dirichlet problem for convection–diffusion–reaction inside a permeable cylindrical porous pellet

International Journal of Engineering Science 49 (2011) 606–624

Contents lists available at ScienceDirect

International Journal of Engineering Science

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

Dirichlet problem for convection–diffusion–reaction inside a permeablecylindrical porous pellet

Jai Prakash a, G.P. Raja Sekhar a,⇑, Sirshendu De b

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, India

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

Article history:Received 28 June 2010Accepted 7 October 2010Available online 22 March 2011

Keywords:Stokes flowDarcy’s lawSaffman conditionOscillatory flowNutrient transportStarvation zone

0020-7225/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.ijengsci.2010.10.006

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

The present article deals with the study of convection and diffusion coupled with eitherzero or first order reaction inside a permeable circular cylindrical porous pellet under oscil-latory flow. Unsteady Stokes equations are used for the flow outside the permeable porouspellet and Darcy’s law is used inside the pellet. We use the stream function approach inorder to solve the hydrodynamic problem. Then the convection–diffusion–reaction prob-lem is formulated and solved analytically for both zero order and first order rate of nutrientuptake. The Dirichlet boundary condition, which can be achieved by neglecting the exter-nal mass transfer resistance, is used at the surface of permeable porous pellet. Also in caseof zero order, an optimality criterion, which is a relationship between the Peclet numberand the Thiele modulus, is proposed to avoid the starvation everywhere inside the pellet.Based on this criterion, classification is done in order to identify the regions of nutrient suf-ficiency and starvation. A comparison is also made with nutrient transport inside a spher-ical porous pellet. It is observed that in case of zero order, for a fixed combination of otherparameters, spherical pellet demands a higher value of Thiele modulus compared to thecylindrical pellet in order to force starvation. Moreover, in case of first order reaction,one does not witness starvation zones either in cylindrical pellet or in spherical pellet.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Mass transfer processes involve the transfer of various components within a phase and between phases by molecular dif-fusion and natural or forced convection. Mass is transferred by concentration gradient or partial pressure gradients. Initiallydiffusion was considered as the only mechanism for mass transport inside the porous particles (Smith, Van Ness, & Abbott,2004; Wakao & Smith, 1962). But in case of large–pore materials, convection cannot be ignored. Mass transfer also plays avery important role in operations of food processing, such as drying, extraction, distillation, and absorption. Mass transfer isalso involved in several physical, chemical and biological food processes, such as salting, sugaring, oxygen absorption, etc. Inmany of these processes, mass transfer takes place through different porous geometries called porous catalysts. Porous cat-alyst particles are widely used in the chemical industry and are extensively treated in the chemical engineering literature.The catalyst pellets are, in most cases, fluidized by the action of a gas or liquid flowing through a reactor. The fluid enters atthe bottom of a bed of catalyst particles and the particles are fluidized by the shear force that the fluid exerts on their surface.Intense research has been carried out on diffusion phenomena in porous bodies for many years by using diffusion model todifferent materials. The present study is focused on the interaction between convection, diffusion and reaction inside acylindrical porous pellet.

. All rights reserved.

Raja Sekhar).

http://dx.doi.org/10.1016/j.ijengsci.2010.10.006

mailto:[email protected]

http://dx.doi.org/10.1016/j.ijengsci.2010.10.006

http://www.sciencedirect.com/science/journal/00207225

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

Nomenclature

a radius of the porous pellet [m]k permeability of the porous pellet [m2]r radial distanceve 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 [N/m2]U1 magnitude of the far field uniform velocity [m/s]In modified Bessel function of first kindKn modified Bessel function of second kindci concentration inside the porous pellet [mole/m3]S uptake rate [mole/m3s]k0 rate constant [s�1]D diffusivity [m2/s]DG Change in Gibbs free energy [J]DH Change in Enthalpy [J]T Absolute temperature [K]DS Change in entropy [J/K]c0 concentration at the surface of the porous pellet [mole/m3]~c dimensionless concentrationDa Darcy numberPe Peclet numberjVij magnitude of the internal velocityG, H dimensionless parameters

Greek symbolsh inclinationw stream functiona slip coefficientk dimensionless parameterx frequency of oscillation [s�1]- dimensionless frequency of oscillationq density of the fluid [kg/m3]l dynamic viscosity [kgm�1s�1]m kinematic viscosity [m2/s]/ Thiele modulus

Subscript/Superscripte external to the pelleti internal to the pellet

J. Prakash et al. / International Journal of Engineering Science 49 (2011) 606–624 607

Pan and Zhu (1998) studied the reaction–diffusion process inside the cylindrical catalyst pellet. They obtained the effec-tive diffusivities and Thiele modulus. The concentration distribution inside a cylindrical pellet depends on the value of Thielemodulus. For large values of Thiele modulus the concentration drops rapidly inside a pellet due to dominance of reactionover diffusion. The phenomenon of mass transfer is seen during the vegetables drying. There are several studies wherethe Dirichlet boundary condition has been used in drying of vegetables and grains (Mulet, 1994; Mulet, Berna, & Rosello,1989; Sokhansanj, 1987). A number of studies have been done where diffusion is accompanied by chemical reaction Aris(1975, 1967). Aris (1975) obtained analytical expressions for reaction in an isothermal finite cylinder with no external trans-port resistances, i.e., Dirichlet boundary condition at the pellet surface. Gunn (1967) solved the corresponding problem forhollow cylindrical pellets and gave effectiveness factor versus Thiele modulus curves for several values of length to diameterratio of the cylinder. Ho and Hsiao (1977) obtained an approximate effectiveness factor for an isothermal first order reactionfor finite cylindrical pellet using singular perturbation on infinite cylinder. The problem of diffusion and reaction in a non-isothermal finite cylindrical porous pellet was studied by Mukkavilli, Tavlarides, and Wittmann (1987a, 1987b) using inte-gral equation method in presence/absence of external transport resistances. Sorensen, Guertin, and Stewart (1973) used theorthogonal collocation technique to solve the problem of first order chemical reaction in a non-isothermal finite cylindricalcatalyst pellet with Dirichlet boundary conditions.

608 J. Prakash et al. / International Journal of Engineering Science 49 (2011) 606–624

In all the above cited studies, the problem of diffusion and reaction has been discussed corresponding to first order iso-thermal/ non–isothermal reaction for a finite cylindrical pellet. Stephanopoulos and Tsiveriotis (1989) took convection intoconsideration and studied the problem of convection and diffusion coupled with zero order isothermal reaction inside aspherical porous pellet. They have discussed the interaction of the external free fluid flow around a porous catalyst particleand analyzed the intraparticle flow field. They have also shown that the transport outside the catalyst particle is unaffectedby the intraparticle fluid flow. The contribution of intraparticle convection to the total mass transfer increases with the mag-nitude of the intraparticle mass Peclet number (Cardoso & Rodrigues, 2007). Prakash, Raja Sekhar, De, and Böhm (2010b)extended the study of Stephanopoulos and Tsiveriotis to the study of convection–diffusion problem coupled with isothermalzero order reaction under oscillatory Stokes flow. They have shown a significant effect of oscillations on nutrient transportinside the porous pellet. The present study aims to deal with the convection–diffusion–reaction problem inside a permeablecylindrical porous pellet under oscillatory flow for isothermal zero and first order reaction, respectively. Mass transfer of aporous species is enhanced by several orders of magnitude when it is present in a fluid medium subjected to oscillatory mo-tion. There are number of applications where oscillatory flow is used such as, waste water treatment, heat exchange, etc.

The present study aims at understanding the mass transport inside a permeable cylindrical porous pellet in an oscillatoryflow in which the convection–diffusion process is coupled with isothermal zero or first order reaction kinetics. For the flowinside the porous pellet one may use either Darcy’s equation or Brinkman equation. Employing each of them bring its ownadvantages and disadvantages. The scope of the present work is more analytical where stream function approach is used tosolve the hydrodynamic problem and then the combined convection–diffusion–reaction problem will be solved in terms ofFourier series solution. It is more likely that employing Brinkman equation does not allow the present analytical approachand may force numerical solution of the convection–diffusion–reaction problem. Hence, we concentrate on the Darcy’s equa-tion. However, using Darcy’s equation would bring in several restrictions. But, nevertheless the problem is of interest for aclass of important technological applications. In this context, it is worth addressing some of the main limitations in employ-ing Darcy’s equation.

Though Darcy’s law is an empirical derivation, its extensive use in literature tempts one to consider it as a law of physics.The treatises by Muskat (1946), and Scheidegger (1974) provide the detailed discussion in the context of problems dealingwith the flow of homogeneous fluids through porous media. But, there do exist other interesting approaches where Darcy’slaw is derived. Several authors have obtained Darcy’s equation as an approximation from the balance of linear momentum inthe context of mixture theory (Atkin & Craine, 1976a, 1976b; Bowen, 1976; Rajagopal & Tao, 1995). Also Gray (1983) andHassanizadeh and Gray (1979a, 1979b) used an averaging technique from an engineering prospective to derive Darcy’s equa-tion. Hornung (1996) has derived Darcy’s law via homogenization of Stokes equation in porous media. Rigorous proofs arealso presented based on asymptotic expansions.

In the context of mixture theory, Rajagopal (2007) described in detail the assumptions under which Darcy’s law isachieved as an approximation to the balance of linear momentum for the flow through porous media. These assumptionsmake Darcy’s law more restrictive. The most important assumptions being (i) ignoring the balance of linear momentumof the solid considering that the solid is a rigid porous body, (ii) the only frictional forces are due to the interaction of thefluid at the boundaries of the pore. In fact these are the damping forces that depend on the permeability, (iii) frictional forcesdue to viscosity are neglected and finally (iv) the nonlinear inertial terms are neglected. These restrictions imply that thesolid matrix is non-deformable and the strains are negligible. Hence, while using Darcy’s equation one has to keep theserestrictions in view and interpret the physical meaning of the results in hand.

For the present problem, we use unsteady Stokes equations outside the porous pellet and Darcy’s law inside the porouspellet. The stream function approach is used to obtain the velocity external to the pellet. The computed hydrodynamic part isused to evaluate the nutrient transport via the combined convection–diffusion–reaction equation. It is well known that masstransfer resistance is expressed in terms of mass transfer coefficient. The non-dimensional form of mass transfer coefficientis Sherwood number which is proportional to Peclet number (Sh � Pe1/3) (Stephanopoulos & Tsiveriotis, 1989). It is estab-lished in Stephanopoulos and Tsiveriotis (1989) that the ratio of internal to external mass transfer resistances for a perme-able spherical particle is much greater than 1. Following similar arguments, one may neglect the external mass transferresistance in the treatment of convection–diffusion–reaction problem inside a permeable particle. The combined convec-tion–diffusion–reaction problem is solved for zero or first order reaction occurring inside the permeable cylindrical porouspellet subject to the Dirichlet boundary condition. Analytical expressions are obtained for the concentration profiles. Also, wederive necessary and sufficient conditions ensuring the non-occurrence of starvation zones in case of zero order reaction.

2. Oscillatory Stokes flow past a permeable cylindrical porous pellet

2.1. Governing partial differential equations

A permeable circular cylindrical porous pellet of radius a and permeability k occupying the domain Xi � R2 is consideredin an arbitrary oscillatory flow of a viscous incompressible fluid (Fig. 1). Let the boundary r = a be denoted by C. It is assumedthat the flow inside the permeable porous pellet, i.e., Xi is described by Darcy’s law, and that the flow outside the pellet, i.e.,Xe ¼ R2 nXi is governed by unsteady Stokes flow. Homogenization of unsteady Stokes equations inside porous mediumleads to steady Darcy’s law (Looker & Carnie, 2004), consequently, the flow inside the pellet is governed by

0c=c Ω

Ωi

Γ ei

x

00U i

Fig. 1. Geometry of the problem.


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. We assume that the flow outside thepellet is described by 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. We consider oscillatory flow with frequency x as ve = Vee�ixt and pe = Pee�ixt. Thus, thegoverning equations transform to

�iqxVe ¼ �rPe þ lr2Ve in Xe; ð5Þ

r � Ve ¼ 0 in Xe: ð6Þ

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

a .The magnitude of the far field uniform velocity is U1. Correspondingly, the non-dimensional equations for the flow insidethe porous pellet are

Vi ¼ �DarPi in Xi; ð7Þ

r � Vi ¼ 0 in Xi ð8Þ

and for the flow outside are

ðr2 � k2ÞVe 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, andr2 ¼ @2

@r2 þ 1r@@r þ 1

r2@2

@h2. The symbol� from the Eqs. (7)–(10) is droppedfor notational simplicity.

2.2. Boundary conditions

Continuity of normal component of velocity is an obvious choice at a permeable interface. In order to have a completelydetermined flow of the free fluid, an additional condition on the tangential component of the free fluid velocity needs to bespecified at the interface. Based on experiments, Beavers and Joseph (BJ) (1967) proposed a condition that can be applied at aporous–liquid interface. Further, Taylor (1971), and Richardson (1971) provided support justifying this BJ condition. Amathematical justification of this interface condition was obtained by Saffman (1971). A common choice while matchingDarcy’s law with the Stokes equation is, continuity of pressure and continuity of normal velocity components along withthe Saffman (1971) slip condition for tangential velocity components (Raja Sekhar & Amaranath, 1996). When the flow isof oscillatory nature, applicability of Saffman condition at a porous–liquid interface is validated by Looker and Carnie(2004), however under low frequency. It may be noted that the non-dimensionalized Eqs. (7) and (8) contain a parameter


called Darcy number which is k/a2. However a=ffiffiffikp

is called the Brinkman parameter. The investigations of Looker and Carnie(2004) on the hydrodynamics of oscillatory weakly permeable sphere use Saffman condition which limits the validity ofsolution only for larger values of the Brinkman parameter. More detailed discussion on such limitations is given towardsthe end of this section. Keeping these limitations in view, the boundary conditions on C are.

(i) Continuity of pressure: Pe = Pi.(ii) Continuity of normal velocity component: Ve

r ¼ Vir .

(iii) Saffman condition for tangential velocity components:

Veh ¼

ffiffiffiffiffiffiDap

a@Ve

h

@r; Ve

u ¼ffiffiffiffiffiffiDap

a@Ve

u

@r;

where a is the dimensionless slip coefficient.

2.3. Method of solution

Using polar coordinates (r,h), the momentum equations (9) and (10) are written as

ðr2 � k2ÞVer �

Ver

r2 �2r2

@Veh

@h¼ @Pe

@r; ð11Þ

ðr2 � k2ÞVeh �

Veh

r2 þ2r2

@Ver

@h¼ 1

r@Pe

@h; ð12Þ

together with the equation of continuity

@Ver

@rþ Ve

r

rþ 1

r@Ve

h

@h¼ 0: ð13Þ

The equation of continuity (13) enables us to introduce stream function w as

Vr ¼ �1r@w@h

; Vh ¼@w@r: ð14Þ

Substituting the velocity components in Eqs. (11) and (12) and eliminating the pressure, we get

r2ðr2 � k2Þwe ¼ 0: ð15Þ

One can also introduce the stream function wi corresponding to the Darcy equation to realize that r2wi = 0. However, sincethe Darcy pressure is Laplacian, we use this fact while expressing the solution inside the porous cylinder.

The general solution of Eq. (15) can be expressed as

we ¼ w1 þ w2; ð16Þ

where w1 and w2 satisfy r2w1 = 0, (r2 � k2)w2 = 0. Therefore, the general solution for we can be written as

we ¼X1n¼0

Anrn þ Bn

rnþ CnInðkrÞ þ DnKnðkrÞ

� �ðan cos nhþ bn sin nhÞ; ð17Þ

where In and Kn are modified Bessel functions of first and second kind, respectively. An, Bn, Cn, Dn, an, bn are arbitrary constantsthat depend on the flow. We proceed (for convenience) with the terms involving sinnh and scale the constants with bn. Onusing the stream function given in Eq. (17), we obtain the components of velocity and pressure in the fluid region as

Ver ¼ �

X1n¼0

Anrn�1 þ Bn

rnþ1 þ CnInðkrÞ

rþ Dn

KnðkrÞr

� �n cos nh; ð18Þ

Veh ¼

X1n¼0

Anrn�1 � nBn

rnþ1 þ Cn kInþ1ðkrÞ þ InðkrÞr

� �þ Dn

KnðkrÞr� kKnþ1ðkrÞ

� �� sin nh; ð19Þ

Pe ¼ p0 þX1n¼0

k2 Anrn � Bn

rn

� �cos nh: ð20Þ

For a given basic flow, the coefficients An and Cn are known and the coefficients Bn and Dn are to be determined.Since the Darcy pressure satisfies the Laplace equation r2Pi = 0, we have

Pi ¼ p0 þX1n¼0

Enrn þ Fn

rn

� �cos nh: ð21Þ


However, not to allow any singularities inside the pellet, Pi needs to be finite at origin leading to Fn = 0, n P 0. Therefore thevelocity components inside the pellet are given by

Vir ¼ �Da

X1n¼0

nEnrn�1 cos nh; ð22Þ

Vih ¼ Da

X1n¼0

nEnrn�1 sin nh; ð23Þ

where En are unknown constants. Once the velocity components are known, stream function inside the porous pellet can beobtained as

wi ¼ DaX1n¼0

Enrn sin nh: ð24Þ

In the expressions given in Eqs. (18)–(23), velocity and pressure due to a given basic flow determine the coefficients An andCn, whereas the coefficients Bn, Dn, and En are determined from the boundary conditions and are obtained in terms of theknown coefficients An and Cn as follows:
� i Bn ¼
�kKn�1 lþKnnal2k2�2Knna�2KnnlþKnn2l3k2�Knk2 lþKnk

4 l3þKnnl3k2�kKn�1aþk3Kn�1al2þk3Kn�1l3�

Anþ �KnkIn�1l�kKn�1aIn�KnkIn�1a�kKn�1 lInð ÞCn

hk Knnal2kþKnk

3 l3þKnklþKn�1aþk2Kn�1l3þKn�1lþKnnl3kþKnn2l3kþk2Kn�1al2� � ;

ð25Þ

Dn ¼ ��2na� 2nlþ 2n2l3k2� �

An þ �l2k3In�1aþ Ink2nl3 þ Ink

2n2l3 � kIn�1lþ Ink2l� kIn�1a� l3k3In�1 þ Ink

4l3 þ Ink2nal2

� �Cn

h ik Knnal2kþ Knk

3l3 þ Knklþ Kn�1aþ k2Kn�1l3 þ Kn�1lþ Knnl3kþ Knn2l3kþ k2Kn�1al2� � ;

ð26Þ

En ¼k 2kKn�1lþ 2Knnaþ 2Knnlþ 2Knk

2lþ 2kKn�1a

An þ KnkIn�1lþ kKn�1aIn þ KnkIn�1aþ kKn�1lInð ÞCn� �

Knnal2kþ Knk3l3 þ Knklþ Kn�1aþ k2Kn�1l3 þ Kn�1lþ Knnl3kþ Knn2l3kþ k2Kn�1al2

; ð27Þ

where l ¼ffiffiffiffiffiffiDap

.The velocity components both outside and inside the pellet can be obtained using the above expressions. In case of uni-

form flow along x-axis, we have Ve ¼ U1 i and the corresponding stream function is given by we = U1rsinh. Correspondingly,the coefficients An and Cn in Eqs. (18)–(20) are determined to be A1 = 1, C1 = 0 and An = 0, Cn = 0, n P 2. In this case the remain-ing coefficients take the following form

B1 ¼�kK0aþ K1al2k2 � 2K1lþ k3K0al2 � 2K1aþ 2K1l3k2 � K1k

2lþ K1k4l3 � kK0lþ k3K0l3

k K1al2kþ k2K0al2 þ K1k3l3 þ K1klþ 2K1l3kþ K0aþ k2K0l3 þ K0l

� � ; ð28Þ

D1 ¼ ��2a� 2lþ 2l3k2

k K1al2kþ k2K0al2 þ K1k3l3 þ K1klþ 2K1l3kþ K0aþ k2K0l3 þ K0l

� � ; ð29Þ

E1 ¼k 2K1k

2lþ 2kK0aþ 2K1lþ 2kK0lþ 2K1a

K1k3l3 þ K1klþ 2K1l3kþ K0aþ k2K0l3 þ K0lþ K1al2kþ k2K0al2 : ð30Þ

Accordingly, the velocity components inside the porous region (Xi) are

Vir ¼ �l2E1 cos h; ð31Þ

Vih ¼ l2E1 sin h: ð32Þ

The explicit solution calculated above for the hydrodynamic problem will be used in the next section in order to evaluate thenutrient transport via the combined transport–reaction equation.

The use of Saffman condition brings limitation on the permeability range. Looker and Carnie (2004) concluded thatSaffman condition is applicable under low frequency. Vainshtein and Shapiro (2009) calculated the force acting on a perme-able particle in oscillatory flow using Brinkman or Darcy equation. It may be noted that in case of Brinkman equation it iscustomary to use continuity of velocity components together with the continuity of stress components and these boundaryconditions are accepted by a large community. Vainshtein and Shapiro (2009) have also used the same boundary conditionsin case of Brinkman equation. However, in case of Darcy equation, the continuity of tangential velocity needs to be replacedby Beavers–Joseph/ Saffman type slip condition which is not the case in Vainshtein and Shapiro (2009). Vainshtein and


Shapiro (2009) reported a critical value of the Brinkman parameter, a=ffiffiffikp

, which is expected to control the applicability ofDarcy equation. They also observed that this critical value diminishes with decreasing frequency of oscillations and reachesthat of a non-oscillating particle �10. It seems that this critical value can be as large as 200 for high frequency of oscillations.For low and moderate values of frequency this critical value can be readily identified. Hence, the hydrodynamic problem ofoscillatory flow past a porous body considering Darcy equation inside together with Saffman condition on the porous–liquidinterface brings in such trade-off between frequency and the range of permeability.

3. Nutrient transport inside a permeable cylindrical porous pellet

3.1. Zero order reaction case

We present here the problem of convection and diffusion coupled with isothermal zero order reaction inside a permeableporous pellet. For this purpose, we consider the following transport-reaction equation

Vi � rci ¼ Dr2ci � S in Xi; ð33Þ

as the governing equation for the nutrient transport inside the permeable porous pellet. Where S is the uniform consumptionrate for the limiting nutrient, i.e., zero order reaction kinetics and D is the diffusivity.

Assuming that the reaction front moving inward maintains a circular shape, we treat the concentration profile of the reac-tant inside the cylinder to be in steady state. This kind of pseudo steady state approximation is common in chemical engi-neering applications (Schmidt, 2005). The external mass transfer resistance is neglected accounting for strong agitation. Itmay be noted that for a spontaneous forward reaction, the change in Gibbs free energy (DG) given by DG = DH � TDS hasto be negative, where DH is the change in enthalpy, T is the absolute temperature and DS is the change in entropy. The reac-tion kinetics in the present study is a spontaneous forward isothermal reaction and change in enthalpy is assumed to be mar-ginal. Therefore, DG < 0 and hence, the non-negativity of DS is ensured (Smith et al., 2004; Wakao & Smith, 1962). Most ofthe biological reaction processes fall under this class. Because a catalyst generally does not perturb the enthalpy change va-lue for a reaction (Waite & Waite, 2007). The combined mass transport Eq. (33) needs to be supplemented with a suitableboundary condition at the porous–liquid interface. In the present investigation we use the Dirichlet boundary condition, i.e.,the internal concentration is equal to the bulk nutrient concentration given by

ci ¼ c0 on C: ð34Þ

The physical quantities are non-dimensionalized as in subsection 2.1 together with ~c ¼ c0�ci

c0, as a result the governing equa-

tions transform to

PeVi � rc ¼ r2cþ /2 in Xi; ð35Þ

c ¼ 0 on C ð36Þ

with

/2 ¼ Sa2

c0D; Pe ¼ U1a

D; ð37Þ

where / is the Thiele modulus and Pe is the Peclet number. Note that we have omitted the symbol � from Eqs. (35) and (36).The velocity Vi given in Eqs. (31) and (32) reduces Eq. (35) to

�Pel2E1 i � rc ¼ r2cþ /2 in Xi: ð38Þ

We proceed further to present an analytical solution for the above convection–diffusion–reaction problem.

3.2. Analytical solution

The governing Eq. (38) and the corresponding boundary condition (36) can be expressed respectively as

Pe2l4E21

4c ¼ r2c in Xi; ð39Þ

c ¼ /2 cos h

Pel2E1

expPel2E1 cos h

2

!; on C ð40Þ

with the help of following transformation

c ¼ c exp � Pel2E1x2

!� /2x

Pel2E1

; x ¼ r cos h: ð41Þ


The Eq. (39) in polar coordinates (r,h) becomes

Pe2l4E21

4c ¼ @

2c@r2 þ

1r@c@rþ 1

r2

@2c@h2 in Xi: ð42Þ

By using separation of variables the solution of Eq. (42), which is finite at origin is obtained as

c ¼X1n¼0

bnInPel2E1r

2

!cos nh; ð43Þ

where In are modified Bessel functions of first kind, which are finite at origin and bn are unknown constants to be determinedusing the boundary condition (40). The Eq. (43) can be expressed in terms of usual Fourier series notation as

c ¼b02I0

Pel2E1r2

� �2

þX1n¼1

bnInPel2E1r

2

!cos nh: ð44Þ

Further, on the boundary C, we get

c ¼b02I0

Pel2E12

� �2

þX1n¼1

bnInPel2E1

2

!cos nh: ð45Þ

The unknown coefficients bn are determined using the Eqs. (40) and (45), and we have

b0 ¼/2

2pI0ðGÞG

Z p

0expðG cos hÞ cos hdh;

bn ¼/2

pInðGÞG

Z p

0expðG cos hÞ cos h cos nhdh; n P 1;

ð46Þ

where G ¼ Pel2E12 . Therefore, we have calculated an expression for the nutrient concentration inside the pellet given by

ciðr; hÞ ¼ c0 þ c0/2 r cos h

Pel2E1a� exp

�Pel2E1r cos h2a

!X1n¼0

bnInPel2E1r

2a

!cos nh

" #: ð47Þ

3.3. First order reaction case

In subsection 3.1, we have presented the solution of a zero order convection–diffusion–reaction problem inside a cylin-drical porous pellet subject to the Dirichlet boundary condition. In this subsection, we consider the corresponding problemof first order reaction kinetics. Correspondingly, the nutrient mass balance can be written as

Vi � rci ¼ Dr2ci � k0ci in Xi ð48Þ

and the boundary condition is given by

ci ¼ c0 on C: ð49Þ

The non-dimensionalization as in subsection 2.1 together with ec ¼ ci

c0leads to

PeVi � rc ¼ r2c� /2c in Xi; ð50Þ

c ¼ 1 on C: ð51Þ

On using the velocity Vi computed as in Eqs. (31) and (32), the mass balance Eq. (50) becomes

�Pel2E1 i � rc ¼ r2c� /2c in Xi: ð52Þ

In order to solve Eq. (52) together with the boundary condition (51), we use the following transformation

c ¼ c exp �Pel2E1x2

!; x ¼ r cos h; ð53Þ

which reduces the Eq. (52) to the form

Pe2l4E21

4þ /2

!c ¼ r2c in Xi: ð54Þ

The corresponding boundary condition takes the form


c ¼ expPel2E1 cos h

2

!; on C: ð55Þ

The solution methodology is same as given in subsection 3.1 and hence we do not describe here. The solution of the Dirichletproblem Eqs. (54) and (55) is given by

ciðr; hÞ ¼ c0 exp�Pel2E1r cos h

2a

!X1n¼0

bnInHra

�cos nh; ð56Þ

where Z
b0 ¼
1pI0ðHÞ

p

0expðG cos hÞdh;

bn ¼2

pInðHÞ

Z p

0expðG cos hÞ cos nhdh; n P 1

ð57Þ

and H ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG2 þ /2

q.

It may be noted that the present solution methodology suffers to handle other common kinetics such as, Michaelis–Menten kinetics, Monod kinetics, which occur in immobilized enzymes and growing populations of immobilized cells or micro-organisms. Because these are non–linear in nature. Due to non-linearity of these kinetics the present method fails to handlethem. But it may be mentioned that the kinetics dealt in the present study are a special case of the Michaelis–

0 0.002 0.004 0.006 0.008 0.010

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Da

|Vi |

ϖ = 3

ϖ = 5

ϖ = 7

ϖ = 9

Da = k/a2

ϖ = ω a2/να = 0.5

1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

ϖ

|Vi |

Da = 0.003

Da = 0.005

Da = 0.007

Da = 0.009

Da = k/a2

ϖ = ω a2/να = 0.5

a

b

Fig. 2. Variation in magnitude of the internal velocity (Vi) with (a) frequency (-), (b) Darcy number (Da).


Menten and Monod kinetic models. When the concentration tends to very large value or zero then these kinetics reduce to zeroorder and first order kinetics, respectively.

4. Results and discussion

The closed form solutions obtained in the previous sections in terms of the velocity and nutrient concentration inside thepellet are considered for numerical simulations. The frequency of oscillation is assumed ranging between 1 KHz and 10 KHzand a2/m = 10�3 s. The nutrient concentration profiles along the x-axis are shown for various combination of the parametersinvolved, like Darcy number (Da), frequency (-), slip coefficient (a), Thiele modulus (/) and Peclet number (Pe). The generalimpact of the reaction inside the pellet is to reduce the concentration inside compared to the bulk nutrient concentration.The flow direction is from right to left and hence shifts the concentration minimum to the left.

4.1. Internal velocity profiles

Fig. 2 shows the variation in magnitude of the internal velocity with frequency and Darcy number. It is observed that theinternal velocity increases with both frequency and Darcy number. Increase in Darcy number offers more volume flow into theporous region for a certain pressure drop. Also, increase in oscillation in the flow field induces enhancement in volumetric

a

b

Fig. 3. Nutrient concentration profile along the x-axis (zero order) (a) Da = 0.003, - = 3, a = 0.5, / = 2.075, (b) Pe = 80, Da = 0.003, - = 3, a = 0.5.


flow inside the pellet for a fixed Darcy number. The corresponding influence on convection may be seen in detail while under-standing the concentration profiles inside the porous pellet.

a

b

Fig. 4. Nutrient concentration profile along the x-axis (zero order) (a) Pe = 80, Da = 0.003, a = 0.5, / = 2.075, (b) Pe = 80, - = 3, a = 0.5, / = 2.075.

Table 1Nutrient concentration profile along the x-axis (zero order) for different frequencies with Pe = 80, Da = 0.003, a = 0.5, / = 2.

x/a - = 3.4 - = 3.8 - = 4.2 - = 4.6 - = 5.5 - = 6.5 - = 7.5 - = 8.5 - = 9.5

�1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000�0.8 0.4853 0.4763 0.4682 0.4611 0.4481 0.4381 0.4321 0.4294 0.4293�0.6 0.2111 0.2084 0.2072 0.2743 0.2119 0.2225 0.2372 0.2548 0.2743�0.4 0.0965 0.1039 0.1126 0.1225 0.1476 0.1788 0.2115 0.2444 0.2768�0.2 0.0890 0.1052 0.1221 0.1394 0.1791 0.2229 0.2651 0.3049 0.3419

0.0 0.1542 0.1760 0.1976 0.2189 0.2653 0.3135 0.3576 0.3977 0.43380.2 0.2968 0.2933 0.3160 0.3378 0.3839 0.4298 0.4704 0.5064 0.53810.4 0.4206 0.4424 0.4629 0.4824 0.5224 0.5611 0.5946 0.6236 0.64890.6 0.5969 0.6137 0.6294 0.6441 0.6738 0.7018 0.7257 0.7461 0.76360.8 0.7915 0.8010 0.8097 0.8178 0.8338 0.8487 0.8612 0.8718 0.88091.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Table 2Nutrient concentration profile along the x-axis (zero order) for different frequencies with Pe = 100, Da = 0.003, a = 0.5, / = 2.

x/a - = 3.4 - = 3.8 - = 4.2 - = 4.6 - = 5.5 - = 6.5 - = 7.5 - = 8.5 - = 9.5

�1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9983�0.8 0.4618 0.4537 0.4470 0.4415 0.4332 0.4293 0.4298 0.4336 0.4399�0.6 0.2072 0.2090 0.2126 0.2177 0.2335 0.2560 0.2814 0.3082 0.3353�0.4 0.1213 0.1352 0.1503 0.1662 0.2038 0.2464 0.2879 0.3273 0.3641�0.2 0.1373 0.1601 0.1830 0.2058 0.2555 0.3072 0.3543 0.3968 0.4348

0.0 0.2164 0.2435 0.2697 0.2950 0.3478 0.3999 0.4456 0.4854 0.52030.2 0.3353 0.3625 0.3882 0.4123 0.4615 0.5084 0.5483 0.5825 0.61190.4 0.4802 0.5040 0.5261 0.5466 0.5873 0.6253 0.6570 0.6838 0.70660.6 0.6425 0.6602 0.6765 0.6913 0.7205 0.7472 0.7692 0.7876 0.80320.8 0.8169 0.8265 0.8353 0.8432 0.8585 0.8724 0.8837 0.8932 0.90111.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

a

b

Fig. 5. /2 as a function of Peclet number at the onset of starvation (zero order) (a) Da = 0.003, a = 0.5, (b) - = 3, a = 0.5.



4.2. Concentration profiles

4.2.1. Zero order reactionThe present study is on Stokes flow past a circular cylindrical porous pellet in the absence of external mass transfer

resistance. The reaction reduces the concentration inside the pellet compared to the bulk value at the surface. The flow con-sidered is from +ve to �ve axis along x direction and influences the concentration minimum (see Figs. 3 and 4). The variationof concentration with Pe for the choice Da = 0.003, - = 3, a = 0.5, / = 2.075 is shown in Fig. 3a. Increasing Pe enhances theoverall nutrient concentration throughout the pellet. Diffusion dominates convection at low Peclet number (Pe) and the pel-let cannot experience significant nutrient transport. However, a gradual increase in Peclet number assists convection and thenutrient transportation inside gets enhanced. As a result, increase in Pe moves the minimum nutrient concentration towardsdownstream while forcing more nutrient transport inside. Fig. 3b depicts the variation of concentration with Thiele modulus(/), which is the ratio of reaction rate to the diffusion. Increasing the value of Thiele modulus increases the rate of reactioninside the pellet which shows that the surface reaction is rapid and the nutrient gets consumed very fast into the interior ofthe pellet. This leads to the nutrient exhaustion inside the pellet. This is termed as starvation in literature (Stephanopoulos &Tsiveriotis, 1989). The threshold values displaying the onset of starvation are: Pe = 80 in Fig. 3a and / = 2.075 in Fig. 3b.

Effect of frequency on the nutrient concentration profile is presented in Fig. 4a for the combination of parameters Pe = 80,Da = 0.003, a = 0.5, / = 2.075. It is evident from Fig. 4a that at - = 3, the pellet is at the onset of starvation and - < 3 may leadto starvation zone. Increasing frequency assists convection, which is responsible for more nutrient transport inside the pellet.

a

b

Fig. 6. Nutrient concentration profile along the x-axis (first order) (a) Da = 0.003, - = 3, a = 0.5, / = 3.5, (b) Pe = 80, Da = 0.003, - = 3, a = 0.5.

a

b

Fig. 7. Nutrient concentration profile along the x-axis (first order) (a) Pe = 80, Da = 0.003, a = 0.5, / = 3.5, (b) Pe = 80, - = 3, a = 0.5, / = 3.5.

Table 3Nutrient concentration profile along the x-axis (first order) for different frequencies with Pe = 80, Da = 0.003, a = 0.5, / = 3.

x/a - = 3.4 - = 3.8 - = 4.2 - = 4.6 - = 5.5 - = 6.5 - = 7.5 - = 8.5 - = 9.5

�1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000�0.8 0.5221 0.5136 0.5054 0.4975 0.4807 0.4637 0.4485 0.4352 0.4235�0.6 0.3143 0.3085 0.3031 0.2983 0.2891 0.2816 0.2767 0.2742 0.2736�0.4 0.2328 0.2314 0.2304 0.2300 0.2307 0.2339 0.2391 0.2461 0.2544�0.2 0.2158 0.2183 0.2212 0.2245 0.2330 0.2440 0.2562 0.2693 0.2829

0.0 0.2386 0.2442 0.2501 0.2561 0.2703 0.2866 0.3033 0.3201 0.33670.2 0.2939 0.3019 0.3098 0.3178 0.3358 0.3555 0.3748 0.3936 0.41160.4 0.3844 0.3937 0.4029 0.4119 0.4317 0.4528 0.4727 0.4917 0.50950.6 0.5195 0.5289 0.5379 0.5467 0.5655 0.5849 0.6030 0.6197 0.63520.8 0.7162 0.7230 0.7296 0.7359 0.7491 0.7624 0.7745 0.7856 0.79561.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000


This fact can be better observed in Table 1 (Pe = 80) and Table 2 (Pe = 100), where the concentration values are tabulated withdifferent frequencies for a fixed combination of the other parameters. Hence, an increase in the nutrient transport is seenthroughout the pellet resulting an increase in the concentration minimum. In conclusion, as - increases, the nutrient

Table 4Nutrient concentration profile along the x-axis (first order) for different frequencies with Pe = 100, Da = 0.003, a = 0.5, / = 3.

x/a - = 3.4 - = 3.8 - = 4.2 - = 4.6 - = 5.5 - = 6.5 - = 7.5 - = 8.5 - = 9.5

�1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0001 1.0002 1.0009�0.8 0.4984 0.4885 0.4791 0.4702 0.4519 0.4344 0.4198 0.4080 0.3987�0.6 0.2988 0.2932 0.2883 0.2843 0.2776 0.2741 0.2739 0.2776 0.2816�0.4 0.2301 0.2301 0.2309 0.2324 0.2378 0.2465 0.2576 0.2702 0.2840�0.2 0.2241 0.2288 0.2339 0.2395 0.2533 0.2701 0.2878 0.3059 0.3240

0.0 0.2554 0.2634 0.2717 0.2801 0.2994 0.3211 0.3425 0.3634 0.38370.2 0.3169 0.3273 0.3376 0.3478 0.3704 0.3947 0.4178 0.4398 0.46050.4 0.4109 0.4224 0.4337 0.4446 0.4682 0.4928 0.5156 0.5367 0.55620.6 0.5457 0.5567 0.5673 0.5774 0.5989 0.6206 0.6404 0.6583 0.67460.8 0.7352 0.7429 0.7503 0.7573 0.7718 0.7862 0.7989 0.8103 0.82051.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000


concentration minimum moves downstream while the overall nutrient content of the pellet increases. Fig. 4b shows the con-centration profile for the combination of parameters Pe = 80, - = 3, a = 0.5, / = 2.075. The pellet is at the onset of starvationwhen Da = 0.003 and for Da < 0.003 one may observe starvation zones. The general trend is that as Da increases the porousregion offers less resistance and the fluid flows easily through the pellet enhancing the overall nutrient. The minimum nutri-ent concentration towards downstream increases with increase in Darcy number.

4.2.2. An optimal criterion to avoid starvationHere we present a criterion to avoid starvation inside a cylindrical pellet in case of zero order reaction. Prakash et al.

(2010b) have investigated convection–diffusion–reaction inside a spherical porous pellet and observed starvation zonesfor a particular combination of parameters. Similar behavior is seen in the present investigations also. Considering the effectof Peclet number on nutrient concentration profile as shown in Fig. 3a for the combination of parameters Da = 0.003, a = 0.5,/ = 2.075, - = 3, it is seen that at Pe = 80, the pellet is at the onset of starvation and for Pe < 80, the pellet experiences star-vation zone. Similar phenomenon is seen while considering the effect of Thiele modulus (see Fig. 3b). One can control thisphysical impossibility by restricting the validity of Eq. (47) to only those situations yielding positive concentrations for theconsumed nutrient. This would give critical values of the parameters in order to use these models more effectively.

In order to calculate the threshold value of the Thiele modulus in terms of the Peclet number, Pe, and hence to avoid star-vation zones, we propose the following optimization problem

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

Since the problem is symmetric with respect to the axis parallel to the flow direction, the minimum is expected to occur onthe x-axis (Fig. 3a) and is located on the downstream portion of the pellet, i.e., in the region �a 6 r 6 0, h = p. The above con-dition with the help of expression (47) and

/20ðPeÞ ¼ max

�16y60� y

Pel2E1

þ exp � Pel2E1y2

!X1n¼0

bnInPel2E1y

2

!" #( )�1

; ð58Þ

solves the optimization problem, where y = r/a. Hence the condition to avoid starvation inside the pellet is

/ < /0ðPeÞ: ð59Þ

The condition (59) tells that the threshold value of the Thiele modulus would be helpful to classify the regions of nutrientsufficiency and starvation as shown in Fig. 5a and b. It is evident from literature Stephanopoulos and Tsiveriotis (1989) thatPe could be in the order of 200. The results presented in Fig. 5a and b underline the criterion in choosing a perfect combi-nation of the parameters involved and the range of Thiele modulus in order to avoid starvation. Fig. 5a and b show regions ofstarvation for different combination of parameters at different y = r/a, and the critical Thiele modulus can be obtained. How-ever, it can be seen that there is a minimum Thiele modulus which can be used for any combination of parameters to avoid star-vation inside the pellet.

4.2.3. First order reactionThe corresponding results in case of first order reaction will be presented here. Nutrient concentration profile along the x-

axis is given for the combination of various parameters involved in Eq. (56), like Darcy number (Da), frequency (-), slip coef-ficient (a), Thiele modulus (/) and Peclet number (Pe).

Fig. 6a depicts the variation of nutrient concentration with Pe for the combination of parameters Da = 0.003, - = 3, a = 0.5,/ = 3.5. Convection dominates diffusion at higher values of Pe and hence as Pe increases the overall nutrient inside the pelletgets enhanced and the minimum nutrient concentration towards the center of the pellet increases. It may be noted that incase of first order reaction, one does not witness any starvation zones. Whereas in case of zero order, starvation zonesoccur due to constant consumption of the nutrient. Effect of Thiele modulus (/) is shown in Fig. 6b for the combination

a

b

c

Fig. 8. Comparison between spherical and cylindrical pellet (zero order) (a) Da = 0.003, - = 3, a = 0.5, / = 2.075, (b) Pe = 80, Da = 0.003, a = 0.5, / = 2.075, (c)Pe = 80, Da = 0.003, - = 3, a = 0.5.


a

b

Fig. 9. Comparison between spherical and cylindrical pellet (first order) (a) Da = 0.003, - = 3, a = 0.5, / = 3.5, (b) Pe = 120, Da = 0.003, a = 0.5, / = 3.5.


of parameters Pe = 80, Da = 0.003, - = 3, a = 0.5. It is seen that the concentration minimum towards the center of the pelletdecreases for large values of the Thiele modulus. Since at lower values of Thiele modulus, diffusion dominates the rate ofreaction, the reactant diffuses well into the interior of the pellet. The concentration minimum towards the center doesnot decrease but as the Thiele modulus increases the reactant is consumed very fast into the interior of the pellet and theconcentration minimum towards the center of the pellet decreases.

Effect of frequency on the nutrient concentration profile is shown in Fig. 7a for the combination of parameters Pe = 80,Da = 0.003, a = 0.5, / = 3.5. It is observed that as - increases the overall nutrient concentration increases inside the pellet(Table 3 (Pe = 80) and Table 4 (Pe = 100)). In conclusion, as - increases, the minimum nutrient concentration moves down-stream while the overall nutrient content of the pellet increases. Fig. 7b shows the concentration profile for the combinationof parameters Pe = 80, - = 3, a = 0.5, / = 3.5 with varying Da. Similar to the zero order case, the nutrient concentration min-imum moves downstream with increasing Darcy number due to smaller resistance.

4.3. Comparison of nutrient transport inside a cylindrical pellet against a spherical pellet

Prakash et al. (2010b) have studied the problem of convection–diffusion–reaction inside a porous spherical pellet underoscillatory flow. Here we present a comparison of the present work with that of Prakash et al. (2010b). Fig. 8a presents thevariation of concentration with Pe for the combination of parameters Da = 0.003, - = 3, a = 0.5, / = 2.075. It is seen that for afixed Pe, the impact of convection is more for a spherical pellet compared to a cylindrical pellet. We may observe that


decreasing Pe forces starvation beyond a particular threshold value. While fixing all other parameters, in case of cylindricalpellet this threshold value is 80, whereas in case of spherical pellet it will be much smaller. Because for a particular value ofPe, spherical pellet receives more volume flow compared to a cylindrical pellet and hence delays the starvation till a smallerPe is reached. Fig. 8b shows the comparison of nutrient transport inside the spherical and cylindrical pellet with - for thecombination of parameters Pe = 80, Da = 0.003, a = 0.5, / = 2.075. Since increase in frequency assists convection, the nutrienttransport increases inside the pellet. In case of cylindrical pellet, the onset of starvation is seen at - = 3 and forces nutrientexhaustion for - < 3. When all other parameters are fixed as in the case of cylindrical pellet, the onset of starvation inside thespherical pellet can either be seen at a smaller frequency or at a higher Thiele modulus. Fig. 8c represents the effect of Thielemodulus on nutrient concentration inside the pellets for the combination of parameters Pe = 80, Da = 0.003, - = 3, a = 0.5.One can see that the onset of starvation in case of cylindrical pellet is at / = 2.075, which is not the case for the sphericalpellet. This concludes that in order to force starvation, the spherical pellet demands a higher value of Thiele modulus com-pared to the cylindrical pellet, when all other parameters remain the same.

We now compare the present study done in case of first order reaction inside a cylindrical pellet with Prakash, Raja Se-khar, De, and Böhm (2010a), where it is done for a spherical pellet. There the convection–diffusion problem coupled withisothermal first order reaction inside a spherical porous pellet under oscillatory flow is treated. Effect of Peclet number(Pe) on nutrient concentration is shown in Fig. 9a for the combination of parameters Da = 0.003, - = 3, a = 0.5, / = 3.5. Itis seen that concentration minimum shifts towards the downstream due to convection and this value is smaller for the cylin-drical pellet compared to that of a spherical pellet. Fig. 9b shows the effect of frequency on nutrient concentration inside thepellets. The qualitative behavior is similar to the zero order case and there is a quantitative difference corresponding to cylin-drical and spherical pellets. It may be noted that no starvation zones are seen both in cylindrical and spherical pellets cor-responding to a first order reaction. The above findings help one to design more effective convection diffusion reactionsystems where porous catalysts are used either in cylindrical or spherical form.

5. Conclusion

The nutrient transport inside a permeable cylindrical porous pellet in an oscillatory flow is studied. The model assumes acylindrical 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 using the stream function approach and theconvection–diffusion–reaction problem is formulated for zero order or first order reaction kinetics. External mass transferresistance is neglected at the surface of the pellet. Effect of various parameters on the velocity as well as concentration pro-files is analyzed. It is observed that the internal velocity increases both with frequency and permeability, hence for a fixedDarcy number, increase in frequency enhances the volume flow inside the pellet. In case of zero order reaction starvationzones occur for a particular choice of parameters. An optimality criterion is proposed to avert this physical impossibility.Based on this criterion, classification is done in order to identify the regions of nutrient sufficiency and starvation. However,in case of first order reaction no such starvation zones occur either in cylindrical pellet or in spherical pellet. The presentinvestigations analyze the sensitivity of Dirichlet type boundary condition in case of oscillatory Stokes flow past a cylindricalporous pellet for both zero order and first order reaction kinetics. A comparison is also made with nutrient transport inside aspherical porous pellet. One can conclude that in order to force starvation, a spherical pellet demands a larger value of Thielemodulus compared to a cylindrical pellet, when all other parameters remain the same.

Acknowledgements

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

References

Aris, R. (1975). The mathematical theory of diffusion and reaction in permeable catalysts. Oxford: Clarcndon Press.Atkin, R. J., & Craine, R. E. (1976a). Continuum theories of mixtures: Applications. IMA Journal of Applied Mathematics, 17(2), 153–207.Atkin, R. J., & Craine, R. E. (1976b). Continuum theories of mixtures: Basic theory and historical development. QJMAM, 29(2), 209–244.Beavers, G. S., & Joseph, D. D. (1967). Boundary conditions at a naturally permeable wall. Journal of Fluid Mechanics, 30, 197–207.Bowen, R. E. (1976). Theory of mixtures. In A. C. Eringen (Ed.). Continuum Physics (Vol. III). New York: Academic Press.Cardoso, S. S. S., & Rodrigues, A. E. (2007). Convection, diffusion and reaction in a nonisothermal, porous catalyst slab. AIChE Journal, 53, 1325–1336.Gray, W. G. (1983). General conservation equations for multiphase systems 4. Constitutive theory including phase change. Advances in Water Resources, 6,

130–140.Gunn, D. J. (1967). Diffusion and chemical reaction in catalyst and absorption. Chemical Engineering Science, 22(11), 1439–1455.Hassanizadeh, M., & Gray, W. G. (1979a). General conservation equations for multiphase systems 1. Averaging procedure. Advances in Water Resources, 2,

131–144.Hassanizadeh, M., & Gray, W. G. (1979b). General conservation equations for multiphase systems 2. Mass, momenta, energy and entropy equations.

Advances in Water Resources, 2, 190–203.Ho, T. C., & Hsiao, G. C. (1977). Estimation of the effectiveness factor for a cylindrical catalyst support: A singular perturbation approach. Chemical

Engineering Science, 32(1), 63–66.Hornung, U. (1996). Homogenization and porous media. Springer.Looker, J. R., & Carnie, S. L. (2004). The hydrodynamics of an oscillating porous sphere. Physics of Fluids, 16, 62–72.


Mukkavilli, S., Tavlarides, L. T., & Wittmann, C. W. (1987a). Integral method of analysis for chemical reaction in a nonisothermal finite cylindrical catalystpellet – I. Dirichlet problem. Chemical Engineering Science, 42(1), 27–33.

Mukkavilli, S., Tavlarides, L. T., & Wittmann, C. W. (1987b). Integral method of analysis for chemical reaction in a nonisothermal finite cylindrical catalystpellet – II. Robin problem. Chemical Engineering Science, 42(1), 35–40.

Mulet, A. (1994). Drying modelling and water diffusivity in carrots and potatoes. Journal of Food Engineering, 22, 329–348.Mulet, A., Berna, A., & Rosello, C. (1989). Drying of carrot I. Drying models. Drying Technology, 7, 537–557.Muskat, M. (1946). The flow of homogeneous fluids through porous media. J.W. Edwards Publication.Pan, T., & Zhu, B. (1998). Study on diffusion–reaction process inside a cylindrical catalyst pellet. Chemical Engineering Science, 53(5), 933–946.Prakash, J., Raja Sekhar, G. P., De, S., & Böhm, M. (2010a). Convection, diffusion and reaction inside a spherical porous pellet in the presence of oscillatory

flow. European Journal of Mechanics – B/Fluids, 29, 483–493.Prakash, J., Raja Sekhar, G. P., De, S., & Böhm, M. (2010b). A criterion to avoid starvation zones for convection–diffusion–reaction problem inside a porous

biological pellet under oscillatory flow. International Journal of Engineering Science, 48, 693–707.Rajagopal, K. R. (2007). On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Mathematical Models and Methods in

Applied Sciences, 17(2), 215–252.Rajagopal, K. R., & Tao, L. (1995). Mechanics of mixtures. Singapore: World scientific.Raja Sekhar, G. P., & Amaranath, T. (1996). Stokes flow past a porous sphere with an impermeable core. Mechanics Research Communications, 23, 449–460.Richardson, S. (1971). A model for the boundary condition of a porous material – Part 2. Journal of Fluid Mechanics, 49, 327–336.Saffman, P. G. (1971). On the boundary condition at the surface of a porous medium. Studies in Applied Mathematics, 50, 93–101.Scheidegger, A. E. (1974). The physics of flow through porous media. University of Toronto Press.Schmidt, L. D. (2005). The engineering of chemical reactions. Oxford University Press.Smith, J. M., Van Ness, H. C., & Abbott, M. M. (2004). Introduction to chemical engineering thermodynamics. U.S.A.: McGraw-Hill.Sokhansanj, S. (1987). Improved heat and mass transfer models to predict grain quality. Drying Technology, 5, 511–525.Sorensen, J. P., Guertin, E. W., & Stewart, W. E. (1973). Computational models for cylindrical catalyst particles. AIChE Journal, 19, 969–975.Stephanopoulos, G., & Tsiveriotis, K. (1989). The effect of intraparticle convection on nutrient transport in porous biological pellets. Chemical Engineering

Science, 44(9), 2031–2039.Taylor, G. I. (1971). A model for the boundary condition of a porous material – Part 1. Journal of Fluid Mechanics, 49, 319–326.Vainshtein, P., & Shapiro, M. (2009). Forces on a porous particle in an oscillating flow. Journal of Colloid and Interface Science, 330, 149–155.Waite, G. N., & Waite, L. (2007). Applied cell and molecular biology for engineers. McGraw-Hill Professional.Wakao, N., & Smith, J. M. (1962). Diffusion in catalyst pellets. Chemical Engineering Science, 17, 825–834.

dirichlet problem for convection–diffusion–reaction inside a permeable cylindrical porous pellet

Documents