International Journal of Heat and Mass Transfer 68 (2014) 21–28

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International Journal of Heat and Mass Transfer

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

Mass/heat transfer through laminar boundary layer in axisymmetricmicrochannels with nonuniform cross section and fixed wallconcentration/temperature

0017-9310/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.08.101

⇑ Corresponding author. Tel.: +39 06 44585609; fax: +39 06 44585451.E-mail addresses: alessandra.adrover@uniroma1.it (A. Adrover), augusta.

pedacchia@uniroma1.it (A. Pedacchia).

Alessandra Adrover ⇑, Augusta PedacchiaLa Sapienza Università di Roma, Dipartimento di Ingegneria Chimica, Materiali e Ambiente, Via Eudossiana 18, 00184 Rome, Italy

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

Article history:Received 2 April 2013Received in revised form 29 August 2013Accepted 29 August 2013

Keywords:Boundary-layer theoryMicrochannelsLaminar forced convectionGraetz–Nusselt

We present a similarity solution for mass/heat transfer in laminar forced convection at high Pecletnumbers. The classical boundary layer solution of the Graetz–Nusselt problem, valid for straight channelsor pipes, is generalized to an axisymmetric microchannel with circular cross-section, whose radius R(z)varies continuously along the axial coordinate z. The case of fixed wall concentration/temperature isanalyzed.

The advection/diffusion transport problem is solved by taking into account both the tangential andnormal velocity components (and their scaling behaviours as a function of the wall normal distance),in order to obtain an accurate description of the concentration/temperature profile in the boundary layer.

The analytical solution of the local Sherwood/Nusselt number is compared with finite elementsnumerical results for a truncated cone and a wavy sinusoidal channel.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

A correct estimation of heat and mass transfer coefficients is apowerful tool in the design of heat exchangers, mass transferequipments and reactors, as well as micro-devices for chemicaland biomedical applications [1-6]. Focusing on laminar forcedconvection of an incompressible fluid in a duct, the estimation oftransport coefficients requires the solution of the classical Gra-etz–Nusselt problem [7,8]. Originally proposed for a sudden stepchange of the wall temperature at some positions along the ductand no axial diffusion [9,10], the Graetz–Nusselt problem is validfor both heat and mass transfer and it has been solved in transientand steady state [11], for Dirichlet and Neumann boundary condi-tions [12], non-Newtonian fluids [13], high viscous dissipation[14], boundary condition of continuity between two counterflowstreams [15], axial diffusion [16,17], simultaneous heat and masstransfer [18–22]. The boundary layer problem, valid in the limitof Pe ?1, has been analytically solved in channels with constantcross-section for a variety of different cross-sections [23]. On theother hand, convection–diffusion transport in converging ordiverging flows has been addressed for Taylor dispersion at lowReynolds number [24], but very few efforts have been devoted tothe Graetz–Nusselt problem in converging–diverging channels.

Non-parallel ducts have been indicated as a possible strategy toenhance heat transfer and numerical examples have been shownto corroborate the so-called ‘‘field synergy principle’’ [25].Castelloes et al. [26] investigated heat transfer enhancement inconverging–diverging channels in laminar flow conditions for1 < Pe < 100. The energy equation was solved using a hybridnumerical-analytical approach based on the Generalized IntegralTransform Technique (GITT) in partial transformation mode for atransient formulation.

Recently, an analytical solution [27] has been proposed for thecombined diffusive and convective mass transport from a surfacefilm of arbitrary shape at a given uniform concentration to a puresolvent, flowing in the creeping regime through converging–diverging microchannels with slender rectangular cross-section.In [27] Adrover and Pedacchia clearly show that, close to thecurved releasing boundary, both the tangential vt and the normalvelocity vn components play a role in the mass transfer process,and their scaling behaviour as a function of wall the normal dis-tance should be taken into account for an accurate description ofthe concentration profile in the boundary layer.

By following similar arguments, we present a similarity solutionfor mass/heat transfer in laminar forced convection at high Pecletnumbers in axisymmetric microchannel with circularcross-section, whose radius R(z) varies continuously along the axialcoordinate z. Creeping flow conditions and fixed concentration/temperature at the channel wall are assumed. High values of Pecletnumber, together with low Reynolds numbers are often

Nomenclature

List of symbolsc concentrationD diffusion coefficientg(z,Pe) rescaling function, Eq. (16)h mass/heat transfer coefficientI(z) integral function, Eq. (21)Lz channel lengthn vector normal to the releasing wall at the point (Rz(z),z)Pe = vRR0/D cross-sectional Peclet numberPeeff = aPe effective Peclet numberPel local Peclet number~r; ~z radial and axial dimensional coordinatesr, z radial and axial dimensionless coordinatesRz(z) dimensionless cross-section radius depending on the

axial position zR0z(z) = dRz/dz first order derivative of Rz(z)R00z (z) = d2Rz/dz2 second order derivative of Rz(z)R0 radius of the inlet sectionR = R(s) = Rz(z(s)) dimensionless cross-section radius depending

on the curvilinear abscissa sR0(s) = dRz(z)/dzjz(s) first order derivative of Rz(z) evaluated at z(s)R00(s) = d2Rz(z)/dz2jz(s) second order derivative of Rz(z) evaluated

at z(s)

s curvilinear abscissaSh = h R0/D Sherwood numberShapp approximate Sherwood number evaluated by neglecting

the nornal convective termt vector tangent to the releasing wall at the point (Rz(z),z)T temperaturev0

nðsÞ prefactor of the quadratic term of the normal velocitycomponent vnðd; sÞ ’ v0

nðsÞd2

v0t ðsÞ prefactor of the linear term of the tangent velocity

component v tðd; sÞ ’ v0t ðsÞd

vr, vz dimensionless radial and axial velocity componentsvn, vt dimensionless normal and tangent velocity componentsvR average inlet axial velocity

Greek symbolsa = R0/Lz channel aspect ratiod wall normal distanceg = dg(s,Pe) similarity variable/ dimensionless scalar field (concentration or

temperature)

z

r

inflow z=0

outflow z=L /Rz 0

nt

R (z)z

s

δ

Fig. 1. Schematic representation of a channel longitudinal section at h = 0. r and zrepresent the dimensionless radial and axial coordinates. d is the dimensionlesswall normal distance. s is a curvilinear abscissa measured from the channel inletsection. n and t are the vectors normal and tangent to the releasing wall.

22 A. Adrover, A. Pedacchia / International Journal of Heat and Mass Transfer 68 (2014) 21–28

encountered in microfluidic applications, especially in connectionwith mass transport problems characterized by low diffusivity val-ues, see e.g. micro-mixing devices [28–31], dispersion problems[32] and wide-bore chromatography [33,34].

2. Statement of the problem and numerical solutions

Let us consider an incompressible fluid moving in creeping flowconditions through an axisymmetric microchannel with circularcross-sections, whose radius varies along the axial coordinate.

Let ~r and ~z be the radial and axial coordinates, R0 the radius ofthe inlet section, Lz the channel length and D the diffusioncoefficient. Let / be a scalar field representing a dimensionlessconcentration or temperature

/ ¼ c � cinlet

cwall � cinlet¼ T � T inlet

Twall � T inlet

In terms of dimensionless spatial coordinatesz ¼ ~z=R0; 0 6 z 6 1=a ¼ Lz=R0 and r ¼ ~r=R0; 0 6 r 6 RzðzÞ (seeFig. 1) the steady-state convection–diffusion transport equationand boundary conditions (fixed wall concentration/temperaturecwall/Twall and Danckwerts inlet–outlet conditions) read as

� Pev rðr;zÞ@/@r� Pevzðr;zÞ

@/@zþ @

2/@z2 þ

1r@

@rr@/@r

� �¼ 0 ð1Þ

Pevz/�@/@z

� �z¼0¼ 0; /ðRzðzÞ;zÞ ¼ 1;

@/@r

����r¼0¼ 0;

@/@z

����z¼1¼ 0 ð2Þ

where a = R0/Lz� 1 is the aspect ratio, Pe = vRR0/D is the cross-sec-tional Peclet number (Pe = Re Sc or Pe = Re Pr) evaluated with re-spect to the average inlet axial velocity vR. Let vr(r,z) and vz(r,z)be the dimensionless velocity components:

v rðr; zÞ ¼2rR0zpR3

z

1� rRz

� �2 !

ð3Þ

vzðr; zÞ ¼2

pR2z

1� rRz

� �2 !

;

Z Rz

0vzðr; zÞ2prdr ¼ 1 ð4Þ

where R0z = d Rz/d z. The parabolic axial velocity profile vz(r,z) isevaluated from lubrication theory by enforcing unitary flow rateand no-slip boundary conditions. The radial velocity component vr

(r,z) is obtained by enforcing the continuity equation in cylindricalcoordinates [27]. Creeping flow conditions are assumed.

The local mass/heat transfer coefficient h can be expressed interms of the Sherwood/Nusselt number Sh = h R0/D, Nu = h R0/kevaluated from the gradient at the releasing wall as:

A. Adrover, A. Pedacchia / International Journal of Heat and Mass Transfer 68 (2014) 21–28 23

Shðz; PeÞ ¼ � 1ðcwall � cinletÞ

rcjr¼RzðzÞ � n ¼ �r/jr¼RzðzÞ � n ð5Þ

Nuðz; PeÞ ¼ � 1ðTwall � T inletÞ

rTjr¼RzðzÞ � n ¼ �r/jr¼RzðzÞ � n ð6Þ

n ¼ ðnr ;nzÞ ¼�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ Rz02p ;

R0zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Rz02

p !

ð7Þ

n being the normal vector to the releasing wall at the point (Rz(z),z).The advection–diffusion equations (1) and (2) with the velocity

profile Eqs. (3) and (4) has been solved by finite elements method(FEM, Comsol Multiphysics 4.2) with a number of finite elementsranging from 105 to 5 � 106 elements, depending on the value ofthe effective Peclet number Peeff = aPe. The larger the Peeff, thesmaller the thickness of the boundary layer, the larger the numberof finite elements required to resolve steep gradients in the bound-ary layer (minimum element dimension 10�4 close to the wall).The shape functions were Lagrange quadratic and the linear solvermethod was PARDISO, with a relative tolerance � = 10�8.

Numerical results for the spatial profiles of / are reported inFig. 2 for aPe = 103 and two different geometries, highlighting theeffect of the wall shape Rz(z) on the thickness and structure ofthe boundary layer.

The truncated-cone Rz(z) = 1 � baz (Fig. 2, top) is a convergingchannel, exhibiting an almost constant thickness of the boundarylayer, because the natural thickening of the boundary layer alongthe axial coordinate z is counterbalanced by the narrowing of thecross-section, determining an increase of the local shear rate.

The wavy sinusoidal channel Rz(z) = 1 � b sin(kpaz) (Fig. 2, bot-tom) is a converging–diverging channel, showing a non monotonicbehaviour of the thickness of the boundary layer.

In order to verify the accuracy of the numerical results we adopttwo different spatial discretizations of the computational domain,solve the transport problem and compare the concentrationprofiles. In Comsol Multiphysics 4.2 we generate a mesh with theoptions (1) ‘‘physics controlled, default’’, (2) ‘‘free triangular’’ and(3) ‘‘extremely fine’’ grid. Then we perform two and three ‘‘regularrefinements’’. In the case of the truncated cone these mesh optionsgenerate a mesh made by 305.632 elements (two refinements) and1.222.528 elements (three refinements). In the case of the sinusoi-dal channel the mesh is made by 737.728 elements (two refine-ments) and 2.950.912 (three refinements). In order to comparethe numerical results for the resulting steady-state concentrationfields we compare the thicknesses of the boundary layer dbl definedin a shortcut way as follows: let q(z) be the radial position of pointsat which the concentration attains a fixed arbitrary low value (weused / = 0.05 in order to identify the thickness of the boundarylayer) and define dbl(z) = Rz(z) � q(z). Fig. 3 shows the thicknessof the boundary layer dbl vs az for the truncated cone (a) and thewavy sinusoidal channel (b) for aPe = 103 (same data reported inFig. 2). Coutinuous lines show the behaviour of dbl(z) with thefinest grid (three refinements). Points show dbl(z) with a morecoarse grid (two refinements). The excellent agreement betweendata obtained with two and three refinements confirms thenumerical reliability of the concentration fields. Numerical resultsare not influenced by the relative tolerance � of the linear solver for� 6 10�6. Similar analysis have been performed for higher values ofaPe.

3. Boundary layer formulation and similarity solution

Let d be the wall normal distance, and s a curvilinear abscissa(see Fig. 1). Close to the channel wall, the tangent and normalvelocity components scale, as a function of the normal walldistance d, as follows

vnðd; sÞ ¼ v0nðsÞd

2 þOðd3Þ; ð8Þ

v0nðsÞ ¼

4

pR4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ R02

q ðR0ð1þ R02Þ � RR0R00Þ

v tðd; sÞ ¼ v0t ðsÞdþOðd

2Þ; v0t ðsÞ ¼

4p

1þ R02

R3 ð9Þ

where R ¼ RðsÞ ¼ RzðzðsÞÞ; R0ðsÞ ¼ dRzdz jzðsÞ and R00ðsÞ ¼ d2Rz

dz2

���zðsÞ

The tangential velocity component scales linearly with d andthe normal component scales quadratically. For a complete deriva-tion of vt and vn, see Appendix A.

In the classical solution of the Graetz–Nusselt problem, forcylindrical channels of constant radius (Rz = 1,R0z = 0), the boundarylayer equation attains the form

Pev0t d@/@z¼ 1ð1� dÞ

@

@dð1� dÞ @/

@d

� �’ @

2/

@d2 ; v0t ¼ 4=p ð10Þ

where the axial dispersion contribution is neglected (large Pevalues) and the remaining part of the Laplacian operator is approx-imated as @2c

@d2 by considering that 1 � d ’ 1 close to wall, where theboundary layer develops.

We make the same assumptions/simplification in the solutionof the generalized Graetz–Nusselt problem, in an axisymmetricchannel with a radius Rz(z) continuously varying along the axialcoordinate z, but we also include the important contribution ofthe normal convective term.

The boundary layer equation in a tangential-normal referencesystem (s,d) can be thus be approximated as

Pev0t d@/@sþ Pev0

nd2 @/@d¼ @

2/

@d2 ð11Þ

/ð0; dÞ ¼ 0; /ðs > 0;0Þ ¼ 1; /ðs;1Þ ¼ 0 ð12Þ

where v0t and v0

n are given by Eqs. (8) and (9).The boundary layer equation Eqs. (11) and (12) for the axisym-

metric channel is identical to the boundary layer problem for thechannel with non-uniform slender rectangular cross-section,which has been presented in [27]. Therefore it can be analogouslysolved by a similarity approach.

By introducing the similarity variable g = dg(s,Pe), the concen-tration/temperature profile can be rewritten in the invariant form/(s,d,Pe) = f(g), where g(s,Pe) satisfies the following ordinary dif-ferential equation:

g0ðs; PeÞ ¼ � v0n sð Þ

v0t sð Þ gðs; PeÞ þ g4ðs; PeÞ

Pev0t ðsÞ

!; 1=gð0; PeÞ ¼ 0: ð13Þ

The invariant profile f(g) attains the form

f ðgÞ ¼ 1� 1A

Z g

0exp½�g03=3�dg0

� �; A ¼ Cð1=3Þ

32=3 ð14Þ

Correspondingly, the dimensionless transfer coefficient is given by

Shðs; PeÞ ¼ gðs; PeÞA

ð15Þ

By solving Eq. (13) with v0t and v0

n Eqs. (8) and (9), one obtains thefollowing expression for the rescaling function g(s,Pe), or equiva-lently for g(z,Pe):

gðz; PeÞ ¼ 4Pe3p

� �13

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ R0z

2q

Rz

Z z

0ð1þ R0z

2Þdz0� ��1

3

: ð16Þ

Fig. 4 shows the behaviour of Sh vs az in a truncated cone withradius Rz(z) = 1 � baz (b = 0.5,a = 0.01), at different values ofaPe 2 [102,106], as obtained by numerical solution of Eqs. (1) and(2), and the comparison with the analytical solution Eq. (16)

Fig. 2. 3-d spatial profiles for / obtained by numerical solution of Eqs. (1)–(4), aPe = 103. Cartesian system of coordinates: x = rcos (h), y = rsin (h), z = z. Top: truncated cone,Rz(z) = 1 � baz, b = 0.5, a = 0.01. Bottom: sinusoidal channel Rz(z) = 1 � bsin(kpaz), b = 0.5, k = 3, a = 0.5.

24 A. Adrover, A. Pedacchia / International Journal of Heat and Mass Transfer 68 (2014) 21–28

Shðz; PeÞ ¼ 4aPe

3pð1þ b2a2Þaz

!13ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ b2a2

p1� baz

ð17Þ

We observe an excellent agreement between numerical andanalytical results for aPe P 104, which represents an extremelyhigh value of Peeff, considering that the tangential dispersion term@2//@s2 can be actually neglected for aPe P 10. Moreover, the

analogous similarity solution presented in [27] for the massboundary layer for a slender rectangular cross-section exhibits anexcellent agreement with simulation results for aPe P 102.

This issue can be easily explained by observing that the approx-imation of the Laplacian operator as @2c/@d2 is actually valid onlyfor very small thicknesses of the boundary layer, thus for very highvalues of aPe. This argument is supported by Fig. 5, showing the

0

0.1

0.2

0.3

0.4

0 0.2 0.4 0.6 0.8 1

δ bl (

φ=0.

05)

α z

a

b

Fig. 3. Thickness of the boundary layer dbl vs the axial coordinate z for the truncatedcone (a) and the wavy sinusoidal channel (b) for aPe = 103. Continuous lines: finestgrid (three refinements). Points: more coarse grid (two refinements).

100

101

102

103

10-2 10-1 100

Sh

α z

Pe α

Fig. 4. Spatial behavior Sh vs az for the truncated cone R(z) = 1 � baz, b = 0.5 atdifferent values of aPe 2 [102,106]. Dashed lines: numerical solution. Continuouslines: analytical solution, Eq. (17).

100

101

102

103

10-2 10-1 100

Sh

α z

Pe α

Fig. 5. Spatial behavior Sh vs az for the cylindrical channel R(z) = 1 at differentvalues of aPe 2 [102,106]. Dashed lines: numerical solution. Continuous lines:analytical solution, Eq. (18).

10-1

100

101

102

103

0 0.2 0.4 0.6 0.8 1

Sh

α z

Pe α

Fig. 6. Spatial behavior Sh vs az for the sinusoidal channel R(z) = 1 � bsin (kpaz),b = 0.5, k = 3, a = 0.5 at different values of aPe 2 [103,106]. Dashed lines: numericalresults. Continuous lines: analytical solution, Eq. (19).

A. Adrover, A. Pedacchia / International Journal of Heat and Mass Transfer 68 (2014) 21–28 25

behaviour of Sh vs az in a cylindrical channel Rz(z) = 1, for a = 0.01and aPe 2 [102,106]. Continuous lines represent the numericalsolution of the convection/diffusion transport problem in cylindri-

cal coordinates, while dashed lines represent the well-knownanalytical solution of the Graetz–Nusselt problem

Shðz; PeÞ ¼ 4Pe3pz

� �1=3 1A

ð18Þ

Also in this simple case, the agreement between numerical solutionand analytical results is satisfactory only for aPe P 104, thusconfirming that the approximation made of the Laplacian term isactually responsible for the deviations observed at lower values ofaPe, in the cylindrical channel as well as in the truncated cone.

As a further numerical confirmation of the validity of theanalytical solution Eq. (16), we consider a more complex channelstructure, which is a sinusoidal converging–diverging channel,R(z) = 1 � bsin (kpaz), with b = 0.5, k = 3, a = 0.5. See Fig. 2(bottom) for the dimensionless concentration profile at aPe = 103.We purposely choose a high value of the aspect ratio (a = 0.5), inorder to show that the analytical solution is quantitatively reliablein the case of long-thin channels (a� 1), as well as short-thickchannels a 2 [0.2,1] characterized by significant and rapidvariations of the channel cross-section along the axial coordinatez, i.e. j R0z j’ Oð1Þ. In the present case, jR0zj 6 3p/4. The only intrinsiclimit to the application of the model is the validity of the lubrica-tion velocity profile Eqs. (3) and (4), thus restricting the soundnessof the analytical solution to creeping-flow conditions (no recircula-tion patterns).

For the sinusoidal channel, the local Sherwood number attainsthe form

Shðz; PeÞ ¼ 43p

aPe

ð1þ c2=2Þazþ c2

4kp sinð2kpazÞ

!13

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ c2 cos2ðkpazÞ

p1� b sinðkpazÞ ð19Þ

where c = b kpa. Fig. 6 shows the comparison between numericaland analytical results for aPe 2 [103,106]. We observe significantdeviations only for aPe = 103 and in the central region (az = 0.5),where the cross-section enlargement is maximum, thus inducinglower values of local tangential Peclet number Pev0

t . The issue ofthe local Peclet number will be addressed in detail in the next sec-tion. The agreement between numerical and analytical results isfully satisfactory for higher values of aPe.

It can be observed that, although the normal velocity compo-nent may be small compared with the tangent velocity component,the normal convective term represents a significant contribution

26 A. Adrover, A. Pedacchia / International Journal of Heat and Mass Transfer 68 (2014) 21–28

for an accurate description of the concentration field and thicknessof the boundary layer. As a numerical confirmation, we considerthe same sinusoidal channel R(z) = 1 � bsin (kpaz) with a = 0.01,which implies a long-thin channel with slow cross-sectionvariations, jR0zj 6 3p10�2/2.

Fig. 7(A) shows that the ratio v0n=v0

t is extremely smalleverywhere along the channel. However, if we neglect the normalconvective term we are not able to accurately describe the spatialbehaviour of the Sherwood number, failing in capturing itscomplex structure of local minima/maxima. This can be readilyunderstood by considering that the normal velocity componentchanges sign moving along the channel: a negative (positive) nor-mal velocity component increases (decreases) the concentrationgradient at the wall, for Dirichlet boundary conditions, thusenhancing (reducing) the mass transfer rate.

Fig. 7(B) shows the comparison between Sh vs az as obtained bythe analytical solution Eq. (19) including the normal convectiveterm (continuous line) and Shapp vs z representing the analytical

-0.1

A

0

0.1

0 0.2 0.4 0.6 0.8 1

v no

/ vto

α z

101

B

102

103

0 0.2 0.4 0.6 0.8 1

Sh

α z

-0.8

-0.6

-0.4

C

-0.2

0

0.2

0.4

0.6

0 0.2 0.4 0.6 0.8 1

I(z)

α z

Fig. 7. (A) v0n=v0

t vs z. (B) Comparison between Sh Eq. (19) and Shapp Eq. (20) foraPe = 106. Sh: Continuous line; Shapp: dashed line. (C) Integral function I(z), Eq. (21).All figures refer to a long-thin sinusoidal channel R(z) = 1 � bsin (kpaz) with b = 0.5,k = 3 and a = 0.01.

solution of the boundary layer equation Eq. (11) neglecting thenormal convective term (dashed line)

ShappðzÞ ¼1A

4Pe3p

� �13Z z

0

R3zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ R0z2

q dz0

0B@

1CA�1

3

ð20Þ

Let us define an integral function

IðzÞ ¼Z z

0

v0nðz0Þ

v0t ðz0Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ R0z

2q

dz0 ð21Þ

representing the integral, along the curvilinear abscissa, of therescaled normal velocity component. This can be assumed as a sim-ple/approximate way to quantify the effect of the normal velocityon the mass transfer rate, from the channel inlet up the currentposition z. Fig. 7(B) and (C) clearly shows that the approximate solu-tion Shapp underestimates the actual value Sh(z) where I(z) < 0 andoverestimates it where I(z) > 0. Negative values of I(z) implyenhancement of the mass transfer rate.

4. Some issues on the analytical solution

The analytic expression for the local Sherwood number Sh(z,Pe)Eq. (16) can be reformulated in terms of a local effective Pecletnumber Pel(z) as follows

Shðz; PeÞ ¼ 31=3

Cð1=3ÞPelðzÞ

z

� �1=3

ð22Þ

PelðzÞ ’4Pe

pR3z

ð1þ R0z2Þ

3=2

hð1þ R0z2Þiz

ð23Þ

hð1þ R0z2Þiz ¼

1z

Z z

0ð1þ R0z

2ðlÞÞdl ð24Þ

The local Pe number Pel takes into account the entire shape of theboundary, from the inlet z = 0 to the current axial position z, bymeans of the integral term h(1 + R02z )iz.

It can be observed that for channels characterized by slowvariations of the radius Rz(z) along the axial coordinate, i.e.R02z � 1, the integral term equals to unity. The analytical expressionof the local effective Peclet number attains the form

PelðzÞ ¼4Pe

pR3z

¼ Pe@v�

z

@r

�����r¼Rz

ð25Þ

Eq. (25) is a shortcut model for Pel(z) and it takes into account in anaive way the local variation of the shear-rate due to cross-sectionvariations. Whenever R02z is of the order of unity, the shortcut modelcan not be adopted.

This observation marks a significant difference between thepresent result and the behaviour we observed for the correspond-ing mass transfer problem in a channel with axial-varying slenderrectangular cross section [27].

Let hBz ðzÞ and hT

z ðzÞ be the expressions for the bottom and topwall, respectively, so that the dimensionless channel height isHðzÞ ¼ hT

z � hBz and varies along the axial coordinate z.

The local Sherwood number for mass transfer at the bottomwall attains the form [27]

Shðz; PeÞ ¼ gðz; PeÞA

¼ 1Að2PeÞ

13

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ hB

z

02qðhT

z � hBz Þ

Z z

0

ð1þ hBz

02Þ

hTz � hB

z

� dz00@

1A�1

3

ð26Þ

Therefore the local Pe number Pel for slowly-varying channelsj h0Bx j� 1 reads as

A. Adrover, A. Pedacchia / International Journal of Heat and Mass Transfer 68 (2014) 21–28 27

PelðzÞ ’6Pe

H3hH�1iz; hH�1iz ¼

1z

Z z

0

dz0

Hðz0Þ ð27Þ

Apart from an obvious difference in the constant coefficientsbetween the axisymmetric and the rectangular channels, one note-worthy aspect must be pointed out. The integral term in Eq. (26)depends on both the channel height H(z) and the first derivativehB

z

0, while in Eq. (16) the corresponding integral term is only

function of the first derivative R0z. Thus, for slowly-varying axisym-metric channels, the local Sherwood number exhibits a classicalboundary layer dependence z�

13 and a shortcut model can be

adopted. On the contrary, for the slender rectangular cross-section,even for j h0Bx j� 1 we need to consider the contribution of theintegral term Eq. (27), accounting for the whole variation of thechannel height, from the inlet z = 0 to the current axial position z.

5. Conclusions

The convection/diffusion problem, equivalent for both heat andmass transfer, has been analytically solved in a boundary layerapproximation with Dirichlet boundary condition, for creepingflow in axisymmetric microchannels with circular cross-sectionvarying along the axial coordinate z as a R0z(z) 2 C2(X). Theproposed solution has been validated by comparing the predicteddimensionless mass transfer coefficient Sh with numerical FEMsolutions and shows to work well for aPe P 104. A simplified mod-el has been suggested in the limit of R0z(z)2� 1. This theoreticalframe may be helpful for a better understanding of heat and masstransport phenomena in microchannels with this type of geome-tries, as well as in natural creeping flow ducts in animals andplants. Moreover, it could be a basis for the future developmentof new kind of optimized mass/heat transport devices.

Appendix A

Let (Rz(z),z) be a point of the releasing wall. Let w(d,z) =(Rz(z) + dnr(z),z + dnz(z)) be the set of points exploring the directionnormal to the wall at the point (Rz(z),z) = w(0,z).

The velocity field v at each point w can be decomposed into anormal vector vn and a tangent vector vt defined as:

vn ¼ ðv � nÞn; vt ¼ ðv � tÞt ðA:1Þ

The normal velocity component vn = (v � n) and the tangent velocitycomponent vt = (v � t), close to w(0,z), vary along the normal coordi-nate d as follows

vnðd; zÞ ¼ v rðwðd; zÞÞnrðzÞ þ vzðwðd; zÞÞnzðzÞ ðA:2Þv tðd; zÞ ¼ v rðwðd; zÞÞtrðzÞ þ vzðwðd; zÞÞtzðzÞ ðA:3Þ

where vr and vz are given by Eqs. (3) and (4), tr, tz are defined as

t ¼ ðtr ; tzÞ ¼R0zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ R0z2

q ;1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ R0z2

q0B@

1CA ðA:4Þ

and nr, nz are given by Eq. (7).By Taylor expanding both vn and vt about d = 0 we obtain:

vnðzÞ ¼4

pR4z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ R0z

2q ðR0zð1þ R0z

2Þ � RzR0zR00z Þd

2 þOðd3Þ ðA:5Þ

v tðzÞ ¼4p

1þ R0z2

R3z

dþOðd2Þ ðA:6Þ

corresponding to Eqs. (8) and (9). It should be observed that Eqs. (8)and (9) represent vn and vt as a function of the curvilinear abscissa

s. Therefore Rz(z) and its derivatives appearing in Eqs. (A.5) and(A.6) are replaced by R = R(s) = Rz(z(s)) (and its derivatives) in Eqs.(8) and (9).

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