Mass transfer through laminar boundary layer in microchannels with nonuniform cross section : The effect of wall shape and curvature

International Journal of Heat and Mass Transfer 60 (2013) 624–631

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Mass transfer through laminar boundary layer in microchannels with nonuniform cross section : The effect of wall shape and curvature Alessandra Adrover ⇑, Augusta Pedacchia La 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

Article history: Received 27 November 2012 Received in revised form 10 January 2013 Accepted 12 January 2013 Available online 12 February 2013 Keywords: Mass/heat transfer Boundary-layer theory Microchannels Laminar forced convection

a b s t r a c t This paper provides an analytical solution for the combined diffusive and convective mass transport from a surface film of arbitrary shape at a given uniform concentration to a pure solvent flowing in the creeping regime through microchannels. This problem arises e.g. in the study of swelling and dissolution of polymeric thin films under the tangential flow of solvent, modeling the oral thin film dissolution for drug release towards the buccal mucosa or oral cavity. We present a similarity solution for mass transfer in laminar forced convection. The classical boundary layer solution of the Graetz-Nusselt problem, valid for straight channels or pipes, is generalized to a microchannel with rectangular cross-section varying continuously along the axial coordinate. Close to the curved releasing boundary hs(s), parametrized by a curvilinear abscissa s, both the tangential vt(r,s) and the normal velocity vn(r,s) components play a role in the dissolution process, and their scaling behavior as a function of wall normal distance r should be taken into account for an accurate description of the concentration profile in the boundary layer. An analytic expression for the local Sherwood number as a function of the curvilinear abscissa and the Peclet number is presented. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Microfluidics is a very promising format to measure the dynamics of different processes: diffusion/dispersion of a solute, protein folding, kinetics of chemical reaction. See reviews [1,2]. A new possible application of microfluidics is the characterization of drug release kinetics from Oral Thin Films (OTF), that are the most advanced form to administer drugs via absorption in the mouth buccally or sublingually, because of theirs compliance and comfort [3–6]. Films are prepared using hydrophylic polymers (HPMC, HPC, etc.) that rapidly dissolve when they come into contact with saliva, delivering the drug locally or to the systemic circulation [7–10]. Typical OTF thickness ranges from 10 to 100 lm. A microfluidic device can be used in order to simulate the laminar tangential flow of saliva on the thin film placed on the tongue or buccal cavity thus investigating, in vitro, the drug release and polymer dissolution processes under specified flow conditions. The flow rate (1 mL/min) and thickness of the salivary film covering the teeth and oral mucosa (70–100 lm) are such that laminar flow conditions must be considered [11–13]. ⇑ Corresponding author. Tel.: +39 348 7267562; fax: +39 06 44585451. E-mail addresses: [email protected] [email protected] (A. Pedacchia).

(A.

Adrover),

0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.01.042

augusta.

The dissolution mechanism of an hydrophylic polymer implies solvent penetration, swelling and phase transition phenomena (with the appearance of a gel state), polymer chains disengagement (after a reptation time) and polymer transport in the fluid outside the film [14–16]. When polymer dissolution is controlled by mass transfer rate at the gel/solvent interface, the presence of a tangential flow of solvent induces nonuniform polymer dissolution (see Fig. 1) and the structure of the gel/solvent interface changes in time, continuously modifying the geometry of the channel. The structure of mass transfer boundary layer evolves in time following the evolution of the gel/solvent interface. The mass transfer rate (and therefore the local Sherwood number) through the laminar boundary layer is influenced by the curvature of the gel/liquid interface. This paper provides an analytical solution for the combined diffusive and convective solute transport from a surface film of arbitrary shape (at a fixed uniform concentration) to a pure solvent flowing, in creeping flow conditions, through a microchannel. The microchannel is delimited by a flat no-slip surface and by the releasing film itself. The case of high Peclet numbers and long-thin channels is considered, so that classic boundary layer approximations can be adopted [17]. We present a similarity solution that generalizes the boundary layer solution of the Graetz–Nusselt problem (valid for straight channels or pipes, [18]) to a solvent flowing in creeping flow

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@c @c @2c @2c  Pev y ðx; yÞ þ þ @x @y @x2 @y2    @c  @c Pev x c  ¼ 0; ¼0  @x x¼0 @xx¼1=a  @c  cðx; hðxÞÞ ¼ cw ; ¼0 @yy¼1

!

 Pev x ðx; yÞ

¼0

ð1Þ ð2Þ ð3Þ

where a = Ly/Lx  1 is the aspect ratio and Pe = vRLy/D = Re Sc is the cross-sectional Peclet number evaluated with respect to the average inlet axial velocity vR = < vxjx=0 > . Inlet Danckwerts conditions are adopted in order to deal with Pe values in the whole range Pe 2 [100  106]. Let vx(x,y) and vy(x,y) be the dimensionless velocity components

2. Statement of the problem and numerical solutions Let us consider a pure solvent, flowing under creeping flow conditions, between two flat plates (distance Ly). At ~ x ¼ 0 it suddenly enters a channel of length Lx, where the top flat wall is insoluble and the bottom wall, of arbitrary shape hð~ xÞ, is a dissolving/releasing wall. The dissolving solute is only sparingly soluble and it dissolves very quickly, so that –(i) the boundary layer for flow is unchanged by the dissolving wall and –(ii) a constant wall concentration cw can be assumed. In terms of dimensionless spatial coordinates x ¼ ~ x=Ly ; 0 6 ~=Ly ; hðxÞ 6 y 6 1, the steady-state solute transx 6 Lx =Ly and y ¼ y port equation and boundary conditions attain the form

6ðy  hðxÞÞð1  yÞ 3

ð1  hðxÞÞ

Z

1

v x ðx; yÞdy ¼ 1

;

ð4Þ

hðxÞ

0

v y ðx; yÞ ¼

6h ðxÞðy  hðxÞÞð1  yÞ2 4

ð1  hðxÞÞ

;

0

h ðxÞ ¼

dhðxÞ : dx

ð5Þ

The parabolic axial velocity profile vx(x,y) is evaluated from lubrication theory by enforcing unitary flow rate and no-slip boundary condition, while the cubic profile for vy(x,y) is obtained by enforcing the continuity equation. Fig. 2 shows the excellent agreement between the spatial behavior of vx(x0,y) and vy(x0,y) for a releasing bottom wall h(x) = (ax)2/2 with a = 0.1 as obtained by the numerical solution of the Navier–Stokes equations in creeping flow conditions (triangles) and the analytical expressions (continuous lines), Eqs. (4) and

3 2.5 2 vx(xo,y)

conditions into a microchannel with rectangular cross-section continuously varying along the axial coordinate. The analytical solution is given for an arbitrary shape of the releasing boundary. The only limitation is the validity of the lubrication approximation adopted for determining the velocity profiles and henceforth for estimating the local shear rate. We show that the shape of the releasing boundary controls the spatial behavior of the local Sherwood number, yet the scaling behavior of Sh as a function of Pe still corresponds to the classic Lévêque solution [17,19,20], i.e. Sh  Pe1/3, for high Peclet values. The article is organized as follows. In Section 2 we present numerical results for the advection– diffusion equation in the presence of a curved releasing boundary. Convex and concave boundaries are considered. In Section 3 we present the boundary-layer formulation of the problem, accounting for both the convective term (due to the velocity component tangent to the releasing wall) and the convective contribution due the normal velocity component, naturally arising for a non-flat shape of the releasing boundary. In Section 4 we solve analytically the boundary-layer advection diffusion equation and present a closed form expression of the local Sherwood number as a function of the shape of the releasing boundary and of the Peclet number. We also propose a simple expression for the local Peclet number, valid for slowly varying channels, that quantifies in a simple way the influence of a curved boundary on the mass transfer rate. We compare analytical and numerical results for both concave and convex releasing boundaries. In Section 5 we present the analytical solution of the same boundary-layer problem in a microchannel where both the top and the bottom walls exhibit a curved shape.

v x ðx; yÞ ¼

1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

y 0.07 0.06 0.05 vy(xo,y)

Fig. 1. Schematic representation of the microdevice for the analysis of drug release and gel erosion from thin polymeric films. The thin polymeric film is initially dry and at the glassy state. When the solvent (water) starts to flow in the microchannel, the solvent penetrates the polymer, that undergoes a glassy-rubbery transition. The polymer swells and, after a reptation time, erodes. When polymer dissolution is controlled by mass transfer rate, the erosion is nonuniform along the channel, and this determines curved shape of the swollen gel/solvent interface.

0.04 0.03 0.02 0.01 0 0

0.2

0.4 y

Fig. 2. Spatial behavior of vx(x0,y) (top) and vy(x0,y) (bottom) for a releasing bottom wall h(x) = (ax)2/2 with a = 0.1. Triangles: numerical solution of the Navier–Stokes equations in creeping flow conditions. Continuous lines: analytical expressions, Eqs. (4) and (5). Five different cross-sections are considered, ax0 = 0.1, 0.3, 0.5, 0.7, 0.9. Arrow indicates increasing values of ax0.

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(5). Five different cross-sections are considered, a x 0 = 0.1, 0.3, 0.5, 0.7, 0.9. The mass transfer rate at the releasing bottom wall is quantified in terms of the local Sherwood number

1 Shðx; PeÞ ¼  rcjy¼hðxÞ  n; cw 0 0

n ¼ ðnx ; ny Þ 1

1 B h C ¼ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 0 02 1þh 1þh

ð6Þ

Fig. 3 shows the behavior of Sh vs (ax) as obtained by numerical solution of the transport equations Eqs. (1)–(5), for different values of the aspect ratio a and increasing values of the effective Peclet number Peeff = aPe. Numerical results have been obtained by finite element method (Comsol 3.5) with a number of finite elements ranging from 105 to 5  106 elements, depending on the value of the Peeff number. The larger the Peeff, the smaller the thickness of the boundary layer, the larger the number of finite elements required to resolve steep concentration gradients in the boundary layer (minimum element dimension 104 close to the releasing wall). From the analysis of Fig. 3 it can be observed that, by lowering the value of a, the curves Sh vs ax, corresponding to the same value of Peeff collapse onto an invariant curve, depending only on Peeff. Fig. 3 clearly shows that for aPe > 102 (sufficiently high effective Pe number) and a < 0.1 (sufficiently long-thin channels) inlet effects related to the Danckwerts inlet condition are neglegible and the standard inlet condition c(0,y) = 0 can be adopted. Moreover, it is evident that Sh profile exhibits a minimum along the axial coordinate. This behavior has to be ascribed to the non-uniform cross-section. Concentration profiles for aPe = 102 and aPe = 104 are shown in Fig. 4, highlighting the effect of the shape and curvature of the bottom wall on the structure and thickness of the mass transfer boundary layer. Two different types of channel are considered: a converging convex channel with h(x) = (ax)2/2 and a converging– diverging concave channel with h = sin(pax)/2. Both configurations will be analyzed in section 4. 3. Boundary layer formulation The simplest shortcut model that can be adopted in order to describe the spatial behavior of the local Sherwood number for a nonflat releasing wall, is a Lévêque like solution

103

10

Shðx; PeÞ ¼

Sh

Pel ðxÞ ¼

6Pe ð1  hðxÞÞ2

ð7Þ

where C(.) is the Gamma function and the local Peclet number Pel(x) takes into account that the axial velocity component vx changes along the axial coordinate and close to the releasing wall can be approximated by a linear function of the vertical wall distance y  h(x)

v x ðx; yÞ ’

6 ð1  hðxÞÞ2

ðy  hðxÞÞ þ Oððy  hÞ2 Þ;

yPh

ð8Þ

This corresponds to adopt a classical Lévêque solution for the concentration field in the boundary layer, including the effect of a local shear rate 6/(1  h)2 that changes along the axial coordinate x because the channel cross-section (1  h) changes along x. Fig. 5 shows the comparison between numerical results (already shown in Fig. 3) for Sh vs ax and the approximate model Eq. (7) for aPe = 103  104 (high effective Pe number) and a = 0.01 (long-thin channel). The approximate model effectively captures the main features of the spatial behavior (rapid decrease and subsequent increase due to the continuous narrowing of the channel cross-section) but it quantitatively fails for ax > 0.4, although the local slope of the releasing wall h0 (x) is extremely small everywhere along the channel h0 (x) = (ax) a 6 a = 0.01. The shortcut model implicitly assumes that both axial dispersion and advection contribution along the vertical direction are neglegible. This is suggested by the observation that the vertical velocity component vy, for the long-thin channel considered, is extremely small, compared to the axial velocity component vy/vx ’ h0 6 a for y ’ h, i.e. in the boundary layer. However, when the releasing wall exhibits a non-flat shape, the small but non-zero vertical velocity components participate with a convective contribution to the vertical flux that cannot be neglected in the mass balance equation because the concentration gradient close to the releasing wall has a significant projection along the vertical axis. Moreover, since the fluid flows almost tangentially to the releasing wall the local Peclet controlling the thickness of the boundary layer is Pe v  t, i.e. the projection of Pe v onto the direction t tangent to the releasing wall. Therefore, the effective local Pe number depends, at each axial coordinate x, on the cross section height 1  h(x) and on the local slope of the releasing wall h0 (x). Therefore, in order to correctly frame the boundary layer problem, we need to change our reference system (x  y) to a tangential-normal reference system (s  r) attached to the releasing bottom wall [21]. Let s be the curvilinear abscissa defined on the bottom wall h(x), i.e.

2

α

 1=3 31=3 Pel ðxÞ ; x Cð1=3Þ

sðxÞ ¼

Z

x

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 1 þ ðh ðx0 ÞÞ2 dx

ð9Þ

0

10

1

10

0

α Pe -4

10

10

-3

-2

10 αx

10

-1

0

10

Fig. 3. Sh vs ax for a dissolving bottom wall h(x) = (ax)2/2 and different values of a = 0.01,0.05,0.1. The arrow indicates increasing values of a. Different sets of curves correspond to different values of the effective Peclet number Peeff = aPe = 101  105. For Peeff P 102 different curves, corresponding to different values of a, are almost indistinguishable.

and r the wall normal distance (see Fig. 6). Let hs(s) = h(x(s)) be the shape function of the bottom wall expressed in terms of the curvi0 00 2 2 linear abscissa, and hs ðsÞ ¼ dh=dxjxðsÞ ; hs ðsÞ ¼ d h=dx jxðsÞ . Close to the releasing wall, we expect a balance of diffusion normal to the wall (along r) and streamwise convection (along s). But, since the tangential velocity vt(r,s) changes moving along the releasing wall, we also need to take into account the contribution of the velocity component vn(r,s) normal to the wall (see Fig. 6 for a schematic representation of the normal and tangential velocity components). One can observe that the structure and thickness of the boundary layer that develops on the bottom wall are controlled by:

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Fig. 4. Concentration profiles for aPe = 102 (top) and aPe = 104 (bottom) as obtained by numerical solution of the transport equations Eqs. (1)–(3) with the approximated velocity profile Eqs. (4) and (5). Left panel: converging channel h(x) = (ax)2/2. Right panel: converging–diverging channel h = sin(pax)/2.

102

y=1

4

Sh

α Pe=10

r

α Pe=103

y

hx

x=0 10

1

0

0.2

0.4

0.6

0.8

Fig. 5. Sh vs ax for a releasing bottom wall h(x) = (ax)2/2 with a = 0.01. Points: numerical results; Continuous line: Shortcut model Eq. (7). a Pe = 103  104.

(i) the tangential velocity component vt(r,s) that, close to the releasing wall, scales linearly with the wall normal distance r, i.e.

ð1  hs Þ2

0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 02 00 0 1 þ h h h h s B s C s s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ ¼ 6@ 02 ð1  hs Þ3 ð1  hs Þ2 1 þ hs

v n ðr; sÞ ¼ v

3

þ Oðr Þ;

ð11Þ

ð10Þ

(ii) the normal velocity component vn(r,s) that, close to the releasing wall, scales quadratically with the wall normal distance r, i.e. 0 2 n ðsÞr

x=1

02 v 0n ðsÞ ¼ v 01 n ðsÞ þ v n ðsÞ

02

ð1 þ hs Þ

_ x

t

Fig. 6. Schematic representation of the tangential-normal reference system and velocity components.

1

αx

v t ðr; sÞ ¼ v 0t ðsÞr þ Oðr2 Þ; v 0t ðsÞ ¼ 6

s

n vt vn v

For a complete derivation of v 0t and v 0n see Appendix A. The boundary-layer equation in the tangential-normal reference system (s  r) can be approximated by

Pev 0t ðsÞr

@c @c @ 2 c  Pev 0n ðsÞr 2 þ 2 ¼ 0 @s @r @r

ð12Þ

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where v 0t ðsÞ; v 0n ðsÞ are given by Eqs. (10) and (11), and the diffusion term along the tangential direction has been neglected being, for high Pe values, small compared with the tangential convective term. By dividing all the terms in Eq. (12) by Pev 0t ðsÞ we obtain

r

@c @c 1 @2c ¼0  RðsÞr 2 þ 0 @s @r Pev t ðsÞ @r 2 0

RðsÞ ¼

v v

ð13Þ

0 hs

0 n ðsÞ 0 t ðsÞ

00 hs

1

B C 0 þ hs  ¼ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3=2 A 02 02 1 þ hs 1 þ hs ð1  hs Þ 0 1

Rs 0 exp  0 Rðs0 Þds R h i 1 : R s0 s 00 0 3 3 exp 3 0 Rðs00 Þds v 01ðs0 Þ ds 0 1

gðs; PeÞ ¼

Pe3 1 3

t

The invariant solution f(g), satisfying Eqs. (18) and (19) attains the form

  Z 1 g f ðgÞ ¼ cw 1  exp½g03 =3dg0 ; A 0

ð14Þ

where R(s) is the ratio between the normal and tangent velocity components, which explicitly depends on the local curvature j(s) of the releasing boundary, and where Pev 0t represents the local tangential Peclet number. In the case of h(x) = hs(s) = C = constant (flow between parallel plates) it follows that s ¼ x; RðsÞ ¼ 0; v 0t ¼ 6=ð1  CÞ2 and the classical Lévêque boundary layer problem can be recovered. 0 It can be observed that for convex-converging channels (hs > 0 00 02 and hs > 0) both contributions v 01 and v to the normal velocity n n component are negative, thus increasing the concentration gradient at the releasing wall and enhancing the mass transfer rate. 0 00 For concave converging channels (hs > 0 and hs < 0) there is a 01 competition between the negative contribution v n and the positive one v 02 n related to wall curvature. This observation may be important in order to define new criteria for optimizing wall shape that maximizes heat or mass transfer. 4. Similarity solution of the boundary layer problem Eqs. (13) and (14) for the steady-state concentration profile c(s,r,Pe) can be solved analytically with the standard boundary layer boundary conditions (for a pure inlet solvent cinlet = 0)

cð0; r; PeÞ ¼ 0;

cðs; 1; PeÞ ¼ cw ;

cðs; 0; PeÞ ¼ 0:

ð15Þ

We introduce a new dimensionless wall distance (similarity variable [22])

g ¼ rgðs; PeÞ

ð16Þ

where g(s,Pe) is a function to be determined in such a way that the concentration profile c(s,r,Pe) can be rewritten in the invariant form

cðs; r; PeÞ ¼ f ðgÞ

ð17Þ

The function g(s,Pe) should satisfy the singularity condition 1/ g(0,Pe) = 0, so that the boundary conditions for c(r,s,Pe) can be recast in the following form for the similarity concentration function

f ð0Þ ¼ cw ;

f ð1Þ ¼ 0

ð18Þ

In terms of f(g), the boundary layer equation Eq. (13) attains the form

  g 0 ðs; PeÞ þ RðsÞgðs; PeÞ f 00 ðgÞ  g2 f 0 ðgÞ Pev 0t ðsÞ ¼ 0: g 4 ðs; PeÞ

ð19Þ

where g0 (s,Pe) = @g(s,Pe)/@s. In order to have an invariant solution f(g), g(s,Pe) should satisfy the following ODE 0

g ðs; PeÞ ¼  RðsÞgðs; PeÞ þ

g 4 ðs; PeÞ Pev 0t ðsÞ

! ;

1=gð0; PeÞ ¼ 0

that can be solved in closed form, thus obtaining

A¼

Cð1=3Þ 32=3

ð22Þ

and the pointwise Sherwood number, evaluated along the releasing boundary

0

h B 0C þ jðsÞhs A ¼ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis 02 1 þ hs ð1  hs Þ

ð21Þ

ð20Þ

Shðs; PeÞ ¼ 

  1 @c gðs; PeÞ @f  gðs; PeÞ : ¼  ¼ cw @r r¼0 cw @ gr¼0 A

ð23Þ

The shape of the releasing boundary hs controls the spatial behavior of the Sherwood number Sh(s,Pe) but the scaling behavior of Sh as a function of Pe still corresponds to the classic Lévêque solution, i.e. Sh(s,Pe)  Pe1/3 for high Peclet values. By substituting Eqs. (10) and (14) for v 0t ðsÞ and R(s) into Eq. (21) one obtains the following general expression for g(s,Pe)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 113 Z s 02 02 1 þ hs 1 þ hs 0 @ gðs; PeÞ ¼ ð2PeÞ ds A : 1  hs ð1  hs Þ 0 1 3

ð24Þ

Accordingly the function g(x,Pe), describing the spatial evolution of the Sherwood number as a function of the axial coordinate x, can be expressed as

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !13 Z x 02 02 1þh ð1 þ h Þ 0 gðx; PeÞ ¼ ð2PeÞ dx ð1  hÞ ð1  hÞ 0 1 3

ð25Þ

It can be observed that, although the normal to tangent velocity factor R(s) depends explicitly on the local curvature of the releasing boundary j(s), the Sherwood profile depends exclusively on the 0 shape hs and slope hs of the releasing boundary, and not on the local curvature. Fig. 7 shows the spatial behavior Sh vs ax for h(x) = (ax)2/2 at different values of Peeff as obtained by numerical solution of Eqs. (1) and (5) and the comparison with the analytical solution Eqs. (23) and (25) reading as 1

gðx;PeÞ ¼ ð2aPeÞ3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !!13 pffiffiffi pffiffiffi 1 þ a2 ðaxÞ2 2 þ ax 2 2 2a ðaxÞ þ 2ð1 þ 2a Þlog pffiffiffi  2  ax ð1  ðaxÞ2 =2Þ ð26Þ Numerical and analytical results shown in Fig. 7 (top) refer to a convex slowly varying channel cross-section, because a = 0.01 and h0 (x) = (ax)a 6 a  1. But the analytical solution Eq. (25) is valid for a generic releasing boundary and no upper bounds for h0 (x) are required. Fig. 7 (bottom) shows the agreement between numerical and analytical results for Sh vs ax for a = 0.01 (slowly varying boundary, long-thin channel) and for a = 1 (thick channel, rapidly varying boundary, h0 (x) = x 2 [0, 1]. The agreement, for high values of aPe and a = 1, is fairly good in the region where h0 (x) approaches unity. Moreover, a simple analysis of the analytic expression Eq. (26) clearly shows that, for a 6 0.1 the curves Sh vs ax are exclusively 0 a function of aPe because 1 þ h ðxÞ2 ’ 1 and this is in agreement with numerical results shown in Fig. 3. For a > 0.1, the term 0 1 þ h ðxÞ2 cannot be approximated by unity and the spatial behavior Sh vs ax depends explicitly on aPe and a, as confirmed by numerical results shown in Fig. 7 (bottom), showing that the two curves for a = 0.01 and a = 1 and the same value of aPe differ significantly.

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10

3

10

2

α=0.01 2

10

1

10

0

Sh

Sh

10

4

α Pe=10

3

α Pe=10

α Pe

10-3

10-2

10-1

10

100

1

0

0.2

0.4

0.6

αx

0.8

1

αx

100

Fig. 8. Log-normal plot of Sh vs ax for h(x) = (a x)2/2 with a = 0.01 for two different values of a Pe = 103  104. Open points: numerical solution of Eqs. (1)–(5); Continuous line: analytical solution Eqs. (23)–(26); Dashed line: shortcut model Eq. (7); Triangles: approximate solution neglecting the normal velocity component vn (dashed lines), Eq. (27).

4

αPe=10 80

Sh

60 The analytic expression for the local Sh number, Eqs. (23) and (25) can be reformulated in terms of a local effective Peclet number Pel(x) as follows

α=1

40

α=0.01

20

Shðx; PeÞ ¼

0 0

0.2

0.4

0.6

0.8

1

αx Fig. 7. Spatial behavior Sh vs ax for h(x) = (ax)2/2 at different values of aPe 2 (101,105) as obtained by numerical solution of Eqs. (1)–(5), compared to the analytical solution Eqs. (23)–(26). Open points: numerical solution; Continuous lines: analytical solution. Top figure: a = 0.01, aPe 2 (101,105). Bottom figure: a = 0.01 and a = 1, aPe = 104.

The excellent agreement between numerical and analytical solution for Peeff P 102 confirms that -(i) the tangential diffusion term @ 2c/@s2 can be neglected starting from aPe ’ 102 and -(ii) the importance of taking into account the contribution of the normal velocity component vn. Fig. 8 shows the behavior of Sh vs ax as obtained by neglecting the normal convective contribution in the boundary layer equation Eq. (12) i.e. by enforcing R(s) = 0 in Eq. (21) thus approximating g(s,Pe), or equivalently g(x,Pe), as follows 1

gðs; PeÞ ’

Pe3

Z

s

1

33

0

0 B gðx; PeÞ ¼ ð2PeÞ @ 1 3

1 0 ds v 0t ðs0 Þ Z 0

13 113

x

ð1  hÞ2 0 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx A 02 1þh

ð27Þ

It can be observed that the approximate model Eq. (27) neglecting the normal velocity component (triangles in Fig. 8) fails for ax P 0.3, where also the shortcut model Eq. (7) quantitatively fails (dashed line in Fig. 8). This is because, although the normal velocity component vn (as well as the vertical velocity component vy) is extremely small, compared with the tangential velocity component, the convective term along the normal direction (as well as the convective term along the vertical direction) gives a significant contribution to mass flow rate at the releasing wall.

Pel ðxÞ ’

 1=3 31=3 Pel ðxÞ x Cð1=3Þ 6Pe

ð28Þ

ð1  hÞ3 < ð1  hÞ1 ix Z 0 1 x dx < ð1  hÞ1 >x ¼ x 0 1  hðx0 Þ

where Pel is reported in a simplified form, by assuming h0 2  1 in the entire domain 0 6 x 6 1/a. In our example h0 2 = a4x2 6 a2  1 by the assumption of long-thin channel.1 By comparing the analytic expression of the local Peclet number Eq. (28) with that of the shortcut model Eq. (7) we observe that they actually differ by a factor

31=3 6Pef ðxÞÞ Shðx; PeÞ ¼ Cð1=3Þ xð1  hÞ2 ¼

!1=3 ;

1

f ðxÞ ð29Þ

ð1  hÞ < ð1  hÞ1 >x

that, at each point x, takes into account the entire shape of the releasing boundary, from the inlet x = 0 to the current axial position x by means of the integral term <(1  h)1 > x. Fig. 9 shows the spatial behavior of the factor f(x) for a converging channel h(x) = (ax)2/2 (curve a) and for a converging- diverging channel h = sin(pax)/2 (curve b). In both cases, the channel height is reduced by a factor 1/2 and the local slope of the releasing boundary h0 is such that h0 ’ a  1. We observe that, for the converging channel, the factor f(x) is always greater than unity and the shortcut model underestimates the local Peclet number in the entire domain (see also Fig. 8) as expected, since the normal velocity component is negative in the whole range 0 6 ax 6 1. 1 The complete expression of Pel without the simplifying assumption of slowly varying channel section h0 2  1 is given by

02

Pel ðxÞ ’

6Peð1 þ h Þ3=2 3

02

ð1  hÞ < ðð1 þ h Þð1  hÞ1 >x

:

630

A. Adrover, A. Pedacchia / International Journal of Heat and Mass Transfer 60 (2013) 624–631

and the tangential and normal velocity components associated with the releasing bottom wall are given by

1.6 1.4

b

v t ðx; rÞ ¼ 6 1.2

T

B0

2

Þ

B 2

ðh  h Þ 0

v n ðx; sÞ ¼ 6B @

a f (x)

ð1 þ h

B0

r þ Oðr 2 Þ T

ðh  h 0Þ 1 þ h T

B 3

ðh  h Þ

1

ð32Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B0

2

1

B 00 B 0

þ T

h h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC Ar2 þ Oðr 3 Þ B

ðh  h Þ2

1þh

B0

ð33Þ

2

0.8

By substituting Eqs. (32) and (33) into Eq. (21) for the similarity function one obtains

0.6

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 113 2 2 Z B0 B0 1 þ h @ x ð1 þ h Þ 0 A dx gðx; PeÞ ¼ ð2PeÞ T B T B 0 ðh  h Þ ðh  h Þ 1 3

0

0.2

0.4

0.6

0.8

1

αx 2

Fig. 9. Factor f(x) vs ax for a converging channel h(x) = (ax) /2 (curve a) and for a converging–diverging channel h = sin(pax)/2 (curve b).

70 60

Sh

50 40

By comparing Eq. (34) with Eq. (25) it can be observed that Eq. (25) represent a particular case of Eq. (34) for a flat top wall hT = 1. This type of channel configuration highlights the role of the normal convective term in a straightforward way. Let us consider the simple case of a flat bottom releasing wall (hB = 0, hB0 = 0) and a non-flat top impermeable wall, e.g. hT = 1  kax with k = 0.5 determining a linearly converging channel. In this case the tangential-normal reference system s-r coincides with the x-y reference system. The normal (vertical) velocity component close the releasing bottom wall is extremely small compared to the tangential (axial) velocity. Specifically the absolute value of the ratio vy/vx at each point x can be bounded above by

  0 v y  ðy  hB ÞhT ka k  ¼ ay 6 kay 6 y6 v  T B 1k 1  kax x ðh  h Þ

30 20 0

0.2

0.4

0.6

0.8

1

αx Fig. 10. Sh vs ax for the converging–diverging channel h = sin(pax)/2 with a = 0.01 for a Pe = 104. Open points: numerical solution of transport equations Eqs. (1)–(5); Dashed line: Shortcut model Eq. (7); analytical solution Eq. (29).

so that, for high Pe numbers and long-thin channels, when the boundary layer concentrates very close to y = hB = 0, the normal velocity component would be, in principle, neglegible. However, the normal convective term vy@c/@y cannot be neglected in the solution of the boundary-layer equation. The local mass transfer rate accounting for the normal convective term, attains the form

For the converging–diverging channel the shortcut model underestimates the local Peclet number f(x) > 1 for ax 6 0.7 and overestimate it f(x) < 1 for ax P 0.7 as expected because the normal velocity component is negative for ax < 0.5 and positive ax > 0.5. See Fig. 10 for a comparison between Sh vs ax as obtained by numerical solution (open points), the shortcut model (dashed line) and the analytical model Eq. (29) accounting for normal convective term.

By following similar arguments, the analytical expression for g(x,Pe) can be easily generalized to the case in which both the top and the bottom wall exhibit a non flat shape. Let hB and hT be the expressions for the bottom (releasing) wall and top (impermeable) wall, respectively. By lubrication theory and continuity equation the dimensionless velocity components vx and vy attain the form B

T

B

ðh  h Þ3 B

v y ðx; yÞ ¼

Z

T

6ðy  h Þðh  yÞ

;

hT

v x ðx; yÞdy ¼ 1

hB

T

B0

T

B

ð30Þ T0

6ðy  h Þðh  yÞððh  yÞh þ ðy  h Þh Þ T

B

ðh  h Þ4

40 35 30 Sh

5. Arbitrary shape of the top wall

v x ðx; yÞ ¼

ð34Þ

:

ð31Þ

25 20 a 15 b 10 0

0.2

0.4

0.6

0.8

1

αx Fig. 11. Sh vs ax for the converging long-thin channel (a = 0.01, aPe = 103) with a 0 flat bottom releasing wall (hB = 0, hB = 0) and a non-flat top impermeable wall T (h = 1  kax with k = 0.5). Open points: numerical solution of transport equations Eqs. (1)–(3) including (curve a) and neglecting (curve b) the normal (vertical) convective term. Continuous lines: analytical solution of the boundary layer equation. Curve a): Eq. (35); Curve b): Eq. (36);

A. Adrover, A. Pedacchia / International Journal of Heat and Mass Transfer 60 (2013) 624–631

Shðx; PeÞ ¼

1 2aPe A ax½hT 3 < ðhT Þ1 >

!1=3

By Taylor series expansion of

631

vt and vn about r = 0 we get

while, neglecting the normal convective term

0 2 ð1 þ h ðxÞ Þ r þ Oðr 2 Þ ðA:4Þ 2  ð1  hðxÞÞ 0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 00 0 h ðxÞ 1 þ h ðxÞ2 h ðxÞh ðxÞ C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAr 2 þ Oðr 3 Þ ðA:5Þ v n ðx;rÞ ¼ 6B þ @ 3 0 2 2 ð1  hðxÞÞ ð1  hðxÞÞ 1 þ h ðxÞ

1 2aPe Shðx; PeÞ ¼ A ax < ðhT Þ2 >

corresponding to Eqs. (10) and (11) written in terms of the curvilinear abscissa s, hs(s) and its derivatives.

¼

1 2kaPe A ð1  kaxÞ3 logð1  kaxÞ1

v t ðx;rÞ ¼ 6

!1=3 ð35Þ

!1=3

1 2aPe ¼ A ax  kðaxÞ2 þ k2 ðaxÞ3 =3

!1=3 ð36Þ

Fig. 11 shows the behavior of Sh vs ax for a = 0.01 and aPe = 103 as obtained by numerical solution of the transport equations Eqs. (1)– (3) by including (curve a, points) and by neglecting (curve b, points) the normal convecting term. Continuous lines (a) and (b) represent the corresponding analytical expression Eqs. (35) and (36), respectively. As expected, neglecting the normal convective term determines a significant underestimation of the local Sherwood number. 6. Conclusions We presented an invariant solution of the boundary layer equation in a channel with continuously varying cross-section and fixed wall concentration. This represents a generalization of the Graetz– Nusselt boundary layer solution valid for mass or heat transfer in straight channels or pipes. To correctly formulate the problem, the advection diffusion equation must be recast in a tangential-normal system of coordinates relative to the releasing boundary, and the normal convective term must be necessarily accounted for. The shape of the releasing boundary h and its local slope h0 control the spatial behavior of the Sherwood number Sh(x,Pe), yet the scaling law of Sh as a function of Pe still corresponds to the classic Lévêque solution, i.e. Sh(x,Pe)  Pe1/3 for high Peclet values and long-thin channels. The analytical expression for the local Sh as a function of the axial coordinate x has been validated versus numerical results in several geometries and for a wide range of Pe values. A new expression for the local Pe has been presented, accounting for the shape of the releasing boundary. The results presented may be helpful in the design and optimization of microdevices for heat/mass transport [23–25]. Moreover, they can bring a deeper insight into transport problems such as transport in stenotic blood vessels [25] and drug release from swelling/eroding thin polymeric films. Appendix A Let ð x; hð xÞÞ be a point of the releasing wall, corresponding to a curvilinear abscissa s ¼ sð xÞ. Let nð xÞ and tð xÞ be the normal and tangent vectors, respectively, to the releasing wall at the point ð x; hð xÞÞ. Let zð x; rÞ ¼ ð x þ rnx ð xÞ; hð xÞ þ rny ð xÞÞ be the set of points exploring the direction nð xÞ normal to the bottom wall at the point ð x; hð xÞÞ ¼ zð x; 0Þ. The tangential vt and normal vn velocity components close to zð x; 0Þ are defined as

v t ðx; rÞ ¼ v x ðzðx; rÞtx ðxÞ þ v y ðzðx; rÞty ðxÞ v n ðx; rÞ ¼ v x ðzðx; rÞnx ðxÞ þ v y ðzðx; rÞny ðxÞ 0

0

1

1 h B C t ¼ ðtx ; t y Þ ¼ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 02 02 1þh 1þh

ðA:1Þ ðA:2Þ ðA:3Þ

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