European Journal of Mechanics B/Fluids 29 (2010) 483–493

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European Journal of Mechanics B/Fluids

journal homepage: www.elsevier.com/locate/ejmflu

Convection, diffusion and reaction inside a spherical porous pellet in thepresence of oscillatory flowJai Prakash a, G.P. Raja Sekhar a,∗,1, Sirshendu De b, Michael Böhm ca Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721 302, Indiab Department of Chemical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721 302, Indiac Centre for Industrial Mathematics, University of Bremen, 28334 Bremen, Germany

a r t i c l e i n f o

Article history:Received 15 July 2009Received in revised form22 April 2010Accepted 18 May 2010Available online 31 May 2010

Keywords:Stokes flowDarcy’s lawPorous pelletThiele modulusOscillatory flowNutrient transport

a b s t r a c t

The problem of convection, diffusion and reaction inside a spherical porous pellet is investigatedanalytically. Unsteady Stokes equation is used for the flow outside the porous pellet and Darcy’slaw is used inside the pellet. A solenoidal decomposition method is employed for the hydrodynamicproblem. Following the above findings, the convection–diffusion–reaction problem is formulated andsolved analytically for a first order reaction. The behavior of the nutrient transport is discussed withrespect to various parameters like Darcy number, Peclet number, frequency and Thiele modulus. Alsothe effectiveness factor corresponding to the first order reaction is computed.

© 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

Oscillatory flow has been an interesting subject of study foryears. Many applications of oscillatory flow to engineering prob-lems have been investigated. Some of the examples are electro-chemical reactors [1], soils [2], biofilms [3], and ultrafiltration [4].It is well known that the mass transfer of a species is enhanced byseveral orders of magnitude when it is present in a fluid mediumsubjected to oscillatorymotion. There are several examples ofmasstransfer processes where oscillatory flow plays an important role.Mass transport enhancement via oscillatory flow has been stud-ied for numerous geometries: a curved tube (Eckmann and Grot-berg [5]), a flexible tube (Dragon and Grotberg [6]), a tube withconductive walls (Jiang and Grotberg [7]) and many more.Large-pore permeable particles are currently used as catalysts

adsorbents, high performance liquid chromatography packing,supports for biomass growth, ceramic membranes. The conceptbehind various applications is that in large-pore materials

∗ Corresponding author. Tel.: +91 3222 283684.E-mail addresses: rajas@maths.iitkgp.ernet.in, rajasgp@gmail.com

(G.P. Raja Sekhar).1 Part of work is done while the corresponding author is at Institute for AppliedAnalysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, D70569 Stuttgart, Germany as Alexander von Humboldt Experienced Researcher.

0997-7546/$ – see front matter© 2010 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.euromechflu.2010.05.002

intraparticle mass transport is not only due to diffusion but alsodue to convection inside the pores. In the present article, ouraim is to discuss the mass transport inside a spherical porouscatalyst in an oscillatory flow in which the convection–diffusionprocess is coupled with isothermal first order reaction kinetics.The coupling of chemical kinetics and mass transport process inporous catalytic particles has been the subject of intense study fora long time. An extensive study of the earlier work can be foundin the treatise, The Mathematical Theory of Diffusion and Reaction inPermeable Catalysts by Aris [8]. In fact, several studies have shownthat the total transport rate inside the particle can be enhancedsignificantly due to the contribution of intraparticle convection inlarge-pore catalysts. The importance of intraparticle convection oncatalyst particles is shownbyKomiyama and Inoue [9]. They solvedthe problem of intraparticle convection–diffusion and reaction ina finite cylinder parallel to the flow for a first order reactionand provided a technique to obtain the intraparticle velocity.A theoretical justification describing the effect of intraparticleconvection for an irreversible, isothermal first order reaction isgiven by Nir and Pismen [10] for rectangular (slab), cylindrical andspherical geometries.The problem of convection and diffusion in permeable isother-

mal particles was discussed by Rodrigues et al. [11] and the anal-ogy between slab and spherical geometries was established. Luet al. [12] extended this convection–diffusion study to convec-tion–diffusion–reaction study with first order reaction. Convec-tive flow inside non-isothermal catalyst was studied by Rodrigues

484 J. Prakash et al. / European Journal of Mechanics B/Fluids 29 (2010) 483–493

Nomenclature

a radius of the porous pellet [m]k permeability of the porous pellet [m2]r radial distanceX position vectorve oscillatory velocity external to the porous pellet

[m/s]pe oscillatory pressure external to the porous pellet

[N/m2]Ve amplitude of the oscillatory velocity external to the

porous pellet [m/s]Pe amplitude of the oscillatory pressure external to the

porous pellet [N/m2]Vi velocity internal to the porous pellet [m/s]P i pressure internal to the porous pellet [N/m2]p0 constant [N/m2]U characteristic velocity [m/s]V0 basic velocity [m/s]V∗ velocity due to the disturbance [m/s]A, B scalarsfn modified spherical Bessel function of first kindgn modified spherical Bessel function of second kindSn, Tn spherical harmonicsPmn associated Legendre polynomialci concentration inside the porous pellet [mole/m3]k′ rate constant [s−1]D diffusivity [m2/s]c0 concentration at surface of the porous pellet

[mole/m3]c dimensionless concentrationDa Darcy numberPe Peclet number|V eθ | magnitude of external tangential velocity|V i| magnitude of internal velocity

Greek symbols

θ inclinationϕ azimuth angleα slip coefficientλ dimensionless parameterω frequency of oscillation [s−1]$ dimensionless frequency of oscillationρ density of the fluid [kg/m3]µ dynamic viscosity [kgm−1s−1]ν kinematic viscosity [m2/s]φ Thiele modulusχn eigenvalue of differential operatorψn symbol for the eigenfunction of a differential

operatorηc effectiveness factor

and Quinta [13]. They considered a slab geometry and the effectof intraparticle convection on the effective diffusivity measure-ment was analyzed. Lopes et al. [14] have shown that when in-traparticle convection is important, non-isothermal effect plays asignificant role in the intraparticle flow field of catalyst particles.Cardoso et al. [15] have considered the interaction between trans-port of heat by conduction and convection, transport of reactantsby diffusion, convection and chemical reactionwithin a porous cat-alyst particle of slab geometry. They used combination of pertur-bation and integral techniques to derive approximate analyticalsolutions for the concentration profile for a first order, non-isothermal reaction. The coupling of external flow field with

internal flow field and the effect of intraparticle convectionon porous catalyst particles was shown by Stephanopoulosand Tsiveriotis [16]. They considered the combined convec-tion–diffusion–reaction problem with zeroth order kinetics to ob-tain the intraparticle flow field and intraparticle nutrient transportinside the pellet. However, as mentioned earlier, the mass transferof species will be enhanced significantly in the presence of oscil-latory motion. Hence, it is worth investigating the correspondingproblem in case of oscillatory flow. Recently, Khaled and Vafai [17]wrote a review article on the role of porousmedia inmodeling flowand heat transfer in biological tissues. It is mentioned that in appli-cations like osteoinductive devices for repair of bone defects, ad-sorptive separation by high performance liquid chromatography,transport of macromolecules in deformed artery walls convectivetransport needs to be considered.The present article considers the mass transport inside

a spherical porous catalyst in an oscillatory flow in whichthe convection–diffusion process is coupled with isothermalfirst order reaction kinetics. Unsteady Stokes equation is usedoutside the porous pellet and Darcy’s law inside the porouspellet. The intraparticle flow field is obtained by the solenoidaldecomposition of the external velocity field. The computedhydrodynamic part is used to evaluate the nutrient transportvia the combined convection–diffusion–reaction equation. It iswell known that mass transfer resistance is expressed in termsof mass transfer coefficient. The non-dimensional form of masstransfer coefficient is Sherwood number. Sherwood number isproportional to Peclet number (Sh ∼ Pe1/3) [16]. It is establishedin Eq. (36) of Stephanopoulos and Tsiveriotis [16] that theratio of internal to external mass transfer resistances for thepermeable spherical particle is much greater than 1. Therefore,one can neglect the external mass transfer resistance in thetreatment of convection–diffusion–reaction problem inside apermeable particle. The combined convection–diffusion–reactionproblem is solved for first order reaction occurring in thespherical porous pellet subject to theDirichlet boundary condition.Analytical expressions are obtained for concentration profiles andthe behavior of concentration profile with various parametersinvolved is discussed. It is shown in [10] that intraparticleconvection enhances the effectiveness of a first order reaction.Hence, the variation of effectiveness factor with parametersinvolved is analyzed. Some comparisons are donewith the existingresults.

2. Mathematical formulation

A spherical porous pellet of radius a and permeability k isconsidered in an arbitrary oscillatory Stokes flow of a viscousincompressible fluid (Fig. 1). It is assumed that the flow outsidethe porous pellet is governed by unsteady Stokes flow and theflow inside is described by steady Darcy’s law. It may be notedthat Looker and Carnie [18] have shown that homogenizationof unsteady Stokes equations inside porous medium gives timeindependent Darcy’s equation and hence the flow inside the pelletgoverned by steady Darcy’s law is justified. Consequently, the flowinside the pellet (r < a) is governed by the Darcy’s law andcontinuity equation:

Vi = −kµ∇P i, (1)

∇ · Vi = 0, (2)where, k is the permeability of the porous medium, µ is thecoefficient of viscosity of the fluid. The flow outside the pellet (r >a) is described by the unsteady Stokes and continuity equations:

ρ∂ve

∂t= −∇pe + µ∇2ve, (3)

∇ · ve = 0, (4)

J. Prakash et al. / European Journal of Mechanics B/Fluids 29 (2010) 483–493 485

Fig. 1. Geometry of the problem.

where ρ is the density of the fluid. For the case of oscillatory flowwith frequency ω, we set the velocity and pressure fields ve and peas ve = Vee−iωt and pe = Pee−iωt . Thus, the governing equationstransform to

−iρωVe = −∇Pe + µ∇2Ve, (5)

∇ · Ve = 0. (6)

Here Ve and Pe represent the velocity and pressure fields outsidethe porous pellet, and Vi and P i are those of the flow inside theporous pellet. The physical quantities are non-dimensionalized byusing the variables X = X

a , V = VU , P =

Pµ Ua. Here U is the

magnitude of the far field uniform velocity. Therefore, the non-dimensional equations for the flow inside the porous region (r <1) take the form

Vi = −Da∇P i, (7)

∇ · Vi = 0, (8)

and the corresponding equations for the fluid region (r > 1)reduce to

(∇2 − λ2)Ve = ∇Pe, (9)

∇ · Ve = 0, (10)

where λ2 = − iωa2

ν, and Da = k

a2is the Darcy number. Note that

we have omitted the symbol ˜ from Eqs. (7)–(10).

2.1. Boundary conditions

The obvious boundary condition at a permeable interface isthe continuity of the normal velocity, which is consequence ofthe incompressibility. In order to have a completely determinedflow of the free fluid, some condition on the tangential componentof the free fluid velocity needs to be specified at the interface.Classically, vanishing of the tangential velocity of the free fluidat the porous interface was supposed. However, this conditionis not satisfactory for porous interface. Beavers and Joseph [19]proposed a new condition postulating that the difference betweenthe slip velocity of the free fluid and the tangential componentof seepage velocity is proportional to the shear rate of freefluid. They have verified this law experimentally and found thatthe proportionality constant depends linearly on square root ofpermeability. Saffman further studied the experimental boundarycondition of Beavers and Joseph and pointed out that the seepagevelocity was much smaller than the other quantities hence maybe dropped [20]. In general, while matching Darcy’s law with theStokes equation, continuity of pressure and continuity of normalvelocity components are used along with Saffman’s slip conditionfor tangential velocity components [21]. Looker and Carnie [18]showed that Saffman’s boundary condition can be applied foroscillatory Stokes flows at least under low frequency. The

corresponding non-dimensionalization forces that the solution ofLooker and Carnie is valid for larger values of Brinkman parametera/√k. Similar limitations hold for the present investigation as

we employ Saffman’s condition at the porous–liquid interfacetogether with the continuity of pressure and continuity of normalvelocity. Hence, the boundary conditions on r = 1 are given by:

(i) Continuity of pressure field: Pe = P i(ii) Continuity of normal velocity component: V er = V

ir

(iii) Saffman’s boundary condition for the tangential componentsof velocity field:

V eθ =

√Daα

∂V eθ∂r, V eϕ =

√Daα

∂V eϕ∂r,

where α is the dimensionless slip coefficient.

2.2. Method of solution

It may be noted that Eq. (9) which is the Stokes equation in thecase of oscillatory flow is mathematically similar to the Brinkmanequation that is frequently used for porous media except that themeaning of the parameter λ is different. Hence, some features ofthe existence theory for the Brinkman equation can be adopted.Padmavathi et al. [22] have assumed that any velocity vector andpressure scalar satisfying equations of the form Eqs. (9) and (10)can be expressed as

Ve = ∇ × ∇ × (AeX)+∇ × (BeX), (11)

Pe = p0 +∂

∂r[r(∇2 − λ2)Ae], (12)

where X is the position vector of the current point, p0 is aconstant, and Ae and Be are unknown scalar functions satisfying theequations

∇2(∇2 − λ2)Ae = 0, (∇2 − λ2)Be = 0. (13)

Moreover, Raja Sekhar et al. [23] have shown via solenoidaldecomposition of the velocity vector that the expressions givenin Eqs. (11)–(13) form the general solution of Brinkman equation.Furthermore, this complete general solution has been used to solveproblems dealing with viscous flow past porous objects [24–26].In the present investigation, we employ this complete generalsolution. Let us now assume that the velocity field V0 of the basicflow, i.e., of the unperturbed flow in the absence of any boundariesis given by

V0 = ∇ × ∇ × (A0X)+∇ × (B0X), (14)

A0 =∞∑n=1

[αnrn + βnfn(λr)

]Sn(θ, ϕ),

B0 =∞∑n=1

γnfn(λr)Tn(θ, ϕ), (15)

where Sn(θ, ϕ) and Tn(θ, ϕ) are spherical harmonics of the form

Sn(θ, ϕ) =n∑m=0

Pmn (ζ )(Anm cosmϕ + Bnm sinmϕ), ζ = cos θ, (16)

Tn(θ, ϕ) =n∑m=0

Pmn (ζ )(Cnm cosmϕ + Dnm sinmϕ), ζ = cos θ, (17)

where Pmn are associated Legendre polynomials and Anm, Bnm, Cnm,Dnm are the known coefficients. The coefficients αn, βn, γn arearbitrary constants and corresponding to a given basic flow in theabsence of any boundaries, αn, βn, γn take a suitable form. Forexample in case of uniform flow along the z-axis, we have α1 =

486 J. Prakash et al. / European Journal of Mechanics B/Fluids 29 (2010) 483–493

1/2, β1 = 0, γ1 = 0. In addition, the scalar functions A0 and B0satisfy Eq. (13). Itmay be noted that the scalars Ae, Be represent theflow field and the vector equations are now reduced to equivalentscalar equations.On the other hand, if the basic flow with the velocity field V0

is perturbed by the presence of a stationary porous pellet with theradius r = 1, then the velocity fieldVe of the resulting flow outsidethe porous pellet is given by Ve = V0+V∗, where V∗ is the velocitydue to the disturbance flow such that V∗ → 0 as r →∞. It maybe noted that in the above decomposition, the basic flow as well asthe perturbed flow are oscillatory in nature. Hence, the resultingflow in the exterior region (r > 1) is given by

Ae =∞∑n=1

[αnrn +

α′n

rn+1+ βnfn(λr)+ β ′ngn(λr)

]Sn(θ, ϕ), (18)

Be =∞∑n=1

[γnfn(λr)+ γ ′ngn(λr)

]Tn(θ, ϕ), (19)

where fn(λr) and gn(λr) are modified spherical Bessel functionsof first and second kind, respectively. Since in the porous region(r < 1) the pressure field is harmonic and finite at the origin, itcan be expressed as

P i = p0 +∞∑n=1

δnrnSn(θ, ϕ), (20)

where (r, θ, ϕ) are spherical coordinates with respect to the originchosen at the center of the sphere r = 1. In the above expressionsα′n, β

′n, γ′n and δn are unknown constants that are determined from

the boundary conditions and are obtained in terms of the knowncoefficients αn, βn and γn as follows:

α′n = (n+ 1)

×

[{Xngn(λ)+ λ(α + l)(1− l2λ2)gn+1(λ)}αn + λ(α + l)Ynβn

]Zn

,

β ′n = −(n+ 1){(1− l2λ2)αn + fn(λ)βn} + (n+ 1+ nl2λ2)α′n

(n+ 1)gn(λ),

γ ′n =γn{(α − nl)fn(λ)− lλfn+1(λ)}(nl− α)gn(λ)− lλgn+1(λ)

,

δn = λ2{nα′n − (n+ 1)αn},

where

Xn = λ2{l2(n+ 1)α − l3(n2 + λ2 − 1)− l},Yn = fn(λ)gn+1(λ)+ fn+1(λ)gn(λ),

Zn =[l{n(n+ 1)(n+ 2)− (n2 + λ2 − 1)(nλ2l2 + n+ 1)}

+α(n+ 1)(nλ2l2 + 2n+ 1)]gn(λ)

− λ(α + l)(nλ2l2 + n+ 1)gn+1(λ),

l =√Da.

The velocity components both outside and inside the pellet canbe obtained using the above expressions. In case of uniformflow along z-axis, setting V0 = Uk, we may notice that thecorresponding expressions for A0 and B0 in dimensionless form areA0 = 1

2 r cos θ, B0 = 0. Comparing with the general expressionsgiven in Eq. (15), we have α1 = 1

2 , β1 = 0 and γ1 = 0. In this case,the corresponding velocity components in fluid region (r > 1) are

V er =

1+ X1g1(λ)+ λ(α + l)(1− l2λ2)g2(λ)r3Z1

−

[(1− l2λ2)Z1 + (l2λ2 + 2){X1g1(λ)+ λ(α + l)(1− l2λ2)g2(λ)}

]g1(λr)

2rZ1g1(λ)

× cos θ, (21)

V eθ = −

1− X1g1(λ)+ λ(α + l)(1− l2λ2)g2(λ)r3Z1−

2rg1(λr)− λg2(λr)

×

[(1− l2λ2)Z1 + (l2λ2 + 2){X1g1(λ)+ λ(α + l)(1− l2λ2)g2(λ)}

]2Z1g1(λ)

× sin θ, (22)

and inside the porous pellet (r < 1)

V ir = −l2δ1 cos θ, (23)

V iθ = l2δ1 sin θ, (24)

where

δ1 =λ2[(lλ2 − 6α − 6l)g1(λ)+ 3λ(α + l)g2(λ)

]Z1

,

and δ1, X1, and Z1 correspond to δn, Xn, and Zn when n = 1. Theexplicit solution calculated above for the hydrodynamic problemhas been used in order to evaluate the nutrient transport via thecombined convection–diffusion–reaction equation (25).The use of Saffman condition brings limitations on the

permeability range. Looker and Carnie [18] concluded that Saffmancondition is applicable under low frequency. Vainshtein andShapiro [27] calculated the force acting on a permeable particlein oscillatory flow using the Brinkman and the Darcy equation.However, they have used continuity of velocity componentstogether with the continuity of stress components. It may benoted that in case of Brinkman equation these boundary conditionsare accepted by a large community, whereas in case of Darcyequation, the continuity of tangential velocity needs to be replacedby Beavers–Joseph/ Saffman type slip condition. Vainshtein andShapiro [27] identified a critical value of the Brinkman parameter,a/√k, that controls the applicability of Darcy equation. They

also observed that this critical value diminishes with decreasingfrequency of oscillations and reaches that of a non-oscillatingparticle ≈10. It seems that critical value can be as large as 200for high frequency of oscillations. For low and moderate valuesof frequency, one can identify the corresponding critical valueof the Brinkman parameter. Hence, the hydrodynamic problemof oscillatory flow past a porous sphere with Darcy equationinside subject to Saffman condition limits one to consider only aparticular range of permeability.

3. Nutrient transport inside the porous pellet

The hydrodynamic velocity components internal to the porouspellet that are computed in the previous section are incorporatedin the definition of the combined transport–reaction problem. Inwhat follows a solution is presented for the case of a sphericalporous catalyst. Assuming an isothermal first order reactionkinetics and diffusivityD, the nutrientmass balance can bewrittenas

Vi · ∇ci = D∇2ci − k′ci. (25)

On using the expressions for velocity Vi given in Eqs. (23) and (24),we have from Eq. (25)

− Ul2δ1k · ∇ci = D∇2ci − k′ci. (26)

J. Prakash et al. / European Journal of Mechanics B/Fluids 29 (2010) 483–493 487

Eq. (25) applies to a spherical pellet with a concentration boundarycondition at the porous–liquid interface equal to the bulk nutrientconcentration

ci = c0 on r = a. (27)

Now, the physical quantities are non-dimensionalized by usingthe variables X = X

a , c = cic0. After non-dimensionalization the

governing equation together with the boundary condition is givenby

−Pe l2δ1k · ∇ c = ∇2c− φ2c, (28)

c = 1, on r = 1, (29)

with

φ2 =k′a2

D, Pe =

UaD, (30)

whereφ is the Thielemodulus corresponding to the above problemand Pe is the Peclet number. The solution of Eq. (28) with boundarycondition (29) is obtained by introducing the transformation

c = c exp

(−Pe l2δ1z2

), z = r cos θ, (31)

which reduces Eq. (28) to the form

Pe2l4δ214

c = (∇2 − φ2)c in r < 1. (32)

The corresponding boundary condition takes the form

c = exp

(Pe l2δ1 cos θ

2

), on r = 1. (33)

In spherical coordinates, (r, θ, ϕ), Eqs. (32) and (33) become(φ2 +

Pe2l4δ214

)c =

1ξ 2

∂

∂ξ

(ξ 2∂ c∂ξ

)

+1ξ 2

∂

∂ζ

[(1− ζ 2)

∂ c∂ζ

]in r < 1, (34)

c = exp

(Pe l2δ1ζ2

), on ξ = 1 (35)

where ξ = r, ζ = cos θ . The differential operator with respect toζ of Eq. (34) has the following form:

(L+ χ)ψ = 0, (36)

where

L =ddζ

[(1− ζ 2)

ddζ

],

subject to the boundary conditions c = 1 at z = ±1.The only physically acceptable eigenfunctions of the above

operator are the Legendre polynomials of first kind:

ψn = Pn(ζ )with eigenvalues χn = n(n+ 1),

(n = 0, 1, 2, . . .). (37)

Thus the following Legendre–Fourier series is assumed for thesolution of Eq. (34)

c =∞∑n=0

Hn(ξ)Pn(ζ ) =∞∑n=0

Hn

(ra

)Pn(cos θ). (38)

With respect to the inner product

〈f , g〉 =∫ 1

−1f (x)g(x)dx, (39)

the differential operator of Eq. (36) is self-adjoint and itseigenfunctions are orthogonal. Using this inner product, Eqs. (34)and (35) can be projected to the eigenfunction space, yielding(φ2 +

Pe2l4δ214

)Hn =

1ξ 2

∂

∂ξ

(ξ 2∂Hn∂ξ

)− n(n+ 1)

1ξ 2Hn, (40)

22n+ 1

Hn(1) =∫+1

−1exp

(Pe l2δ1ζ2

)Pn(ζ )dζ . (41)

The solution of Eq. (40)with Eq. (41) as boundary condition is givenby the following series:

Hn(ξ) = an(ξ)nFn(ξ). (42)

Therefore Eq. (40) reduces to the form

F ′′n +(2n+ 1)

ξF ′n −

(φ2 +

Pe2l4δ214

)Fn = 0, (43)

since ξ is a regular singular point of Eq. (43), we assume thesolution of the form

Fn(ξ) =∞∑m=0

bmξm, (44)

and solve Eq. (43). As a result, we get b1 = 0 and obtain thefollowing recurrence relation

bm =

(φ2 +

Pe2 l4δ214

)m(m+ 2n+ 1)

bm−2. (45)

Since b1 = 0, bm = 0, m = 1, 3, 5, . . . , and for m even saym = 2j, the expression for bm becomes

b2j =

(φ2 +

Pe2 l4δ214

)2j(2j+ 2n+ 1)

b2j−2. (46)

Eq. (46) can be written as

b2j = b0j∏i=1

(φ2 +

Pe2 l4δ214

)(n+ 2i)2 + (n+ 2i)− n(n+ 1)

. (47)

Therefore solution for Fn in terms of the variable r/a is given by

Fn

(ra

)= 1+

∞∑j=1

(ra

)2j

×

j∏i=1

(φ2 +

Pe2 l4δ214

)(n+ 2i)2 + (n+ 2i)− n(n+ 1)

, (48)

here b0 is taken as 1 for simplicity. Now from the boundarycondition given in (41), we have

Hn(1) =2n+ 12

∫+1

−1exp

(Pe l2δ1ζ2

)Pn(ζ )dζ ,

488 J. Prakash et al. / European Journal of Mechanics B/Fluids 29 (2010) 483–493

=2n+ 12

(−1)n

2nn!

∫+1

−1(ζ 2 − 1)n

dn

dζ n

{exp

(Pe l2δ1ζ2

)}dζ ,

=2n+ 12n+1n!

∫+1

−1(1− ζ 2)n

(Pe l2δ12

)nexp

(Pe l2δ1ζ2

)dζ . (49)

We note that Hn(1) = anFn(1), and hence from Eq. (49) we get

an =2n+ 12n+1n!

1Fn(1)

∫+1

−1(1− ζ 2)n

(Pe l2δ12

)n

× exp

(Pe l2δ1ζ2

)dζ . (50)

The expression (38) and the transformed equation (31) are used toobtain the nutrient concentration profile inside the pellet given by

ci(r, θ) = c0 exp

(−Pe l2δ1r cos θ

2a

)

×

∞∑n=0

an

(ra

)nFn

(ra

)Pn(cos θ), (51)

where Fn and an are given by Eqs. (48) and (50). It can be seenthat 0 ≤ ci ≤ 1 in non-dimensional form and it is a convexfunction. The above expression for the concentration correspondsto the Dirichlet boundary condition at the porous pellet surface.Similar computationsmay help in order to use a flux condition andthe corresponding calculation is deferred for a future investigation.The above problem has been solved corresponding to first

order reaction kinetics. There are other common kinetics suchas, Michaelis–Menten kinetics, Monod kinetics, which occur inimmobilized enzymes and growing populations of immobilizedcells or microorganisms. But these kinetics lead to nonlinearproblem and the present solution methodology is no morevalid to handle such type of kinetics. It may be mentionedthat the kinetics dealt in the present article is a special caseof the Michaelis–Menten and Monod kinetic models, when theconcentration tends to zero.

4. Results and discussion

The analytical solution derived in the previous sections allow usto determine the velocity and nutrient concentration profiles. Thecharacteristics of the velocity and concentration profiles are showngraphically. The frequency of oscillation is assumed to be between1 kHz and 10 kHz and a2/ν = 10−3 s.

4.1. Velocity profiles

The tangential velocity component V eθ versus r/a is shown inFig. 2. In general, fluid experiences less resistance in the presenceof the porous pellet in comparison to impermeable particle, asimpermeable particle expects no slip on the surface. However, aporous pellet with smaller Darcy number offers larger resistancecompared to that of with larger Darcy number. It can be seenfrom Fig. 2(a) that when the frequency of oscillation is low, themagnitude of the uniform far field basic velocity slightly increasesand near to the body the velocity reduces due to the resistanceoffered by the particle. However, for large frequencies, the increasein the magnitude of the uniform far field basic velocity is more(Fig. 2(b)). When Da = 0, which corresponds to a solid particle,the uniform far field velocity slightly increases before it realizesthe resistance due to the particle and then decreases to satisfy theno-slip condition. Fig. 3 shows the variation in magnitude of the

internal velocity with frequency and Darcy number. It is observedthat the internal velocity increases with both frequency and Darcynumber. Increase in Darcy number offers more volume flow into the porous region for a fixed pressure drop. Also, increase inoscillation in the flow field induces enhancement in volumetricflow inside the pellet for a fixed Darcy number.

4.2. Concentration profiles

The characteristics of the concentration profile is showngraphically. Here nutrient concentration profile along the z-axisis given for a combination of various parameters involved inEq. (51), like Darcy number (Da), frequency ($), slip coefficient(α), Thiele modulus (φ) and Peclet number (Pe). Due to thereaction inside the porous pellet, the concentration inside reducescompared to the bulk nutrient concentration at the surface. Alsothe concentration contours are influenced by the flow direction.Fig. 4(a) shows the concentration profile in case of steady flowi.e. $ = 0, for a fixed Da, α and φ. It can be seen that forPe = 0, there is a symmetry about the center of the pellet asthis is the case of diffusion only and the minimum concentrationoccurs at z = 0. But for a non-zero Pe, the concentrationminimum is shifted in the direction of the flow due to convection.The increase in minimum concentration with increasing Pe ismarginal. Effect of Peclet number on nutrient concentration profileis shown in Fig. 4(b) for the combination of parameters Da =0.005, α = 0.5, φ = 1, $ = 3. It is seen that increasing Peincreases the overall nutrient concentration throughout the pellet.Since at low Peclet number diffusion dominates convection, thepellet cannot experience significant nutrient transport. However,as Peclet number increases convection becomes a dominatingfactor and hence nutrient transport increases inside the pellet. Inconclusion, as Pe increases, the nutrient concentration minimummoves downstreamwhile the overall nutrient content of the pelletincreases.A comparison between steady and oscillatory flow is shown

in Fig. 5(a). It can be easily seen that oscillatory flow assistsconvection and hence more nutrient is transported inside thepellet compared to steady flow. Though the general tendency ofPe is to assist convection, in case of steady flow the effect of Peis marginal whereas in case of oscillatory flow it is substantial.Effect of frequency on the nutrient concentration profile is shownin Fig. 5(b) for the combination of parameters Pe = 100, Da =0.005, α = 0.5, φ = 1. It is observed that as $ increasesthe overall nutrient concentration increases inside the pellet. Inconclusion, as $ increases, the nutrient concentration minimummoves downstreamwhile the overall nutrient content of the pelletincreases.Fig. 6(a) shows the concentration profile for the combination of

parameters Pe = 100, $ = 3, α = 0.5, φ = 1 with varyingDa. Increasing Da offers less resistance and fluid flows easilythrough the pellet resulting enhancement of the overall nutrient.The nutrient concentration minimum moves downstream withincreasing Darcy number. Fig. 6(b) shows the concentration profilefor the combination of parameters Pe = 100, Da = 0.005, $ =3, α = 0.5 with varying φ. When Thiele modulus is small,the reaction rate is less compared to the diffusion rate, and thepellet concentration becomes nearly uniform. For large valuesof the Thiele modulus, the reaction rate is large compared tothe diffusion rate and one can see sharp concentration gradientsnear the surface of catalyst. Stephanopoulos and Tsiveriotis [16]considered the steady Stokes flow outside the porous pellet andformulated the convection–diffusion–reaction problem for zerothorder reaction. They have obtained analytical expression for thenutrient transport inside the pellet and observed starvation zoneinside the pellet for a particular choice of parameters involved. In

J. Prakash et al. / European Journal of Mechanics B/Fluids 29 (2010) 483–493 489

a b

Fig. 2. Variation in magnitude of the external tangential velocity (V eθ ) with r/a.

a b

Fig. 3. Variation in magnitude of the internal velocity (V i) with (a) frequency ($). (b) Darcy number (Da).

a b

Fig. 4. Nutrient concentration profile along z-axis, Da = 0.005, α = 0.5, φ = 1. (a) steady case. (b)$ = 3.

order to avoid starvation a relationship between Peclet numberand the Thiele modulus was established. Prakash et al. [28] haveextended the study of Stephanopoulos and Tsiveriotis [16] to thecase of oscillatory flow and discussed the effect of frequency ofoscillation on nutrient transport and observed similar starvationzones for some choice of parameters. It is noted that in the currentstudy which corresponds to first order reaction, we do not see

any such starvation zones. In case of zeroth order reaction, forlarge values of the Thiele modulus, surface reaction is rapid. Thereactant is consumed fast into the interior of the pellet leadingto the exhaustion of the nutrient resulting to a starvation zone.Whereas, in case of first order reaction kinetics, nutrient does notget exhausted as the reaction proceeds and therefore no starvationzones occur. It may be noted here that the starvation region

490 J. Prakash et al. / European Journal of Mechanics B/Fluids 29 (2010) 483–493

a b

Fig. 5. Nutrient concentration profile along the z-axis. (a) comparison between steady and oscillatory cases. (b) Pe = 100,Da = 0.005, α = 0.5, φ = 1 with$ .

a b

Fig. 6. Nutrient concentration profile along the z-axis. (a) Pe = 100,$ = 3, α = 0.5, φ = 1. (b) Pe = 100,Da = 0.005,$ = 3, α = 0.5.

mentioned above for zeroth order kinetics is a mere mathematicaldescription to limit the solution physically meaningful. In realitythe concentration at a location may not be zero or extremelylow, specially in cases like endogenous metabolism and substrateuptake for cellular energetic, where the nutrient concentrationmay be a finite value. Another idea to avoid such starvation isto introduce the concept of dead zone. Valdés–Parada et al. [29]derived an approximate solution for a stirred tank bioreactor withnonlinear kinetics. The approximate solution contains a zero orderterm that gives negative values of the concentration. In order toavoid this they introduced a non-reaction zone called dead zone.These kind of criterion [28,29] to avoid starvation display thelimitations of the models used.In Fig. 7, we have shown the variation of concentration for the

combination of parameters Da = 0.005, $ = 3, α = 0.5, φ =2.5 with varying Pe. It is seen that the concentration minimumtowards the center of the pellet decreases for large values of theThielemodulus.When Thielemodulus is equal to 1, rate of reactionand diffusion are comparable and the concentration minimumtowards the center does not decrease much but as the Thielemodulus increases the reactant is consumed very fast into theinterior of the pellet and the concentration minimum towardsthe center of the pellet decreases. In Fig. 8, we have shown acomparison with Da in the range similar to that of coarse gels(fibers) and polymers with that of a representative other material.It may be noted that since fibers have low permeability, in order

Fig. 7. Nutrient concentration profile along the z-axis Da = 0.005,$ = 3, α =0.5, φ = 2.5 with Pe.

to have convection the range of Pe is very high compared toa typical other material having large permeability. Hence, therewould be a trade-off between Da and Pe as depending on thematerial properties one has to choose suitable range. However,since a particular material may have a fixed permeability, theremay be a limitation in order to vary permeability keeping effective

J. Prakash et al. / European Journal of Mechanics B/Fluids 29 (2010) 483–493 491

a b

Fig. 8. Nutrient concentration profile along the z-axis for polymers, fibers with$ = 5, α = 0.5, φ = 1.

a b

Fig. 9. Variation of effectiveness factor (ηc)with φ. (a) Pe = 200,Da = 0.007, α = 0.5. (b) Pe = 200,$ = 5, α = 0.5.

a b

Fig. 10. Variation of effectiveness factor (ηc)with frequency ($). (a) Da = 0.005, α = 0.5, φ = 1. (b) Pe = 100, α = 0.5, φ = 1.

diffusivity and vice versa. Hence, the simulations shown may beunderstood keeping this limitation in view.

4.3. Effectiveness factor

Strieder and Aris [30] have shown via variational inequalitiesthat the effect of first order reaction will be enhanced byintraparticle convection and the same fact has been explained

briefly by Nir and Pismen [10]. Hence, it is worth investigatingthe impact of various parameters on the effectiveness factor.The ratio of the observed reaction rate to the rate in theabsence of intraparticle mass transfer resistance is defined as theeffectiveness factor. The effectiveness factor in case of first orderreaction for a spherical geometry is defined as [10]

ηc =32φ2

∫ 1

−1

∂ c∂r

∣∣∣∣r=1dζ . (52)

492 J. Prakash et al. / European Journal of Mechanics B/Fluids 29 (2010) 483–493

Fig. 11. Variation of Cmin with Darcy number (Da).

After some algebra we obtain

ηc =3π

φ2Pe l2δ1

∞∑n=0

(−1)n(2n+ 1)I2n+1/2(Pe l2δ12

)

×

β2In+3/2

(β

2

)In+1/2

(β

2

) − Pe l2δ12

In+3/2(Pe l2δ12

)In+1/2

(Pe l2δ12

) , (53)

with β =√Pe2l4δ21 + 4φ2.

Fig. 9 depicts the variation of effectiveness factor with Thielemodulus for various values of$ andDa. It is clear that in the rangeof φ < 1, the effect on ηc is marginal whereas the drop is strikingin the more practical range of 1 < φ < 10. Fig. 10(a) showsthe effectiveness factor for the combination of parameters Da =0.005, α = 0.5, φ = 1 with varying $ . It is observed thateffectiveness factor increases with increase in Pe for a fixed $ .This is because as Pe increases the surface reaction rate decreasesas a result effectiveness factor increases. Fig. 10(b) shows theeffectiveness factor for the combination of parameters Pe =100, α = 0.5, φ = 1 with varying Da. Increasing Da enhancesthe effectiveness factor for a fixed Pe and φ.

4.4. Variation of minimum concentration

For a fixed combination of other parameters, the variation ofminimum concentration with Da is shown in Fig. 11. It is seen thatthe concentration minimum increases with increasing Da, due toincreased volume flow inside the pellet.

5. Conclusion

The nutrient transport inside a spherical porous pellet in anoscillatory flow has been studied. The model assumes a sphericalporous pellet placed in an oscillatory Stokes flow. Darcy’s law isused inside the porous pellet and unsteady Stokes equation is usedoutside the pellet. Velocity and pressure fields are obtained andthe convection–diffusion–reaction problem is formulated for a firstorder reaction kinetics. Effect of various parameters on the velocityas well as concentration profile is shown. It is observed that theinternal velocity increases both with frequency and permeability,hence for a fixed Darcy number, increase in frequency enhancesthe volume flow inside the pellet. The effect of frequency on thenutrient concentration profile is discussed for various combinationof parameters involved and a significant effect of frequency of

oscillation is seen on the nutrient transport inside the pellet. Effectof the Thiele modulus on concentration profile is shown. It is seenthat for small values of the Thiele modulus nutrient diffuses verywell into the interior of the pellet and for large values of theThiele modulus the surface reaction is rapid and the concentrationminimum decreases towards the center of the pellet. Also theeffectiveness factor is obtained and the variation with variousparameters is shown graphically. A significant enhancement in theeffectiveness factor is observed.

Acknowledgements

The first author (JP) would like to recognize the support ofCouncil of Scientific and Industrial Research (CSIR), India. Thesecond author (GPRS) acknowledges the support by Alexander vonHumboldt Foundation, Germany for the Fellowship.

References

[1] N. Carpenter, E. Roberts, Mass transport and residence time characteristics ofan oscillatory flow electrochemical reactor, Chem. Eng. Res. Des. 77 (1999)212–217.

[2] D. Neeper, A model of oscillatory transport in granular soils, with applicationto barometric pumping and earth tides, J. Cont. Hyd. 48 (2001) 237–252.

[3] H. Nagaoka, Mass transfer mechanism in biofilms under oscillatory flowconditions, Water Sci. Technol. 36 (1996) 329–336.

[4] S. Ilias, R. Govind, Potential applications of pulsed flow for minimizingconcentration polarization in ultrafiltration, Sep. Sci. Tech. 25 (2001)1307–1324.

[5] D. Eckmann, J. Grotberg, Oscillatory flow andmass-transport in a curved tube,J. Fluid Mech. 188 (1988) 509–527.

[6] C. Dragon, J. Grotberg, Oscillatory flow and mass-transport in a flexible tube,J. Fluid Mech. 231 (1991) 135–155.

[7] Y. Jiang, J. Grotberg, Bolus contaminant dispersion in oscillatory tube flowwithconductive walls, J. Biomech. Eng. Trans. ASME 115 (1993) 424–431.

[8] R. Aris, The Mathematical Theory of Diffusion and Reaction in PermeableCatalysts, Clarendon, Oxford, 1975.

[9] H. Komiyama, H. Inoue, Effects of intraparticle convective flow on catalyticreactions, J. Chem. Eng. Jpn. 7 (1974) 281–286.

[10] A. Nir, L. Pismen, Simultaneous intraparticle forced convection, diffusion andreaction in a porous catalyst, Chem. Eng. Sci. 32 (1977) 35–41.

[11] A.E. Rodrigues, J.C.B. Lopes, M.M. Dias, G. Carta, Diffusion and convection inpermeable particles: Analogy between slab and sphere geometries, Sep. Tech.2 (1992) 208–211.

[12] Z.P. Lu, M.M. Dias, J.C.B. Lopes, G. Carta, A.E. Rodrigues, Diffusion, convection,and reaction in catalyst particles: Analogy between slab and spheregeometries, Ind. Eng. Chem. Res. 32 (1993) 1839–1852.

[13] A.E. Rodrigues, R. Quinta Ferreira, Convection, diffusion and reaction in alarge–pore catalyst particle, AIChE Symp. Ser. 84 (1988) 80–87.

[14] J.C.B. Lopes, M.M. Dias, V.G. Mata, A.E. Rodrigues, Flow field and non-isothermal effects on diffusion, convection, and reaction in permeablecatalysts, Ind. Eng. Chem. Res. 34 (1995) 148–157.

[15] S.S.S. Cardoso, A.E. Rodrigues, Diffusion and reaction in a porous catalyst slab:perturbation solutions, AIChE J. 52 (2006) 3924–3932.

[16] G. Stephanopoulos, K. Tsiveriotis, The effect of intraparticle convection onnutrient transport in porous biological pellets, Chem. Eng. Sci. 44 (1989)2031–2039.

[17] A.R.A. Khaled, K. Vafai, The role of porous media in modeling flow and heattransfer in biological tissues, Int. J. Heat Mass Transfer 46 (2003) 4989–5003.

[18] J.R. Looker, S.L. Carnie, The hydrodynamics of an oscillating porous sphere,Phys. Fluids 16 (2004) 62–72.

[19] G.S. Beavers, D.D. Joseph, Boundary conditions at a naturally permeable wall,J. Fluid Mech. 30 (1967) 197–207.

[20] P.G. Saffman, On the boundary condition at the surface of a porous medium,Stud. Appl. Math. 50 (1971) 93–101.

[21] G.P. Raja Sekhar, T. Amaranath, Stokes flow past a porous sphere with animpermeable core, Mech. Res. Comm. 23 (1996) 449–460.

[22] B.S. Padmavathi, T. Amaranath, S.D. Nigam, Stokes flow past a porous sphereusing Brinkman model, Z. Angew. Math. Phy. 44 (1993) 929–939.

[23] G.P. Raja Sekhar, B.S. Padmavathi, T. Amaranath, Complete general solution ofthe Brinkman equations, Z. Angew. Math. Mech. 77 (1997) 555–556.

[24] P. Levresse, I. Manas–Zloczower, D.L. Feke, Hydrodynamic analysis of porousspheres with infiltrated peripheral shells in linear flow fields, Chem. Eng. Sci.56 (2001) 3211–3220.

J. Prakash et al. / European Journal of Mechanics B/Fluids 29 (2010) 483–493 493

[25] Anindita Bhattacharyya, G.P. Raja Sekhar, Viscous flow past a porous sphere-effect of stress jump condition, Chem. Eng. Sci. 59 (2004) 4481–4492.

[26] M.K. Partha, P.V.S.N. Murthy, G.P. Raja Sekhar, Viscous flow past a porousspherical shell-effect of stress jump boundary condition, J. Eng. Mech. 131(2005) 1291–1301.

[27] P. Vainshtein, M. Shapiro, Forces on a porous particle in an oscillating flow,J. Colloid Interface Sci. 330 (2009) 149–155.

[28] Jai Prakash, G.P. Raja Sekhar, Sirshendu De, Michael Böhm, A criterion to

avoid starvation zones for convection–diffusion–reaction problem inside aporous biological pellet under oscillatory flow, Internat. J. Engrg. Sci. 48 (2010)693–707.

[29] F.J. Valdés–Parada, J. Álvarez–Ramírez, J.A. Ochoa–Tapia, An approximatesolution for a transient two–phase stirred tank bioreactor with nonlinearkinetics, Biotechnol. Prog. 21 (2005) 1420–1428.

[30] W. Strieder, R. Aris, Variational Methods Applied to Problems of Diffusion andReaction, Springer–Verlag, Heidelberg, 1973.