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LETTER TO THE EDITOR
To the Editor:
The purpose of this letter is to comment on
the boundary-layer model used by Vennela
et al.1 in their article entitled “Sherwood
Number in Flow Through Parallel Porous
Plates (Microchannel) due to Pressure and
Electroosmotic Flow.”
The transport of an electrically neutral sol-
ute that is retained by a semipermeable wall
of the slit microchannel is modeled within the
mass-transfer boundary layer as
ðA1yÞ @c
@x2
Pew
4
@c
@y5@2c
@y2;
A153
16ReSc
de
L
� �mðkh;RÞ; Pew5
vwde
D(1)
where ðA1yÞ is the dimensionless axial electro-
lyte solution velocity (assumed a linear func-
tion of wall distance y and independent of the
axial position x) and PewðxÞ is the dimension-
less permeation velocity assumed inversely
proportional to the thickness of the mass-
transfer boundary layer dðxÞ5ðx=A1Þ1=3.
The authors derive to the following expression
for the Sherwood number
ShðxÞ5 4
I
A1
x
� �1=3
; I5
ð10
exp ð2g3=92BgÞdg
(2)
where B5Pew=ð6A1=31 Þ is a constant depending
on the length averaged permeation parameter
Pew .
We would like to point out that the solvent
permeation through the porous wall may affect
significantly the axial flow rate (for a fixed
pressure gradient) for high values of the ratio
a5Pew=ðReScde=LÞ like those investigated by
the authors (e.g., Pew 51000 and
ðReScde=LÞ54000Þ so that the dependence of
the axial velocity on the axial position must be
taken into account.
Therefore, we solve the boundary layer
equation Eq. 1 by replacing ðA1yÞ with
A1ysðxÞ, s(x) being a function of the axial posi-
tion and the ratio a. We adopt the same bound-
ary conditions as Vennela et al.1 We introduce
a similarity variable g5y=dðxÞ where
dðxÞ5A21=31 =gðxÞ is the thickness of the bound-
ary layer which depends on a function g(x) to
be determined to have an invariant solution
cðx; yÞ5f ðgÞ
d2f
dg25
df
dg2
Pew
4A1=31 gðxÞ
1g2sðxÞgðxÞ4
dg
dx
!;
Pew
4A1=31 gðxÞ
f j02df
dg
����0
50; f jg!151 (3)
If we assume, according with Vennela
et al.,1 that the permeation flux through the
porous wall (Pew) is inversely proportional to
d, we have Pew
4A1=3
1gðxÞ
5B5constant . To have an
invariant solution satisfying the boundary con-
ditions, we enforce2
dg
dx52
1
3
gðxÞ4
sðxÞ ; 1=gð0Þ50
) gðxÞ5ðx
0
dx0
sðx0Þdx0� �21=3
(4)
thus, obtaining for the pointwise Sherwood
number the following expression
ShðxÞ5 4
IðA1Þ1=3gðxÞ;
I5
ð10
exp ð2g3=92BgÞdg;
B5Pew
4A1=31
�ð1
0
gðx0Þdx0�21
(5)
It should be observed that, by assuming
sðxÞ51, that is, by neglecting the variation of
the axial velocity along the axial coordinate x,
one recovers the result by Venuela et al.1
because gðxÞ5x21=3.
From a simple material balance, if we
assume a uniform permeation velocity equal to
the average permeation velocity Pew , the axial
velocity function s(x) turns out to be a linear
decreasing function x, that is, sðxÞ5112að122xÞ.
Figure 1 shows the comparison between
Eq. 2 derived from Vennela et al.1 and Eq. 5
for two different values of a. It can be
observed that Eq. 2 neglecting spatial variation
of the axial velocity (i.e., sðxÞ51) significantly
overestimates the pointwise Sh for x > 0:5.
Therefore, Eq. 5 must be adopted for a � 0:1.
Alessandra Adrover and Augusta PedacchiaLa Sapienza Universit�a di Roma
Dipartimento di Ingegneria Chimica, Materiali,Ambiente
Via Eudossiana 18, 00184 Rome, ItalyE-mail: [email protected]
Literature Cited
1. Vennela N, Mondal S, De S, BhattacharjeeS. Sherwood number in flow through paral-lel porous plates (Microchannel) due topressure and electroosmotic flow. AIChE J.2012;58:1693–1703.
2. Adrover A, Pedacchia A, Mass transferthrough laminar boundary layer in micro-channels with nonuniform cross section:the effect of wall shape and curvature, IntJ Heat Mass Transfer. 2013;60:624–631.
Figure 1. Log-normal plot of Sh(x) versus x for two different values ofa5Pew=ðReScde=LÞ. R 5 0 (purely pressure driven flow).
[Color figure can be viewed in the online issue, which is available at
wileyonlinelibrary.com.]
VC 2013 American Institute of Chemical EngineersDOI 10.1002/aic.14272Published online November 4, 2013 in WileyOnline Library (wileyonlinelibrary.com).
AIChE Journal December 2013 Vol. 59, No. 12 Published on behalf of the AIChE DOI 10.1002/aic 4887
