a criterion to avoid starvation zones for convection–diffusion–reaction problem inside a porous...

International Journal of Engineering Science 48 (2010) 693–707

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International Journal of Engineering Science

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A criterion to avoid starvation zones for convection–diffusion–reactionproblem inside a porous biological pellet under oscillatory flow

Jai Prakash a, G.P. Raja Sekhar a,*,1, Sirshendu De b, Michael Böhm c

a 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 30 October 2009Received in revised form 7 January 2010Accepted 11 February 2010Available online 6 March 2010Communicated by K.R. Rajagopal

Keywords:Stokes flowDarcy’s lawSaffman conditionOscillatory flowNutrient transportStarvation zone

0020-7225/$ - see front matter � 2010 Elsevier Ltddoi:10.1016/j.ijengsci.2010.02.004

* Corresponding author. Tel.: +91 3222 283684.E-mail address: [email protected] (G.P.

1 Part of work is done while the corresponding aut57, D 70569 Stuttgart, Germany as Alexander von Hu

a b s t r a c t

The behavior of nutrient transport inside a porous spherical pellet in an oscillatory Stokesflow is investigated analytically. Unsteady Stokes equations are used for the flow outsidethe porous pellet and Darcy’s law is used inside the pellet. A solenoidal decompositionmethod is employed for the derivation of the flow field outside the pellet. The correspond-ing convection–diffusion–reaction problem is formulated and solved analytically for a zer-oth-order nutrient consumption rate. From the obtained solution a general conditionbetween the Peclet number and Thiele modulus is derived to obviate the nutrient reduc-tion everywhere in the pellet. For the correct modeling of the processes involving flowthrough biological catalysts this becomes a necessary and sufficient condition.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

It is well known that the mass transfer of a porous species is enhanced by several orders of magnitude when it is presentin a fluid medium subjected to oscillatory motion. This enhancement takes place even if there is no net total flow over a cycleof the oscillation. The physics of enhanced mass transfer is explained clearly by Thomas and Narayanan [1]. Vibrating eitherthe surface or the fluid surrounding it may cause an oscillatory flow field [2–5]. This oscillatory motion approach is very use-ful particularly in diffusion limited processes such as electrochemical reaction as well as biomedical and biochemical engi-neering, membrane filtration process, etc. In the field of biochemical engineering, oscillatory motion plays a significant roleto enhance the mass transfer rate in biofilms and bioreactors [6,7]. There are other examples where oscillatory motion hasshown significant performance improvement such as ultrafiltration, microfiltration, electrodialysis, and electrophoresis [8–11]. Although the significant effect of oscillatory motion on mass transfer enhancement has been demonstrated, an under-standing of the mechanisms and their contribution to the enhancement factor still is a point of concern in porous catalysts.Porous biocatalysts such as immobilized enzymes or cells are used as bioreactors in biochemical processes. A bioreactor maybe defined as a biocatalyst in a container. The biocatalyst may be anyone of a wide variety of chemically active biologicalsubstances, such as, enzymes, hormones and other non-living substances and living microorganisms, such as, bacteria,viruses, yeast, plant and animal cells, etc. Many biological processes involve the aggregation of cells in to a cluster and porous

. All rights reserved.

Raja Sekhar).hor is at Institute for Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldringmboldt Researcher.

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Nomenclature

a radius of the porous pellet (m)k permeability of the porous pellet (m2)X 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)Pi pressure internal to the porous pellet (N/m2)p0 constant pressure (N/m2)U1 magnitude of the far field uniform 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 harmonicsPm

n associated Legendre polynomialci concentration inside the porous pellet (mole/m3)S nutrient uptake rate (mole/m3 s)D diffusivity (m2/s)c0 concentration at surface of the porous pellet (mole/m3)c dimensionless concentrationDa Darcy numberPe Peclet numberjVe

hj magnitude of external tangential velocityjVij magnitude of internal velocity

Greek symbolsa slip coefficientk dimensionless parameterx frequency of oscillation (s�1)- dimensionless frequency of oscillationq density of the fluid (kg/m3)l dynamic viscosity (kg m�1 s�1)m kinematic viscosity (m2/s)/ Thiele modulus

694 J. Prakash et al. / International Journal of Engineering Science 48 (2010) 693–707

pellets are formed by cell aggregation during growth. The coupling of chemical reaction and diffusive transport in porouscatalytic particles has been the subject of intense investigation in chemical engineering for many years.

In The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Aris [12] presented a comprehensive review ofthe literature up to 1975. Since then, a number of scientific publications have considered the effect of transport by convec-tion, in addition to diffusion, inside porous catalyst particles undergoing chemical reaction [13–16]. Large-pore permeableparticles are currently used as catalysts adsorbents, high performance liquid chromatography packings, supports for biomassgrowth, ceramic membranes, permeable packings in perfusion chromatography for protein separation, bioseparations. Theconcept behind the various applications is that in large-pore materials intraparticle mass transport is not only due to diffu-sion but also to convection inside the pores. The importance of intra-particle convection when large-pore materials are usedhas been considered in bioreaction engineering [15,17,18]. The problem of convection, diffusion and reaction in porous cat-alysts of different geometries has been studied by showing the equivalence between results corresponding to differentgeometries [16]. Because of its symmetry characteristics the spherical geometry has been widely used. In situation in whichintraparticle convection is important, some analytical solutions for linear systems for the spherical geometry have been ob-tained [13,15,19]. The contribution of intraparticle convection to the total mass transfer increases with the magnitude of theintraparticle mass Peclet number. Nir and Pismen [13] considered intraparticle forced convection on a heterogeneous reac-tion within a porous catalyst pellet and observed a significant enhancement of the catalyst performance. They have consid-ered first order reaction in a slab, cylinder and spherical geometries. Lu et al. [16] considered intraparticle convection–diffusion problem in porous slab and porous sphere with first order reaction kinetics. An equivalence ratio between charac-teristic dimensions of the slab and the sphere was obtained as a function of intraparticle Peclet number and Thiele modulus.In the limiting case of Peclet number and Thiele modulus, these two geometries show analogous behavior. In many

J. Prakash et al. / International Journal of Engineering Science 48 (2010) 693–707 695

situations where the external mass transfer is neglected, the Dirichlet boundary condition is employed, i.e., internal concen-tration is equal to the external bulk nutrient concentration on the boundary, cf. [13,15,16,20]. It is interesting to note thatwhen this boundary condition is used together with a first order reaction kinetics, the internal concentration lies between0 and 1, while in case of zeroth-order kinetics, we can observe starvation zones for some particular combination of param-eters like Peclet number and Thiele modulus. This physical ambiguity needs to be addressed and a reasonable criterionensuring avoidance of such starvation zones is required. In [15], Stephanopoulos and Tsiveriotis have considered theintraparticle convective flow inside a porous spherical pellet. They have obtained the intraparticle flow field by a regular per-turbation method which has been used to evaluate external mass transfer. It has been shown that the external mass transferresistance is much less than the internal mass transfer resistance hence can be ignored. A convection–diffusion–reactionproblem is solved for zeroth-order reaction inside the pellet. Also a criterion is proposed to avoid starvation.

The purpose of the present paper is to investigate the impact of the Dirichlet condition in connection with zero orderkinetics inside a spherical porous biocatalyst in presence of oscillatory Stokes flow. As a by product we obtain a criterionfor (non-) existence of starvation zones. On the other hand, the influence of convective transport on the total rate of masstransfer into porous biological catalysts deserve special attention. In particular, for large biocatalyst particles, diffusion alonemay not account completely for nutrient transport. In such cases convective mass transfer might play an important role. Theimportance of convective mass transfer in a characterizable root culture system has been obtained by Prince et al. [18],where it is experimentally verified that for larger catalyst particles inertial effects (convection) must be taken into account.Ferguson et al. [21] investigated fluid flow and convective transport of solutes within the intervertebral disc. AssumingDarcy’s law inside the tissue they have calculated the fluid flow patterns within the intervertebral disc subject to an inducedload and determined the relative contribution of diffusion and convection to solute transport. They have observed that dur-ing swelling of the tissue under load, convection enhanced the penetration of solutes. Mattern et al. [22] estimated Darcypermeability of Agarose–Glycosaminoglycan gels using a fiber mixture model. The idea is to model permeability of biologicaltissues where the flow resistance is largely due to gel like materials. They used Darcy’s law and predicted the effects of fibersize, volume fraction on the Darcy permeability. Frey et al. [23] investigated the effect of intraparticle convection on thechromatography of biomacromolecules. They developed a model for the analysis of mass transfer in spherical particles ofbidisperse pore structures when both convection and diffusion takes place in the larger pores but only diffusion occurs inthe smaller pores. However, on the macro scale, if we consider larger pores, the effect of intraparticle convection can beunderstood by approximate models like spherical porous pellets.

In this work, we discuss the nutrient transport inside a spherical porous pellet in a viscous flow. We assume unsteadyStokes flow outside the porous pellet and Darcy’s law inside the pellet. The intraparticle flow field is obtained by the sole-noidal decomposition of the external velocity field. The explicitly calculated hydrodynamic part is used to evaluate the nutri-ent transport via the combined convection–diffusion–reaction equation. The external mass transfer has been neglected. Thecombined convection–diffusion–reaction problem is solved for a zeroth-order biological reaction occurring in porous biocat-alyst placed in an oscillatory Stokes flow. We follow the ansatz [15] to solve the nutrient transport equation. Analyticalexpressions are obtained for the concentration profiles. Also, we derive necessary and sufficient conditions ensuring thenon-occurrence of starvation zones.

2. Oscillatory Stokes flow past a spherical porous pellet

2.1. Governing partial differential equations

A spherical porous pellet of radius a and permeability k occupying the domain Xi � R3 is considered in an arbitrary oscil-latory flow of a viscous incompressible fluid (Fig. 1). Let the boundary r ¼ a be denoted by C. It is assumed that the flow in-side the porous pellet, i.e., Xi is described by Darcy’s law, and that the flow outside the pellet, i.e., Xe ¼ R3 nXi is governed byunsteady Stokes flow. It has been shown by homogenization that the macroscopic unsteady Stokes equations in a periodic

0c=c

V0

z

Ω

Ωi

Γ e

Fig. 1. Geometry of the problem.


porous medium have the same form as for steady Stokes flow i.e., Darcy’s law (see [24]). Hence addition of the time deriv-ative term is redundant. Consequently, the flow inside the pellet is governed by the Darcy’s law and continuity equation:

Vi ¼ � klrPi in Xi; ð1Þ

r � Vi ¼ 0 in Xi; ð2Þ

where k is the permeability of the porous medium and l is the viscosity of the fluid. The flow outside the pellet is describedby the unsteady Stokes and continuity equations:

q@ve

@t¼ �rpe þ lr2ve in Xe; ð3Þ

r � ve ¼ 0 in Xe; ð4Þ

where q is the density of the fluid. For the case of oscillatory flow with frequency x, we set the velocity and pressure fields ve

and pe as ve ¼ Vee�ixt and pe ¼ Pee�ixt . Thus, the governing equations transform to

� iqxVe ¼ �rPe þ lr2Ve in Xe; ð5Þr � Ve ¼ 0 in Xe: ð6Þ

Here Ve and Pe represent the velocity and pressure fields outside the porous sphere, and Vi and Pi are those of the flow insidethe porous sphere. The physical quantities are non-dimensionalized by using the variables~Vi ¼ Vi=U1; ~X ¼ X=a;~r ¼ r=a; eP ¼ P= lU1

a . Here U1 is the magnitude of the far-field uniform velocity. Therefore, the non-dimensional equations for the flow inside the porous region take the form

Vi ¼ �DarPi in Xi; ð7Þr � Vi ¼ 0 in Xi; ð8Þ

and the corresponding equations for the fluid region reduce to

r2 � k2� �

Ve ¼ rPe in Xe; ð9Þ

r � Ve ¼ 0 in Xe; ð10Þ

where k2 ¼ � ixa2

m , and Da ¼ ka2 is the Darcy number. Note that we have omitted the symbol � from the Eqs. (7)–(10).

It may be noted that the Eq. (9), which is the Stokes equation in the case of oscillatory flow, is mathematically similar tothe Brinkman equation that is frequently used for porous media except that the meaning of the parameter k is different.Hence, some features of the existence theory for the Brinkman equation can be adopted. Padmavathi et al. [25] have assumedthat any velocity vector and pressure scalar satisfying the equations of the form Eqs. (9) and (10) can be expressed as

Ve ¼ r�r� ðAexÞ þ r � ðBexÞ; ð11Þ

Pe ¼ p0 þ@

@rrðr2 � k2ÞAeh i

; ð12Þ

where x is the position vector of the current point, p0 is a constant, and Ae and Be are unknown scalar functions satisfying theequations

r2ðr2 � k2ÞAe ¼ 0; ðr2 � k2ÞBe ¼ 0: ð13Þ

Moreover, Raja Sekhar et al. [26] have shown via solenoidal decomposition 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 usedto solve problems dealing with viscous flow past porous objects [27–29]. In the present paper, we employ this general solu-tion. Let us now assume that the velocity field V0 of the basic flow, i.e., of the unperturbed flow in the absence of any bound-aries is given by

V0 ¼ r�r� ðA0xÞ þ r � ðB0xÞ; ð14Þ

A0 ¼X1n¼1

anrn þ bnfnðkrÞ½ �Snðh;uÞ; B0 ¼X1n¼1

cnfnðkrÞTnðh;uÞ; ð15Þ

where Snðh;uÞ and Tnðh;uÞ are spherical harmonics of the form

Snðh;uÞ ¼Xn

m¼0

Pmn ðnÞ Anm cos muþ Bnm sin muð Þ; n ¼ cos h; ð16Þ

Tnðh;uÞ ¼Xn

m¼0

Pmn ðnÞ Cnm cos muþ Dnm sin muð Þ; n ¼ cos h; ð17Þ


where Pmn are associated Legendre polynomials and Anm;Bnm;Cnm;Dnm are the known coefficients. The coefficients an; bn; cn are

arbitrary constants and corresponding to a given basic flow in the absence of any boundaries, an; bn; cn take a suitable form.For example in case of uniform flow along the z-axis, we have a1 ¼ 1=2; b1 ¼ 0; c1 ¼ 0. In addition, the scalar functions A0 andB0 satisfy the Eq. (13). It may be noted that the scalars Ae

;Be represent the flow field and the vector equations are now re-duced to equivalent scalar equations.

On the other hand, if the basic flow with the velocity field V0 is perturbed by the presence of a stationary porous pellet Xi,then the velocity field Ve of the resulting flow outside the porous pellet is given by Ve ¼ V0 þ V�, where V� is the velocity dueto the disturbance flow such that V� ! 0 as jxj ! 1. It may be noted that in the above decomposition, the basic flow as wellas the perturbed flow are oscillatory in nature. Hence, the resulting flow in the exterior region Xe is given by

Ae ¼X1n¼1

anrn þ a0nrnþ1 þ bnfnðkrÞ þ b0ngnðkrÞ

� �Snðh;uÞ; ð18Þ

Be ¼X1n¼1

cnfnðkrÞ þ c0ngnðkrÞ� �

Tnðh;uÞ; ð19Þ

where fnðkrÞ and gnðkrÞ are modified spherical Bessel functions of first and second kind, respectively. Since in the porous re-gion Xi the pressure field is harmonic and finite at the origin, it can be expressed as

Pi ¼ p0 þX1n¼1

dnrnSnðh;uÞ; ð20Þ

where ðr; h;uÞ are spherical coordinates with respect to the origin chosen at the center of the sphere. In the above expres-sions a0n; b

0n; c0n and dn are unknown constants that are to be determined from the boundary conditions.

2.2. Boundary conditions

In general, for matching Darcy’s law with the Stokes equation, continuity of pressure and continuity of normal velocitycomponents are used along with Saffman’s slip condition for tangential velocity components (see Raja Sekhar and Amar-anath [30]). Looker and Carnie [24] showed that Saffman’s boundary condition can be applied for oscillatory Stokes flowsat least under low frequency. Recently, Raja Sekhar et al. [31] have used Saffman’s condition while discussing oscillatory flowwithin a porous particle contained in a fixed or fluidized bed, in which the porous particle is placed in a spherical fluid enve-lope. Therefore, we consider the following boundary conditions on the boundary between the porous and fluid regions, i.e.,on C:

(i) Continuity of the pressure field: Pe ¼ Pi.(ii) Continuity of the normal velocity component: Ve

r ¼ Vir .

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

Veh ¼

ffiffiffiffiffiffiDap

a@Ve

h

@r; Ve

u ¼ffiffiffiffiffiffiDap

a@Ve

u

@r;

where a is the dimensionless slip coefficient.

(iv) The far-field condition Ve ! V0 as jxj ! 1.

2.2.1. Closing the problemNow, using these boundary conditions the unknown coefficients a0n; b

0n; c0n and dn are determined in terms of the known

coefficients an; bn and cn, and are given by

a0n ¼ðnþ 1Þ XngnðkÞ þ kðaþ lÞð1� l2k2Þgnþ1ðkÞ

n oan þ kðaþ lÞYnbn

h iZn

; ð21Þ

b0n ¼ �ðnþ 1Þ ð1� l2k2Þan þ fnðkÞbn

n oþ nþ 1þ nl2k2� �

a0nðnþ 1ÞgnðkÞ

; ð22Þ

c0n ¼cn ða� nlÞfnðkÞ � lkfnþ1ðkÞf gðnl� aÞgnðkÞ � lkgnþ1ðkÞ

; ð23Þ

dn ¼ k2 na0n � ðnþ 1Þan

; ð24Þ


where

Xn ¼ k2fl2ðnþ 1Þa� l3ðn2 þ k2 � 1Þ � lg; ð25ÞYn ¼ fnðkÞgnþ1ðkÞ þ fnþ1ðkÞgnðkÞ; ð26Þ

Zn ¼ lfnðnþ 1Þðnþ 2Þ � ðn2 þ k2 � 1Þðnk2l2 þ nþ 1Þghþaðnþ 1Þðnk2l2 þ 2nþ 1Þ

ignðkÞ � kðaþ lÞðnk2l2 þ nþ 1Þgnþ1ðkÞ; ð27Þ

and l ¼ffiffiffiffiffiffiDap

. The velocity components both outside and inside the pellet, can be obtained using the expressions above. Incase of uniform flow along the z-axis, setting V0 ¼ U1k, we may notice that the corresponding expressions for A0 and B0

in dimensionless form are A0 ¼ 12 r cos h;B0 ¼ 0: Comparing with the general expressions given in Eq. (15), we have

a1 ¼ 12 ; b1 ¼ 0 and c1 ¼ 0. In this case, the corresponding velocity components in Xe are

Ver ¼ 1þX1g1ðkÞ þ kðaþ lÞð1� l2k2Þg2ðkÞ

r3Z1�ð1� l2k2ÞZ1 þ ðl2k2 þ2Þ X1g1ðkÞþ kðaþ lÞð1� l2k2Þg2ðkÞ

n oh ig1ðkrÞ

2rZ1g1ðkÞ

24 35cosh;

ð28Þ

Veh ¼� 1�X1g1ðkÞþkðaþ lÞð1� l2k2Þg2ðkÞ

r3Z1� 2

rg1ðkrÞ�kg2ðkrÞ

� ��ð1� l2k2ÞZ1þðl2k2þ2Þ X1g1ðkÞþkðaþ lÞð1� l2k2Þg2ðkÞ

n oh i2Z1g1ðkÞ

24 35sinh;

ð29Þ

and in Xi

Vir ¼ �l2d1 cos h; ð30Þ

Vih ¼ l2d1 sin h; ð31Þ

where

d1 ¼k2 ðlk2 � 6a� 6lÞg1ðkÞ þ 3kðaþ lÞg2ðkÞ� �

Z1;

and X1 and Z1 correspond to Xn and Zn when n ¼ 1. The explicit solution calculated above for the hydrodynamic problemhas been used in order to evaluate the nutrient transport via the combined convection–diffusion–reaction Eq. (32).

3. Nutrient transport inside the porous pellet

Here we discuss the combined transport-reaction problem. In what follows a solution is presented for the case of a spher-ical biological catalyst. Assuming a uniform consumption rate �S for the limiting nutrient, where S is positive, i.e., zeroth-order reaction kinetics and diffusivity D, the nutrient mass balance can be written as

Vi � rci ¼ Dr2ci � S in Xi: ð32Þ

It is assumed that in the spherical biocatalyst pellet, the reaction front moves inward maintaining the spherical shape. It is acommon approximation in chemical engineering applications that the concentration profile of the reactant inside the sphereremains in steady state, even though the interface of the spherical shrinking core moves slowly inward. This is known aspseudo steady state approximation and therefore the concentration field can be considered as steady [32,33]. It is also as-sumed that the external mass transfer resistance is absent due to strong agitation. Eq. (32) is applied to the spherical pelletXi with the Dirichlet boundary condition at the porous–liquid interface, i.e., the internal concentration is equal to the bulknutrient concentration given by

ci ¼ c0 on C: ð33Þ

The physical quantities are non-dimensionalized as in Section 2, together with ~c ¼ c0�ci

c0. Non-dimensionalization of the gov-

erning equations yield

Pe Vi � rc ¼ r2cþ /2 in Xi; ð34Þc ¼ 0 on C; ð35Þ

with

/2 ¼ Sa2

c0D; Pe ¼ U1a

D; ð36Þ


where / is the Thiele modulus and Pe is the Peclet number. Note that we have omitted the symbol � from the Eqs. (34) and(35). On using the expressions for the velocity Vi given in (30) and (31), we have from Eq. (34)

�Pel2d1 cos h@c@r� sin h

r@c@h

� �¼ r2cþ /2 in Xi: ð37Þ

Now we present an analytical solution for the above convection–diffusion–reaction problem.

3.1. Analytical solution

In order to solve the problem above, let us introduce the following transformation

c ¼ c exp � Pel2d1z2

!� /2z

Pel2d1

; z ¼ r cos h; ð38Þ

which reduces the Eq. (37) to

Pe2l4d21

4c ¼ r2c in Xi: ð39Þ

The corresponding boundary condition takes the form

c ¼ /2z

Pel2d1

expPel2d1z

2

!on C: ð40Þ

In spherical coordinates ðr; h;uÞ, Eqs. (39) and (40) become

Pe2l4d21

4c ¼ 1

n2

@

@nn2 @c@n

� �þ 1

n2

@

@f1� f2 � @c

@f

� �in Xi; ð41Þ

c ¼ /2f

Pel2d1

expPel2d1f

2

!on C; ð42Þ

where n ¼ r; f ¼ cos h. The differential operator with respect to f of Eq. (41) has the following form:

ðLþ vÞw ¼ 0; ð43Þ

where

L ¼ ddf

1� f2 � ddf

� �;

subject to the boundary conditions c ¼ 0 at z ¼ 1.The only physically acceptable eigenfunctions of the above operator are the Legendre polynomials of first kind

wn ¼ PnðfÞ with eigenvalues vn ¼ nðnþ 1Þ; ðn ¼ 0;1;2; . . .Þ: ð44Þ

Thus the following Legendre–Fourier series expansion is assumed for the solution of Eq. (41)

c ¼X1n¼0

HnðnÞPnðfÞ ¼X1n¼0

Hnra

� �Pnðcos hÞ; ð45Þ

with respect to the inner product

hf ; gi ¼Z þ1

�1f ðxÞgðxÞdx; ð46Þ

the differential operator of Eq. (43) is self-adjoint and its eigenfunctions are orthogonal. Using this inner product, Eqs. (41)and (42) can be projected to the eigenfunction space, yielding

Pe2l4d21

4Hn ¼

1n2

@

@nn2 @Hn

@n

� �� nðnþ 1Þ 1

n2 Hn in Xi; ð47Þ

22nþ 1

Hnð1Þ ¼/2

Pel2d1

Z þ1

�1f exp

Pel2d1f2

!PnðfÞdf on C: ð48Þ

The solution of Eq. (47) with Eq. (48) as boundary condition is given by the series

HnðnÞ ¼ /2anðnÞnFnðnÞ: ð49Þ


Therefore Eq. (47) reduces to

F 00n þð2nþ 1Þ

nF 0n �

Pe2l4d21

4Fn ¼ 0: ð50Þ

Since n is a regular singular point of Eq. (50), we assume the solution of the form

FnðnÞ ¼X1m¼0

bmnm: ð51Þ

After solving Eq. (50) with the help of Eq. (51), we get b1 ¼ 0 and a recurrence relation is obtained as

bm ¼Pe2 l4d2

14

� �mðmþ 2nþ 1Þ bm�2: ð52Þ

Since b1 ¼ 0; bm ¼ 0 for m ¼ 1;3;5; . . . ; and for m even, say m ¼ 2j, the expression for bm becomes

b2j ¼Pe2 l4d2

14

� �2jð2jþ 2nþ 1Þ b2j�2: ð53Þ

Eq. (53) can be written as

b2j ¼ b0

Yj

i¼1

Pe2 l4d21

4

� �ðnþ 2iÞ2 þ ðnþ 2iÞ � nðnþ 1Þ

: ð54Þ

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

Fnra

� �¼ 1þ

X1j¼1

ra

� �2jYj

i¼1

Pe2 l4d21

4

� �ðnþ 2iÞ2 þ ðnþ 2iÞ � nðnþ 1Þ

; ð55Þ

here b0 is taken as 1 for simplicity. From the boundary condition (48), we have

Hnð1Þ ¼/2

Pel2d1

2nþ 12

Z þ1

�1f exp

Pel2d1f2

!PnðfÞdf ¼ /2

Pel2d1

2nþ 12

ð�1Þn

2nn!

Z þ1

�1ðf2 � 1Þn dn

dfn f expPel2d1f

2

!( )df

¼ /2

Pel2d1

2nþ 1

2nþ1n!

Z þ1

�1ð1� f2Þn f

Pel2d1

2

!n

þ nPel2d1

2

!n�124 35� exp

Pel2d1f2

!df: ð56Þ

It can be seen from Eq. (49) that Hnð1Þ ¼ /2anFnð1Þ. Hence from Eq. (56) we get

an ¼1

Pel2d1

2nþ 1

2nþ1n!

1Fnð1Þ

Z þ1

�1ð1� f2Þn f

Pel2d1

2

!n

þ nPel2d1

2

!n�124 35� exp

Pel2d1f2

!df: ð57Þ

The expression (45) and the transformed Eq. (38) are used to obtain the nutrient concentration profile inside the pellet givenby

ciðr; hÞ ¼ c0 þ c0/2 r cos h

Pel2d1a� exp

�Pel2d1r cos h2a

!X1n¼0

anra

� �n

Fnra

� �Pnðcos hÞ

" #; ð58Þ

where Fn and an are as per Eqs. (55) and (57), respectively. The above expression for the concentration has been obtainedcorresponding to the Dirichlet boundary condition (33) at the surface of the porous pellet. Similar computations may helpin order to use a flux condition and the corresponding calculation is deferred for a future investigation.

4. An optimal criterion to avoid starvation

The variations in the velocity and concentration profiles will be discussed with respect to the parameters involved. Thefrequency of oscillation is assumed in the range of 1 KHz and 10 KHz and a2=m ¼ 10�3 s.

The nutrient concentration profiles along the z-axis are shown for various combination of the parameters involved, likeDarcy number ðDaÞ, frequency ð-Þ, slip coefficient ðaÞ, Thiele modulus ð/Þ and Peclet number ðPeÞ (see Figs. 2 and 4). Thegeneral trend is that the concentration inside the pellet reduces compared to the bulk nutrient concentration at the surface.This is due to the reaction inside the pellet. We may also observe that the concentration contours are influenced by the flowdirection. Detailed discussion on these variations will be presented in Section 5. For the case of zeroth-order kinetics con-sidered here, it is noteworthy that the constant nutrient consumption rate may lead to negative nutrient concentration at

Fig. 2. Nutrient concentration profile along the z-axis: (a) Da ¼ 0:003;- ¼ 3;a ¼ 0:5;/ ¼ 2:47 and (b) Pe ¼ 80;Da ¼ 0:003;- ¼ 3;a ¼ 0:5.


selected points inside the pellet. This is termed as starvation in literature. Let us observe this in particular two situations. Thefirst one is while considering the effect of Peclet number on nutrient concentration profile as shown in Fig. 2a for the com-bination of parameters Da ¼ 0:003;a ¼ 0:5;/ ¼ 2:47;- ¼ 3. It is seen here that at Pe ¼ 80, the pellet is at the onset of star-vation and for Pe below 80, the concentration profiles take negative values which is not meaningful. And the second case istreating the effect of the Thiele modulus on nutrient concentration for a particular combination of the other parameters in-volved (see Fig. 2b). Even in this case one can observe such starvation. This physical impossibility can be obviated by restrict-ing the validity of Eq. (58) to only those situations yielding positive concentrations for the consumed nutrient. This can bedone by choosing an appropriate combination of the parameters. Negative concentrations then would indicate that the mod-el is not applicable to the corresponding situation.

Now we discuss the relationship between the Peclet number, Pe, and the Thiele modulus, /, at the onset of nutrientexhaustion. This relationship is obtained by specifying that the minimum nutrient concentration anywhere in the pelletbe equal to zero

Min ciðr; hÞ ¼ 0; 0 6 r 6 a; 0 6 h 6 p:

Due to the symmetry with respect to the axis parallel to the flow direction, the above minimum is expected to occur on the z-axis (Fig. 2a). Furthermore, the above minimum is located on the downstream portion of the pellet, i.e., in the region0 6 r 6 a. The condition at the onset of nutrient exhaustion then becomes

Min ciðr; h ¼ 0Þ ¼ 0; 0 6 r 6 a:

Fig. 3. /2 as a function of Peclet number at the onset of starvation: (a) Da ¼ 0:003;a ¼ 0:5, (b) - ¼ 3;a ¼ 0:5 and (c) Da ¼ 0:005;- ¼ 3.


Fig. 4. Nutrient concentration profile along the z-axis: (a) Pe ¼ 80;Da ¼ 0:003;a ¼ 0:5;/ ¼ 2:47, (b) Pe ¼ 80;- ¼ 3;a ¼ 0:5;/ ¼ 2:47 and (c)Pe ¼ 80;Da ¼ 0:003;- ¼ 3;/ ¼ 2:47.



The above condition with the support of the expression given in (58) turns out to be

Table 1Nutrien

z/a

�1�0.8�0.6�0.4�0.2

0.00.20.40.60.81

/20ðPeÞ ¼ max

06x61� x

Pel2d1

þ exp � Pel2d1x2

!X1n¼0

anðxÞnFnðxÞ" #( )�1

; ð59Þ

where Fn and an are given by Eqs. (55) and (57), respectively. Here /0 is the critical Thiele modulus in terms of Pe. Hence, weconclude that the condition to avoid starvation inside the pellet is

/ < /0ðPeÞ: ð60Þ

The regions of nutrient sufficiency and starvation, as determined by condition (60), are shown in Fig. 3a–c. Considering thefact that Pe in several cases [15] can be as high as 200, Fig. 3a–c underline the criteria in choosing a perfect combination ofparameters involved in the Eq. (58), the range of Thiele modulus can be found in order to avoid starvation. In Fig. 3a–c re-gions of starvation for different combination of parameters are obtained at different x ¼ r=a, and the corresponding values ofthe Thiele modulus are obtained to avoid the starvation. But from Fig. 3a–c it can be seen that there is a minimum Thiele mod-ulus which can be used for any combination of parameters to avoid starvation inside the pellet. It may be noted that the authorsmade a similar investigation in case of a first order reaction also employing the Dirichlet boundary condition [34]. However,in this case it is seen that the concentration inside the spherical porous pellet lies between 0 and 1 resulting no such star-vation zones.

5. Results and discussion

5.1. Concentration profiles

As mentioned in Section 4, the general trend is that the concentration inside the pellet reduces compared to the bulknutrient concentration at the surface due to the reaction inside the pellet. It is seen that the concentration contours are influ-enced by the flow direction (see Figs. 2 and 4). Increasing Pe increases the overall nutrient concentration throughout the pel-let. Since at low Peclet number, diffusion dominates convection and the pellet cannot experience significant nutrienttransport. However, as Peclet number increases convection becomes a dominating factor and hence nutrient transport in-creases inside the pellet. In conclusion, as Pe increases, the nutrient concentration minimum moves downstream whilethe overall nutrient content of the pellet increases. Effect of the Thiele modulus on the nutrient concentration profile isshown in Fig. 2b for the combination of the parameters Pe ¼ 80;- ¼ 3;Da ¼ 0:003;a ¼ 0:5. The onset of starvation insidethe pellet is seen at / ¼ 2:47. The nutrient concentration minimum moves downstream with decreasing /, which is the ratioof the reaction rate to the diffusion in the porous pellet. Since the present problem considers zeroth-order convection–dif-fusion–reaction, for higher /, the nutrient transport becomes reaction dominated and due to constant consumption nutrientgets starved whereas for lower /, nutrient transport becomes diffusion dominated avoiding any such starvation zones.

Effect of frequency on the nutrient concentration profile is presented in Fig. 4a for the combination of parametersPe ¼ 80;Da ¼ 0:003;a ¼ 0:5;/ ¼ 2:47. Fig. 4a shows that at - ¼ 3, the pellet is at the onset of starvation and - < 3 may leadto nutrient starvation. Increasing frequency assists convection hence an increase in nutrient transport is seen throughout thepellet. This fact can be better observed in Table 1 ðPe ¼ 80Þ and Table 2 ðPe ¼ 120Þ, where the concentration values are tab-ulated with different frequencies for a fixed combination of the other parameters. It is seen that the concentration minimumis increasing with increasing -. In conclusion, as - increases, the nutrient concentration minimum moves downstreamwhile the overall nutrient content of the pellet increases. Fig. 4b shows the concentration profile for the combination ofparameters Pe ¼ 80;- ¼ 3;a ¼ 0:5;/ ¼ 2:47. At Da ¼ 0:003, the pellet is at the onset of starvation and for Da < 0:003 onemay observe starvation zones. The general trend is that as Da increases the porous region offers less resistance and the fluidflows easily through the pellet resulting enhancement in the overall nutrient. The nutrient concentration minimum movesdownstream with increasing Darcy number.

t concentration profile along the z-axis for different frequencies with Pe ¼ 80;Da ¼ 0:003;a ¼ 0:5.

- ¼ 3:4 - ¼ 3:8 - ¼ 4:2 - ¼ 4:6 - ¼ 5:5 - ¼ 6:5 - ¼ 7:5 - ¼ 8:5 - ¼ 9:5

1.00016 1.00011 0.999934 0.99999 0.999997 1 1 1 10.729823 0.735915 0.74171 0.747297 0.758996 0.770789 0.781477 0.791199 0.8000730.491112 0.501048 0.510633 0.519891 0.539602 0.559838 0.57849 0.595707 0.6116190.290187 0.301605 0.312764 0.323672 0.347314 0.372161 0.39556 0.417563 0.4382340.134979 0.145528 0.156028 0.166466 0.189662 0.214836 0.239243 0.262781 0.2853850.0356474 0.0431539 0.0508757 0.0587822 0.077102 0.0980397 0.119278 0.140555 0.1616650.00526767 0.00803295 0.0112483 0.0148742 0.0243495 0.0366613 0.0504409 0.0653281 0.08101850.0607783 0.0580529 0.0558952 0.0542687 0.0523827 0.0528445 0.0556191 0.0603605 0.0667610.224256 0.217 0.210238 0.203959 0.191457 0.180015 0.170921 0.163958 0.1589260.524704 0.516767 0.508966 0.50157 0.485678 0.469322 0.454274 0.440483 0.4279011.00096 1.00084 0.999431 0.999911 0.999958 1 1.00001 0.999997 0.999997

Table 2Nutrient concentration profile along the z-axis for different frequencies with Pe ¼ 120;Da ¼ 0:003;a ¼ 0:5;/ ¼ 2:47.

z/a - ¼ 3:4 - ¼ 3:8 - ¼ 4:2 - ¼ 4:6 - ¼ 5:5 - ¼ 6:5 - ¼ 7:5 - ¼ 8:5 - ¼ 9:5

�1 1 1 1 1 1 1 1 1 1�0.8 0.767191 0.774632 0.781601 0.788142 0.801477 0.814382 0.82563 0.83551 0.844248�0.6 0.553626 0.566512 0.57871 0.590268 0.614153 0.637661 0.658443 0.676906 0.693387�0.4 0.364473 0.38048 0.395838 0.410572 0.441555 0.472721 0.500794 0.526115 0.549�0.2 0.206963 0.223438 0.239538 0.255242 0.289059 0.324096 0.356471 0.386288 0.413703

0.0 0.0913806 0.105424 0.119539 0.133659 0.165156 0.199231 0.231907 0.262926 0.2921720.2 0.0325956 0.0413178 0.0506175 0.0603948 0.0836939 0.110888 0.138627 0.166288 0.1934360.4 0.0524312 0.0535676 0.055666 0.0586256 0.0679697 0.0818272 0.098355 0.116728 0.1362920.6 0.183364 0.17659 0.170823 0.165997 0.158262 0.154109 0.15388 0.156883 0.1625170.8 0.474345 0.46393 0.454097 0.444825 0.425917 0.407894 0.392767 0.380281 0.3701851 1 0.999998 0.999995 1.00001 0.999995 0.999981 0.99994 0.999826 0.99954


The effect of the slip coefficient on the nutrient concentration profile is shown in Fig. 4c for the combination of theparameters Pe ¼ 80;Da ¼ 0:005;- ¼ 3;/ ¼ 2:47. At a ¼ 0:1, the pellet is at the onset of starvation. The influence of the slipcoefficient, a on the concentration is marginal. Because this is a characteristic of the porous material and for a fixed Darcynumber varying slip coefficient may not show a significant impact.

Fig. 5. Variation of magnitude of the external tangential velocity ðVehÞ with r=a.

Fig. 6. Variation of magnitude of the internal velocity ðViÞ with (a) frequency ð-Þ and (b) Darcy number ðDaÞ.


5.2. Internal and external velocity profiles

In general, fluid experiences less resistance under the presence of porous pellet in comparison to impermeable particles,as impermeable particle experiences no slip on the surface. However, a porous pellet with smaller Darcy number offers largerresistance compared to that of a larger Darcy number. The magnitude of the tangential velocity component Ve

h versus r=a isshown in Fig. 5. It can be seen from Fig. 5a that when the frequency of oscillation is low, the magnitude of the uniform farfield basic velocity slightly increases and near the body the velocity reduces due to the resistance offered by the pellet. How-ever, for large frequencies, the increase in the magnitude of the uniform far field basic velocity is more (Fig. 5b). WhenDa ¼ 0, which corresponds to a solid particle, the uniform far field velocity slightly increases before it realizes the resistancedue to the solid particle and then decreases to satisfy the no-slip condition. Fig. 6 shows the variation in magnitude of theinternal velocity with frequency and Darcy number. It is observed that the internal velocity increases with both frequencyand Darcy number. An increase of the Darcy number offers more volume flow in the porous region due to increased perme-ability. Also, increase in oscillatory forcing enhances the volume flow inside the pellet for a fixed Darcy number.

6. Conclusion

The nutrient transport inside a spherical porous pellet in an oscillatory flow has been studied. The model assumes aspherical porous pellet placed in an oscillatory Stokes flow. Darcy’s law is used inside the porous pellet and unsteady Stokesequation is used outside the pellet. Velocity and pressure fields are obtained and the convection–diffusion–reaction problem


is formulated. Effect of various parameters on the concentration profile is shown. It is seen that for a fixed Darcy number,increase in frequency enhances the volume flow inside the pellet. The effect of frequency on the nutrient concentration pro-file is discussed for various combination of the parameters involved and a significant effect of frequency of oscillation is seenon the nutrient transport inside the pellet. It is noticed that the constant consumption rate may lead to negative nutrientconcentration at some points inside the pellet. This depends on various combination of the parameters involved. A relationbetween the Thiele modulus and Peclet number is derived. It is a sufficient and necessary criterion for non-negativity of theoverall concentration in case of the Dirichlet boundary condition.

Acknowledgements

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

References

[1] A.M. Thomas, R. Narayanan, Physics of oscillatory flow and its effect on the mass transfer and seperation of species, Phys. Fluids 13 (4) (2001) 859–866.[2] M.E. Mackay, M.R. Mackley, Y. Wang, Oscillatory flow within tubes containing wall or central baffles, Trans. Inst. Chem. Eng. 69 (1991) 506–513.[3] P. Gao, W. Han Ching, M. Herrmann, C. Kwong, P.L. Yue, Photo-oxidation of a model pollutant in an oscillatory flow reactor with baffles, Chem. Eng. Sci.

58 (3–6) (2003) 1013–1020.[4] H.G. Gomaa, A.M. Al-Taweel, J. Landau, Mass transfer enhancement at vertically oscillating electrodes, Chem. Eng. J. 97 (2004) 141–149.[5] M.B. Liu, G.M. Cook, N.P. Yao, Vibrating zinc electrodes in Ni/Zn batteries, J. Electrochem. Soc. 129 (1982) 913–920.[6] H. Nagaoka, Mass transfer mechanism in biofilms under oscillatory flow conditions, Water Sci. Technol. 36 (1) (1997) 329–336.[7] S. Beeton, H.R. Millward, B.J. Bellhouse, A.M. Nicholson, N. Jenkins, C.J. Knowles, Gas transfer characteristics of a novel membrane bioreactor,

Biotechnol. Bioeng. 38 (1991) 1233–1238.[8] V. Perez-Herranz, J. Garcia-Anton, J.L. Guinon, Velocity profiles and limiting current in an electrodialysis cell in pulsed flow, Chem. Eng. Sci. 52 (5)

(1997) 843–851.[9] A.K. Chandhok, D.T. Leighton, Oscillatory cross-flow electrophoresis, AIChE J. 37 (10) (1991) 1537–1549.

[10] S. Najarian, B.J. Bellhouse, Effect of oscillatory flow on the performance of a novel cross-flow affinity membrane device, Biotechnol. Progr. 13 (1997)113–116.

[11] P. Blanpain-Avet, N. Doubrovine, C. Lafforgue, M. Lalande, The effect of oscillatory flow on cross-flow microfiltration of beer in a tubular mineralmembrane system-membrane fouling resistance decrease and energetic considerations, J. Membrane Sci. 152 (1999) 151–174.

[12] R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Clarcndon Press, Oxford, 1975.[13] A. Nir, L.M. Pismen, Simultaneous intraparticle forced convection–diffusion and reaction in a porous catalyst, Chem. Eng. Sci. 32 (1977) 35–41.[14] A.E. Rodrigues, J.M. Orfao, A. Zoulalian, Intraparticle convection, diffusion and zero order reaction in porous catalysts, Chem. Eng. Commun. 27 (1983)

327–337.[15] G. Stephanopoulos, K. Tsiveriotis, The effect of intraparticle convection on nutrient transport in porous biological pellets, Chem. Eng. Sci. 44 (9) (1989)

2031–2039.[16] 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 sphere

geometries, Ind. Eng. Chem. Res. 32 (1993) 1839–1852.[17] M. Young, R. Dean, Optimization of mammalian-cell bioreactors, Bio/Technology 5 (1987) 835–837.[18] C. Prince, U. Bringi, M. Schuler, Convective mass transfer in large porous biocatalysts: plant organ cultures, Biotechnol. Progr. 7 (1991) 195–199.[19] G. Carta, M. Gregory, D.J. Kirwan, H. Massaldi, Chromatography with permeable supports: theory and comparison with experiments, Separ. Technol. 2

(2) (1992) 62–72.[20] H.S. Nan, M.M. Dias, A.E. Rodrigues, Effect of forced convection on reaction with mole changes in porous catalysts, Chem. Eng. J. 57 (1995) 101–114.[21] S.J. Ferguson, K. Ito, L. Nolte, Fluid flow and convective transport of solutes within the intervertebral disc, J. Biomech. 37 (2004) 213–221.[22] K.J. Mattern, C. Nakornchai, W.M. Deen, Darcy permeability of agarose–glycosaminoglycan gels analyzed using fiber-mixture and Donnan models,

Biophys. J. 95 (2008) 648–656.[23] D.D. Frey, E. Schweinheim, C. Horviith, Effect of intraparticle convection on the chromatography of biomacromolecules, Biotechnol. Progr. 9 (1993)

273–284.[24] J.R. Looker, S.L. Carnie, The hydrodynamics of an oscillating porous sphere, Phys. Fluids 16 (2004) 62–72.[25] B.S. Padmavathi, T. Amaranath, S.D. Nigam, Stokes flow past a porous sphere using Brinkman’s model, Z. Angew. Math. Phys. 44 (1993) 929–939.[26] G.P. Raja Sekhar, B.S. Padmavathi, T. Amaranath, Complete general solution of the Brinkman equations, Z. Angew. Math. Mech. 77 (1997) 555–556.[27] P. Levresse, I. Manas-Zloczower, D.L. Feke, Hydrodynamic analysis of porous spheres with infiltrated peripheral shells in linear flow fields, Chem. Eng.

Sci. 56 (2001) 3211–3220.[28] Anindita Bhattacharyya, G.P. Raja Sekhar, Viscous flow past a porous sphere-effect of stress jump condition, Chem. Eng. Sci. 59 (2004) 4481–4492.[29] M.K. Partha, P.V.S.N. Murthy, G.P. Raja Sekhar, Viscous flow past a porous spherical shell-effect of stress jump boundary condition, J. Eng. Mech. 131

(2005) 1291–1301.[30] G.P. Raja Sekhar, T. Amaranath, Stokes flow past a porous sphere with an impermeable core, Mech. Res. Commun. 23 (1996) 449–460.[31] G.P. Raja Sekhar, Jai Prakash, M. Kohr, Steady and oscillatory analysis of porous catalysts in fluidized beds, Proc. Appl. Math. Mech. 8 (2008) 10613–

10614.[32] O. Levenspiel, Chemical Reaction Engineering, second ed., Wiley, New York, 1972.[33] Lanny D. Schmidt, The Engineering of Chemical Reactions, Oxford University Press, 2005.[34] Jai Prakash, G.P. Raja Sekhar, S. De, M. Böhm, Convection, diffusion and reaction in a spherical porous pellet in presence of oscillatory flow,

communicated.

a criterion to avoid starvation zones for convection–diffusion–reaction problem inside a porous...

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