudi-nal transport mechanisms, on the contrary if the reactions are tooslow, their contributions are negligible. Thus all the reactions pres-ent in the model i.e., the irreversible reaction at the boundary andthe two nonlinear reactions-one in the bulk flow and the other atthe boundary are comparable to each other.

The relative significance of the terms in the transport Eq. (2)with the boundary conditions (3) and (4) can now be identifiedas indicated below with the power of �:

@C@tþ �u @C

@x¼ �2D

@2C@x2 þ

Dr@

@rr@C@r

� �� kCn; 0 < r < a; ð5Þ

@C@r¼ 0; r ¼ 0; ð6Þ

� D@C@r� �CC ¼ @Cs

@t¼ kðaC � CsÞ � �ksC

ms ; r ¼ a: ð7Þ

To carry on the perturbation analysis, let us expand the dependentvariables C and Cs in power series of the perturbed parameter �:

Cðx; r; tÞ ¼ C0ðx; r; t1; t2Þ þ �C1ðx; r; t0; t1; t2Þþ �2C2ðx; r; t0; t1; t2Þ þ Oð�3Þ; ð8Þ

Csðx; tÞ ¼ Cs0ðx; t1; t2Þ þ �Cs1ðx; t0; t1; t2Þ þ �2Cs2ðx; t0; t1; t2Þþ Oð�3Þ: ð9Þ

The terms Cn and Csn, (n = 1,2,3 . . .) are purely oscillatory functionsof short time variable t0 as they are subject to advection due to flowoscillation, but oscillatory effect does not show up on the zeroth or-der (i.e., n = 0), and therefore the leading order concentrations aretaken to be independent of the time variable t0 [17,18].

The nonlinear decay term in (5) and (7) can also be expanded inpower series as:

�kCn ¼ �kðC0 þ �C1 þ �2C2 þ � � � Þn

� �kCn0 þ �2nkCn�1

0 C1 þ Oð�3Þ; ð10Þ

�ksCms ¼ �ksðCs0 þ �Cs1 þ �2Cs2 þ � � � Þm

� �ksCms0 þ �2mksC

m�1s0 Cs1 þ Oð�3Þ: ð11Þ

For the multiple-scale asymptotic analysis, the time derivative hasbeen expanded as:

@

@t¼ @

@t0þ � @

@t1þ �2 @

@t2: ð12Þ

Using the expansions (8)–(12) in Eqs. (5)–(7) and equating the coef-ficients of like powers of � from both sides, a system of differentialequations is obtained:

For zeroth order (O(1)) Eqs. (5)–(7) give:

0 ¼ Dr@

@rr@C0

@r

� �; 0 < r < a; ð13Þ

@C0

@r¼ 0; r ¼ 0; ð14Þ

� D@C0

@r¼ kðaC0 � Cs0Þ; r ¼ a: ð15Þ

Eqs. (13) and (14) obviously imply that the leading order concentra-tion is independent of r, i.e.,C0 ¼ C0ðx; t1; t2Þ: ð16ÞThe boundary condition (15) then givesCs0 ¼ aC0: ð17Þ

As expected, at the leading order the mobile phase of the reactivesolute is at local equilibrium with the immobile phase.


For first order (O(�)), Eqs. (5)–(7) give:

@C0

@t1þ @C1

@t0þ u

@C0

@x¼ D

r@

@rr@C1

@r

� �� kCn

0; 0 < r < a; ð18Þ

@C1

@r¼ 0; r ¼ 0; ð19Þ

� D@C1

@r� CC0 ¼

@Cs0

@t1þ @Cs1

@t0¼ kðaC1 � Cs1Þ � ksC

ms0; r ¼ a: ð20Þ

Averaging the Eqs. (18)–(20) w.r.t the fast time variable t0, we get

@C0

@t1þ us

@C0

@x¼ D

r@

@rr@C1

@r

!� kCn

0; 0 < r < a; ð21Þ

@C1

@r¼ 0; r ¼ 0 ð22Þ

and

�D@C1

@r� CC0 ¼

@Cs0

@t1¼ kðaC1 � Cs1Þ � ksC

ms0; r ¼ a; ð23Þ

where the overbar denotes time average over one period of oscilla-tion and us is the steady velocity component. We further take cross-sectional average of (21) subject to the conditions (17), (22) and(23) to obtain

@C0

@t1þ husi

R@C0

@xþ 2C

aRC0 þ

kR

Cn0 ¼ 0; ð24Þ

where

R ¼ 1þ 2aa;

is the retardation factor.Using (17) in (20) and eliminating @C0

@t1from (18)–(20) and (24),

we rewrite the Eqs. (18)–(20) as:

@C1@t0þ u� husi

R

� �@C0@x � 2C

aR C0 þ 2kaaR Cn

0 ¼ Dr@@r r @C1

@r

; 0 < r < a

@C1@r ¼ 0; r ¼ 0

�D @C1@r ¼ �a husi

R@C0@x þ C

R C0 � kaR Cn

0 þ@Cs1@t0

¼ kðaC1 � Cs1Þ � ksamCm0 þ CC0; r ¼ a

9>>>>>=>>>>>;ð25Þ

The structure of (25) suggests the substitutions

C1 ¼ NðrÞ þ ReðBðrÞeixt0 Þ� � @C0

@xþMðrÞC0 þ EðrÞCn

0 þ FðrÞCm0 ð26Þ

and

Cs1 ¼ Ns þ ReðBseixt0 Þ� � @C0

@xþMsC0 þ EsC

n0 þ FsC

m0 ; ð27Þ

where the coefficients N(r), Ns; M(r), Ms; B(r), Bs; E(r), Es and F(r), Fs

satisfy certain boundary value problems that can be obtained bysubstituting (26) and (27) into (25), and equating the coefficientsof analogous terms from both sides, e.g. the steady terms associatedwith @C0

@x prescribes the equation for N(r) and Ns as:

Dr

ddr

rdNdr

� �¼ us �

husiR

; 0 < r < a ð28Þ

with the boundary conditions

dNdr¼ 0; r ¼ 0;

� DdNdr¼ �a

husiR¼ kðaN � NsÞ; r ¼ a:

Again equating the coefficient of C0 from both sides, M(r) and Ms areobtained as:



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15 October 2010

Dr

ddr

rdMdr

� �¼ �2C

aR; 0 < r < a ð29Þ


dMdr¼ 0; r ¼ 0;

� DdMdr¼ C

R¼ kðaM �MsÞ þ C; r ¼ a:

Similarly, equating the unsteady terms linked with @C0@x , we obtain

the following equation for the complex functions B(r) and Bs:

Dr

ddr

rdBdr

� �¼ ixBþ uw; 0 < r < a ð30Þ

with the boundary conditions,

dBdr¼ 0; r ¼ 0;

� DdBdr¼ ixBs ¼ kðaB� BsÞ; r ¼ a:

Equating the coefficient of C0n, we have the following equation for

E(r) and Es:

Dr

ddr

rdEdr

� �¼ 2ka

aRð31Þ


dEdr¼ 0; r ¼ 0;

� DdEdr¼ � ka

R¼ kðaE� EsÞ; r ¼ a:

Finally the coefficient of C0m produces the following equation for

F(r) and Fs:

Dr

ddr

rdFdr

� �¼ 0 ð32Þ


dFdr¼ 0; r ¼ 0; ð33Þ

� DdFdr¼ 0 ¼ kðaF � FsÞ � ksam; r ¼ a: ð34Þ

For second order (O(�2)), (5)–(7) give:

@C0

@t2þ @C1

@t1þ @C2

@t0þ u

@C1

@x¼ D

@2C0

@x2 þDr@

@rðr @C2

@rÞ

� nkCn�10 C1; 0 < r < a; ð35Þ

@C2

@r¼ 0; r ¼ 0 ð36Þ

and

�D@C2

@r� CC1 ¼

@Cs0

@t2þ @Cs1

@t1þ @Cs2

@t0

¼ kðaC2 � Cs2Þ �mksCm�1s0 Cs1 r ¼ a ð37Þ

Averaging the Eq. (35) w.r.t time followed by space subject to theboundary conditions (36) and (37) we have:

@C0

@t2þ 1

R@hC1i@t1

þ 1Rhu @C1

@xi þ 2C

aRC1ðaÞ þ

2aR

@Cs1

@t1þ kn

RCn�1

0 hC1i

¼ DR@2C0

@x2 : ð38Þ

Using (1), (24), (26) and (27), we find each term of (38) as follows:

C1ðaÞ ¼ NðaÞ @C0

@xþMðaÞC0 þ EðaÞCn

0 þ FðaÞCm0


@hC1i@t1

¼ �husihNiR

@2C0

@x2 �1R

khNi þ husihEið Þ @Cn0

@x� husihFi

R@Cm

0

@x

� 1R

2ChNiaþ husihMi

� �@C0

@x� 1

RkhMi þ 2nChEi

a

� �Cn

0

� 2mChFiaR

Cm0 �

nkhEiR

C2n�10 �mkhFi

RCnþm�1

0 � 2ChMiaR

C0

hu @C1

@xi ¼ ðhusNi þ

12

RehuwB�iÞ @2C0

@x2 þ husEi@Cn

0

@xþ husFi

@Cm0

@x

þ husMi@C0

@x

@Cs1

@t1¼ �husiNs

R@2C0

@x2 �1RðkNs þ husiEsÞ

@Cn0

@x� husiFs

R@Cm

0

@x

� 1R

2CNs

aþ husiMs

� �@C0

@x� 1

RkMs þ

2nCEs

a

� �Cn

0

� 2mCFs

aRCm

0 �nkEs

RC2n�1

0 �mkFs

RCmþn�1

0 � 2CMs

aRC0

nCn�10 hC1i ¼ hNi

@Cn0

@xþ nhMiCn

0 þ nhEiC2n�10 þ nhFiCnþm�1

0 ;

where * denotes the complex conjugate.Substitution of these terms in (38) produces

@C0

@t2� D0eff

@2C0

@x2 þ f01@Cn

0

@xþ f02

@Cm0

@xþ f03

@C0

@xþ v10C

n0 þ v02Cm

0

þ v03C2n�10 þ v04Cmþn�1

0 þ v05C0 ¼ 0 ð39Þ

with

D0eff ¼DRþ 1

RNhusi

R� us

� � �þ 2Ns

husiaR2 �

12R

RehuwB0i;

f01 ¼2k

aR2 ahNi � Nsð Þ � 1R

Ehusi

R� us

� � �� 2husiEs

aR2 ;

f02 ¼ �2husiFs

aR2 � 1R

Fhusi

R� us

� � �;

f03 ¼ �1R

Mhusi

R� us

� � �� 2Mshusi

aR2 � 2C

aR2 hNi þ 2Ns

a

� �þ 2C

aRNðaÞ;

v01 ¼nkhMi

Rþ 2C

aREðaÞ � 2nC

aR2 hEi þ 2Es

a

� �� k

R2 hMi þ 2Ms

a

� �;

v02 ¼2CaR

FðaÞ � 2mC

aR2 hFi þ 2Fs

a

� �;

v03 ¼2nk

aR2 ðahEi � EsÞ;

v04 ¼nkhFi

R�mk

R2 hFi þ 2Fs

a

� �;

v05 ¼2CaR

MðaÞ � hMiR

� �� 4CMs

a2R2 :

6. Transport coefficients

Finally, combining the relationships (24) and (39) and recalling(12) one obtains

@C0

@t� Deff

@2C0

@x2 þ f1@Cn

0

@xþ f2

@Cm0

@xþ f3

@C0

@xþ v1Cn

0 þ v2Cm0

þ v3C2n�10 þ v4Cmþn�1

0 þ v5C0 ¼ 0; ð40Þ



Original text:

Inserted Text

give

Original text:

Inserted Text

have

489490

492492

493

494

495

496

497

498

499

500

501

502

503

504505

507507

508

509

510

511

512

513

514

515

516

517519519

520521523523

524525

527527

528

529530

532532

533534

536536

537

538

540540

541542

544544

545

547547

548

550550

551

553553

554

556556

557

559559

560



15 October 2010

where

f3 ¼ f03 þhusi

R; v1 ¼ v01 þ

kR; v5 ¼ v05 þ

2CaR

and the other coefficients are equal to their dashed counterparts.Here Deff is the effective dispersion coefficient, fi’s (i = 1,2,3)

are the effective coefficients for advective terms and vi’s (-i = 1,2, . . . ,5) are the coefficients for the reaction terms. It shouldbe mentioned here that the coefficients v1, v3 and f1 are the resultsof nonlinear reaction in the bulk flow while v2 and f2 signify theeffect of nonlinear reaction at the boundary. The coupled effectof both the reactions can be seen form v4. The linear irreversiblereaction at the boundary is represented by the coefficient v5.

For determination of the coefficients in (40) we need to solve(28)–(32) subject to the respective boundary conditions whichproduces

NðrÞ ¼ Nð0Þ þ husi2D

r2 1� 12R

� �� r2

4a2

� �;

Ns ¼ aNð0Þ þ a 1þ 6aa

� � husia2

8RDþ a

khusi

R;

MðrÞ ¼ Mð0Þ � C2aRD

r2; Ms ¼ aMð0Þ þ �aa2

4Dþ a

k

� �2CaR

;

BðrÞ ¼ Pwx2 1þ ADJ0ðgrÞ � mJ0ðrrÞ

ðm� DÞJ0ðraÞ

� �; Bs ¼

ibx

BðaÞ; where

A ¼ bJ0ðraÞ þ mrJ1ðraÞbJ0ðgaÞ þ DgJ1ðgaÞ ; g2 ¼ � ix

D; b ¼ � ixka

kþ ix;

EðrÞ ¼ Eð0Þ þ ka2aDR

r2; Es ¼ aEð0Þ þ 1kþ aa

2D

� �kaR;

FðrÞ ¼ Fð0Þ; Fs ¼ aFð0Þ � ks

kam;

562562

563

565565

566

568568

569

571571

where N(0), M(0), E(0) and F(0) are undetermined constants. Theseundetermined constants are not required for the evaluation of Deff,fi’s and vi’s when the reactions are linear. Even for nonlinear reac-tions, they are not essential as far as the coefficients Deff, f1, f2, f3, v3

and v5 are concerned for the terms involving those undeterminedconstants in each of the expressions cancel out to zero. But theseconstants are mandatory for the determination of v1, v2 and v4.To evaluate the undetermined constants let us take, without anyloss of generality, that

572

574574

575

576

577

578

579

580

581

hC1ðrÞi ¼ 0;

so that

hNðrÞi ¼ hMðrÞi ¼ hEðrÞi ¼ hFðrÞi ¼ 0;

which yields

Nð0Þ ¼ husia2ð3� 5RÞ24DR

; Mð0Þ ¼ aC4RD

; Eð0Þ

¼ � aka4RD

and Fð0Þ ¼ 0:
582
583

584

585

586

587

588

589

590

591

592

593

Finally when all the coefficients are determined, using the followingnormalised variables

bC0 ¼C0

Cð0Þ; a ¼ a

a; bC ¼ Ca

D; Da ¼

ka2

D;

k ¼ kk

Cn�1ð0Þ ; ks ¼

ks

kam�1Cm�1

ð0Þ

r ¼ ra; g ¼ ga; d ¼ d=a; b ¼ baD; Sc ¼ m

D:

Eq. (40) can be written as


@bC0

@t�1

� DRþ DTs

husi2a2

Dþ DTw

husi2a2

D

" #@2bC0

@x2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}�2

þ husin1@bCn

0

@x�3

þhusin2@bCm

0

@x�4

þhusin3@bC0

@x�5

þ kc1bCn

0�6þ kc2

bCm0

�7

þ kc3bC2n�1

0�8

þ kc4bCmþn�1

0�9

þ kc5bC0

�10¼ 0; ð41Þ

where

DTs ¼1R

1148� 1

3Rþ 1

8R2

� �þ 2a

R3Da

; ð42Þ

DTw ¼8d6w2

ScRRe

d2A� rJ0ðg�ÞJ1ðrÞ � g�J0ðrÞJ1ðg�Þ½ �2ðSc2 � 1ÞjJ0ðrÞj

2 � iArJ1ðgÞ � gJ1ðrÞðSc � 1ÞgrJ0ðrÞ

( );

ð43Þ

A ¼ bJ0ðrÞ þ ScrJ1ðrÞbJ0ðgÞ þ gJ1ðgÞ

; b ¼ �1� iðk=xÞðk=xÞ2 þ 1

Daa;kx¼ Dad2

2Sc

n1 ¼1

6R3 kað2RDa � 3Da � 12aDa � 24Þ; ð44Þ

n2 ¼2R2 ksam; ð45Þ

n3 ¼1R

1þ 2bCR

13� 1

4R

� �� 1� 1

R

� �2

Da

� �" #; ð46Þ

c1 ¼kbC2R3 aR� 2na2 � 8n

aDaþ a� 8

aDaþ 2

R2bC !

; ð47Þ

c2 ¼4mbCDaR2 ks; ð48Þ

c3 ¼ �n

2R3 ak2ðaDa þ 4Þ; ð49Þ

c4 ¼2m

R2 akks; ð50Þ

c5 ¼2R

bCDa

1� 2bCR

18Rþ 1� 1

R

� �1

Da

� �" #ð51Þ

and for convenience, each term of Eq. (41) is marked enclosingnumbers rounded by circle below the individual terms.

Eq. (41) is the one dimensional nonlinear partial differentialequation that models the concentration of the mobile phase of areacting substance which undergoes a nonlinear chemical reactionboth within the flow and at the boundary. The equation clearlyshows how the combined action of two different types of nonlinearreactions changes the transport mechanisms to a great extend.Complex interactions between the mobile phase and immobilephase and fluid dynamic mechanisms that regulate the evolutionof the concentration of the reactive solute, can be accessed fromthe Eq. (41). Recalling the relationship (17), the spatial and tempo-ral behaviour of Cs0(x, t) can also be deduced from (41).

Two special cases can be derived from the general Eq. (41). Inorder to underline the role of nonlinearities, let us explain thesetwo cases separately-.

Case 1: There is no nonlinear reaction in the bulk flow and theboundary, i.e., the solute undergoes only linear irreversible absorp-tion by the wall material and reversible phase exchange with the



Original text:

Inserted Text

1,2,3)

Original text:

Inserted Text

1,2,..,5)

Original text:

Inserted Text

separately-

594

595

596597

599599

600

601

602

603

604

605

606

607608

610610

611

612

613

614615617617

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634634

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672

673

674

675

Fig. 1. Variation of the oscillatory component of the dispersion coefficient (a) withSchmidt number Sc and (b) with oscillation parameter d.



15 October 2010

flowing fluid. In this case k = ks = 0 and consequently the coeffi-cients n1, n2, c1, c2, c3 and c4 as given by the Eqs. (44), (45), (47),(48), (49) and (50) respectively vanish. So Eq. (41) reduces to

@bC0

@t�1

� DRþ DTs

husi2a2

Dþ DTw

husi2a2

D

" #@2bC0


þhusin3@bC0

@x�5

þ kc5bC0

�10¼ 0;

ð52Þ

where DTs, DTw, n3 and c5 are as given by (42), (43), (46) and (51)respectively.

The Eq. (52) coincides with the one obtained by Ng [17] in hisnoteworthy analysis.

Case 2: This case corresponds to when both the reactions occur-ring at the bulk flow and at the boundary are of linear type, i.e.,n = m = 1. In this case the Eq. (41) reduces to a simple equationwhich reads as:

@bC0

@t� Deff

@2bC 0

@x2 þ ðn1 þ n2 þ n3Þ@bC0

@xþ ðc1 þ c2 þ c3 þ c4

þ c5ÞbC0

¼ 0; ð53Þ

where all the coefficients are defined in (42)–(51). According to Gilland Sankarasubramanian [4], the reaction (K0), convection (K1) anddispersion (K2) coefficients (signs are not taking into consideration)can be identified from this model as:

K0 ¼ c1 þ c2 þ c3 þ c4 þ c5; K1 ¼ n1 þ n2 þ n3; K2 ¼ Deff : ð54Þ

It should be mentioned here that K2 is affected only by the revers-ible phase exchange up to the second order of the present analysis.But the first two transport coefficients (K0 and K1) are affected byboth reversible and irreversible reactions. If we proceed to the nextorder (i.e., third order), then only the effects of irreversible reactionon the corresponding K2 can be found out (Ng [17]).

The nature of these coefficients will be discussed in the follow-ing section. In real cases n1, n2 may be neglected with respect to n3

and c3, c4 with respect to other three coefficients (c1,c2,c5). How-ever for this linear case no approximation is made. Approximationsare made only for the case where nonlinearity is concerned.

The event, when n = m, can also be considered as a subcaseresulting from the general Eq. (41). In this case the Eq. (41) reducesto:

@bC0

@t�1

� DRþ DTs

husi2a2

Dþ DTw

husi2a2

D

" #@2bC0


þhusiðn1�3þ n2�4Þ @bCn

0

@x

þ husin3@bC0

@x�5

þkðc1�6þ c2�7ÞbC n

0 þ kðc3�8þ c4�9ÞbC2n�1

0 þ kc5bC0

�10¼ 0 ð55Þ

The coupled effect of the two nonlinear reactions can be observedfrom the Eq. (55).

7. Results and discussions

In the cross sectionally averaged transport equation, the disper-sive, advective and reactive fluxes are controlled by some effectivecoefficients. These transport coefficients are essentially dependenton the factors such as the steady and unsteady flows, cross sec-tional geometry, partition of solute in various phases, chemicalreactions and so on which are need to be prescribed in order tocompute the coefficients. For the present problem the controllingparameters are, among others, a, Da and bC which are respectivelythe phase partition ratio or retention parameter, Damkohler num-ber (rate of reversible reaction at the boundary) and rate of irre-


versible reaction at the boundary. The retention parameter a isthe ratio of solute mass distributed between the phase retainedby the wall and that carried by the flow. In particular, a ¼ 0 for anon-retentive or inert boundary. The number Da is the ratio ofthe phase exchange rate to the diffusion rate. The order of the non-linearities m and n and corresponding rates of the reactions k andks are also required for the determination of the coefficients. Theparameter d is identified as oscillation parameter. It can be notedthat d ¼

ffiffi2p

K , where K ¼ affiffiffixm

pis the Womersley number. Thus d is re-

lated to the reciprocal of the Womersley number. It is an importantnumber used to describe the unsteady nature of fluid flow in re-sponse to an unsteady pressure gradient. It is the measure of theratio of the time a2

m

� �required for viscosity to smooth out the

transverse variation in vorticity to the period of oscillation 1x

. An-

other way of looking at K is that, for periodic flow within a tube, itis the ratio of the tube radius a to the Stoke’s layer thickness

ffiffiffimx

p.

Let us first describe few important features of the two disper-sion coefficients (i.e., DTs and DTw) arising from the steady andoscillatory part of the fluid motion. These features of dispersioncoefficients were reported by Ng [17] in detail. But for the sakeof being self contained, some of them are reproduced in the text.Fig. 1(a) shows the variation of the oscillatory component of thedispersion (DTw) coefficient with the Schmidt number Sc. It is ob-served from the figure that higher value of Sc may lead to smallerDTw, but the effect is significant only when oscillation parameter dis small and a > 0. When Sc is small or d is large enough, DTw nolonger remains sensitive to Sc. Variation of DTw with d can be seenfrom Fig. 1(b). DTw increases monotonically with d and depending



Original text:

Inserted Text

upto

Original text:

Inserted Text

to

676

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678

679

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681

682

683

684

685

686

687

688

689

690

691

692

693

694

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713

714

715

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717

718

719

720

721

722

723

724

725

Fig. 3. Reaction coefficient as a function of a for different values of various reactionrates when n = m = 1.



15 October 2010

on Sc and a, it becomes almost constant beyond a certain d. Herethe limiting value of DTw is taken into account as Sc ? 1 becausethe expression of DTw (Eq. (43)) is indeterminate at Sc = 1. The ef-fects of the phase partitioning and kinetics on the steady and oscil-latory dispersion components DTs and DTw are shown in Fig. 2. It isseen that the phase exchange has quite similar effects on DTs andDTw. The figure confirms that stronger kinetics of the phase ex-change (a slower exchange rate or smaller Da) will give rise to a lar-ger value of either dispersion coefficients. It is remarkable that thedispersion coefficients increase sharply when the wall conditionjust changes from inert to slightly retentive.

When the reactions are linear (i.e., n = m = 1), the reaction coef-ficient K0 and advection/convection coefficient K1 are computedfrom the expression (54). For convenience of discussion, we as-sume X ¼ ks=k so that X < , = , > 1 according to the reaction rateconstant for the mobile phase is greater than, equal to and lessthan the corresponding value for the immobile phase. Variationof the reaction coefficient K0 with the partition coefficient a isshown in Fig. 3(a,b,c). It is clear from the figures that unless therate of reversible or irreversible reaction at the boundary is strong,the reaction coefficient decreases monotonically with a. If either ofthe reaction rate is fast, reaction coefficient decreases sharply foran initial small range of a to reach to a minimum value followedby a steep rise with a over the remaining range. Effect of the reac-tion rates k and ks on the reaction coefficient K0 can be seen fromFig. 3(a). The increase of reaction rates enhances the reaction coef-ficient and for low a this phenomenon is more noticeable. But irre-versible reaction rate at the boundary seems to have quite oppositeeffect on the reaction coefficient that can be seen from Fig. 3(b).Reaction coefficient diminishes as the rate of irreversible reactionbecomes stronger. Also strong kinetics of the phase exchange leadsto decrease of the reaction coefficient. [Fig. 3(c)]. The effects aremore, when a is small.

The isolated effect of the reaction rates can be seen from Fig. 4for different a. It is seen that for all a, reaction coefficient K0 in-creases as ks increases (for fixed k, increase of X leads to increaseof ks). This is not true when ks remains fixed allowing k to vary. Ef-fect of k is found to be negligible on the reaction coefficient. Thusreaction at the boundary is more influential than that in the bulkflow. As demanded earlier, large a inhabits the rising process to agreat extent. It may be mentioned here that the reaction coefficientK0 varies almost linearly with the reaction rate constants k and ks.

The variation of the convection coefficient K1 with the partitioncoefficient a is shown in Fig. 5(a,b). It is observed that, Da should belarge enough for the monotonic decrease of the convection coeffi-cient with a. Otherwise the decrease of K1 is confined within theinitial small range of a and over the remaining range it shows akeen increase [Fig. 5(a)]. The same is true when the rate of irrevers-ible reaction becomes stronger [Fig. 5(b)]. The figure indicates thatK1 decreases with the rate bC of irreversible reaction. But when the

726

727

728

729

730

731

732

733

734

Fig. 2. Variation of the steady (DTs) and unsteady (DTw) components of thedispersion coefficients with the partition coefficient a for different reaction rateparameter Da.

Fig. 4. Variation of the reaction coefficient (K0) with the ratio of the wall-phasereaction rate to the fluid-phase reaction rate (X)(for the continuous lines k is fixedwhile ks is fixed for the dotted lines).


phase exchange rate is very fast (i.e., Da is large), advection coeffi-cient increases with bC [Fig. 6(a)]. Thus reversible and irreversiblereaction processes are not independent. Effect of one can be greatlyinfluenced by the other. The figure also indicates that under thephase exchange with strong kinetics, the advection speed is af-fected by the wall absorption drastically which explains the wellknown result that the reversible phase exchange can retard advec-tion (Ng and Rudraiah [25]). Fig. 6(b) shows the plots of K1 with thereaction rate ratio X. In this case, a has almost negligible effect on



Original text:

Inserted Text

Ω.

735

736

737

738

739

740

741

742

743

744

745

746747

749749

750

751

752

753754

756756

757

758

759

760761

Fig. 5. Convection coefficient as a function of the retention parameter (a) fordifferent values of phase exchange rate Da and (b) for different values of irreversiblereaction rate bC.

Fig. 6. Variation of the convection coefficient K1: (a) with the irreversible reactionrate parameter bC when a ¼ 1, (b) with the ratio of the reaction rates of the wall-phase to the fluid-phase X when Da = 1.

Fig. 7. The nonlinear bulk-flow reaction coefficient c1 as a function of a: (a) fordifferent values of n, (b) for different values of the irreversible reaction rate bC and(c) for different values of Da.



15 October 2010

the rate of rising of K1 with the reaction rates. Although strongerreaction rate leads to an increase of K1, the growth rate is not sosignificant. The linear variation K1 like K0 with respect to the reac-tion rates is also remarkable.

As far as the case of nonlinearity is concerned, numerous non-linear terms in the Eq. (41) indicates how the complex interactionbetween the mobile phase and reaction alters the evolution of thefluid-phase concentration. These terms are important, from a con-


ceptual point of view, as they show the variety of the links betweenthe reactions and transport mechanisms. However for realistic case(i.e. n,m = O(1)), the following conditions may occur (Revelli andRidolfi [22]):

n1@bCn

0

@x; n2

@bCm0

@x

!<< n3

@bC0

@x;

c3bC2n�1

0 ; c4bCmþn�1

0

� �<< c1

bCn0; c2

bC m0 ; c5

bC 0

� �:

As a consequence third, fourth, eighth and ninth terms (numberswithin the circles below the terms indicate its position) of the Eq.(41) can be neglected for application purpose and the equationcan be approximated as

@bC0

@t�1

� DRþ DTs

husi2a2

Dþ DTw

husi2a2

D

" #@2bC0


þ husin3@bC0

@x�5

þ kc1bCn

0�6þ kc2

bCm0

�7þ kc5

bC0�10

¼ 0: ð56Þ

Furthermore, if the nonlinear reaction terms from the Eq. (56) areomitted, the equation agrees with Ng [17].

The behaviour of the three prominent reaction coefficients (i.e.,c1,c2 and c5), which are the functions of strength and kinetics ofthe nonlinear reactions, can be seen here. The ratio X which deter-



762

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811

Fig. 8. Same as Fig. 7, but for the nonlinear wall-phase reaction coefficient c2.Fig. 9. Variation of the reaction coefficients (c1, c2 and c5) with the correspondingrates of reactions (k, ks and bC).



15 October 2010

mines intensity of the boundary reaction over the bulk flow reac-tion is an important factor for the determination of the coefficients.It is interesting to see from Eqs. (47) and (48) that the nonlinearreaction rate constants k and ks are linearly related with their rep-resentative coefficients (i.e., c1 and c2). Consequently the change ofthe reaction rates shows a linear variation with these coefficients,which is not true for the coefficient c5, the result of first order irre-versible reaction at the boundary. The following discussions aredevoted solely for the three nonlinear reaction coefficients. With-out loss of generality, we take m = n = 2.

Fig. 7 shows the behaviour of the reaction coefficient c1 which isassociated with bulk-flow reaction. It is seen that, except for a ini-tial short range, c1 increases with a over a substantiable domainand then it becomes almost constant with respect to a. Whenthe values of n are of O(0,1), the initial decrement of c1 immedi-ately follows stationarity. As the degree of the reaction n increases,the coefficient c1 is found to decrease and this development ismore focused over the transient portion, i.e., where c1 alters itsdirection from lowering down to rising up with a [Fig. 7(a)]. Al-most no exception is found with c1 compared with the coefficientsalready discussed (i.e., K0 and K1) under the variation of the param-eters bC and Da. Stronger bC or weaker Da both causes the coefficientc1 to increase with a beyond a critical value before which the coef-ficient decreases with a [Fig. 7(b,c)].

However behaviour is something exceptional as far as the coef-ficient c2 is concerned which is linked with the nonlinear reversiblereaction at the boundary. Fig. 8(a) shows that c2 decreases mono-


tonically with a for any nonzero value of m, and the decrement issharp when the nonlinearity is of higher order. The monotonicity isstrict for lower range of a and for large a there is no significantchange. It is observed that the nonlinearity of the reaction leadsto increase the coefficient c2. It should be mentioned here that c2

becomes identically equal to zero when m = 0, i.e., the coefficienthas no longer effect on the transport process in presence of thesource term. The effects of reversible and irreversible reaction rates(i.e., Da and bC) on c2 can be seen from the Fig. 8(b,c). Unlike theother coefficients, c2 is found to increase as the rate of irreversiblereaction at the boundary becomes stronger. Also stronger kineticsof the phase exchange give rise to higher value of c2 which is pro-nounced when a is sufficiently small.

The variation of reaction coefficients (c1, c2 and c5) with respectto the corresponding reaction rates (k, ks and bC) are shown inFig. 9(a,b,c) for different values of the retention parameter a. It isremarkable to note that depending upon the retention parameter,the coefficient c1 may decrease with k [Fig. 9(a)]. Figure shows thatlarge a ensures the lowering of c1 with k, which is not the case forthe coefficient c2. It is seen from Fig. 9(b) that for all a, c2 increaseswith ks, though larger a ceases the growth rate of c2. In both casesthe variations are linear. The coefficient c5 is found to decreasemonotonically as the reaction rate bC increases [Fig. 9(c)].



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15 October 2010

8. Conclusions

The purpose of this study is to ascertain the transport coeffi-cients associated with the unsteady flow of a reacting solutethrough a circular tube with a reactive wall layer. The solute is as-sumed to undergo nonlinear reactions both within the flowingfluid and at the boundary. The reaction at the boundary again con-sists of two components due to phase exchange between the flow-ing fluid and the wall layer, which is nonlinear and a linearirreversible component due to absorption into the wall. The moti-vation is to explore the role of nonlinear reactions in the bulk flowand at the boundary on the transport process, which are of funda-mental importance in chemical and biological processes. In the ab-sence of nonlinearity the results are in good agreement with theexisting literature.

Important feature is that as the nonlinearity of the bulk flowreaction increases, its representative coefficient decreases whilethe opposite is true for the nonlinear boundary reaction. Reversiblereaction rate kinetics as well as the rate of irreversible reaction alsoexhibit opposite effect on these two transport coefficients. Thetransport coefficients are largely dominated by the wall-phasereactions compared with the fluid-phase reactions. These twotypes of reactions make a coupled effect on the transport coeffi-cients so that the influence of one may be largely affected by thestrength of the other reactions. The coefficients are found to begreatly motivated by the kinetics of the reactions when the reten-tive effect of the wall is slow.

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Acknowledgement

The authors thank Professor Chiu-On Ng of the University ofHong Kong for providing partial financial support from the Re-search Grants Council of the Hong Kong Special Administrative Re-gion, China, through Project No. HKU 7156/09E during BSM’s visitto his department. Authors wish to acknowledge the anonymousreviewers for their valuable suggestions for bringing the paper intoits present form.

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