Abstract In this study the entropy generation in mi-
crochannels in microdevices induced by the transient
laminar forced convection in the combined entrance
region between two parallel plates has been investi-
gated numerically. The study considers the microscales
in the region of Kn < 0.001. The effects of aspect ra-
tio, Reynolds number, Prandtl number, Brinkman
number, and the motion of the lower plate on the en-
tropy generation during the simultaneously developing
flow in a parallel-plates channel are investigated. The
obtained results addressing all cases are thoroughly in
good agreement with the expectations that the entropy
generation has its highest value at channel with the
smallest aspect ratio at counter motion of the lower
plate with the highest Re, Pr and Br/W values consid-
ered in the problem.
List of symbols
AR aspect ratio (D/L)
Be Bejan number
Br Brinkman number
D hydraulic diameter, m
h film coefficient, W/m2 K
H plate-to-plate spacing, m
k thermal conductivity, W/m K
Kn Knudsen number, k/D
L length, m
Ma Mach number
Ns dimensionless entropy generation number
Nu Nusselt number
NuL average Nusselt number
Nux local Nusselt number
P dimensionless pressure
P* pressure, N/m2
Pr Prandtl number
Re Reynolds number_S000gen entropy generation, W/m3 K
DT temperature difference, K
T dimensionless temperature
Tin inlet temperature, K
Twall wall temperature, K
T* temperature, K
t* time, s
u dimensionless horizontal velocity component
u* horizontal velocity component, m/s
u0 inlet velocity, m/s
v dimensionless vertical velocity component
v* vertical velocity component, m/s
x*, y* coordinates, m
x, y dimensionless coordinates
Greek lettersa thermal diffusivity, m2/s
b coefficient of thermal expansion, 1/K
c specific heat ratio (cp/cv)
k mean free path, m
q dimensionless density
L. B. Erbay (&)Mechanical Engineering Department,Eskisehir Osmangazi University, 26480 Eskisehir, Turkeye-mail: [email protected]
M. M. YalcınTUSAS Engine Industries, Inc., 26003 Eskisehir, Turkey
M. S . ErcanFord - Otosan _Inonu Plant, 26140 Eskisehir, Turkey
Heat Mass Transfer (2007) 43:729–739
DOI 10.1007/s00231-006-0164-0
123
ORIGINAL
Entropy generation in parallel plate microchannels
L. Berrin Erbay Æ M. Murat Yalcın Æ Mehmet S. Ercan
Received: 2 December 2004 / Accepted: 13 June 2006 / Published online: 28 July 2006 Springer-Verlag 2006
q0 reference density, kg/m3
q* density, kg/m3
m kinematic viscosity, m2/s
s dimensionless time
F viscous dissipation function, s–2
l dynamic viscosity, N s/m2
/ irreversibility distribution function
X dimensionless temperature difference
1 Introduction
During the last decade, the interest in the entropy
generation minimization technique has experienced a
huge growth for the thermal analysis of the flow sys-
tems in engineering devices. On the other hand an
important growth has also been realized in the field of
manufacturing and utilizing of the microelectrome-
chanical systems (MEMS) in recent years. Unavoid-
ably, these two worldwide research fields will join with
the aim of designing improved microdevices with high
performance in energy utilization. That being the case,
the introduction has been organized such that the
reader can find the importance of the second law
analysis, a short discussion on the physics of the
microscale thermohydraulics, an explanation for the
parallel plates macrochannels and studies on micro-
scales, and a short regard to simultaneously developing
flow before introducing the purpose and constrains of
the present study.
Efficient energy utilization during the convection in
any fluid flow is one of the fundamental problems of
the engineering processes in practice. The quantities
that describe the energy utilization performance of the
convective flow are the heat transfer rate and the
irreversibility. The heat transfer rate can be predicted
by the analysis of the first law of the thermodynamics
and represented by the heat transfer coefficient. The
destruction of the available work of the system due to
irreversibilities is the measure of the entropy genera-
tion; therefore, the second law analysis is applied to
investigate the entropy generation rate in terms of the
entropy generation number [1–7]. Both the heat
transfer coefficient defined by Nusselt number in
dimensionless form and the entropy generation rate
given by the dimensionless entropy generation number
are important design parameters which establish the
frame of theoretical performance by efficient energy
utilization for obtaining optimum designs.
Some of the MEMS, such as microducts, micronoz-
zles, micropumps, microturbines and microvalves in-
volve fluid flow. The microchannels of microdevices
are defined as flow passages that have hydraulic
diameters in the range of 10–200 lm [8–10]. The
modeling of flow field in microchannels should be
predicted with a special attention to the effects of
friction, roughness, rarefaction, compressibility, tran-
sition from laminar to turbulent, channel size, channel
geometry, and fluid type.
In the open literature there is no exact consensus on
the flow characteristics of the microchannels and
unfortunately there are some conflicting results ob-
tained from the experiments. Numerical solutions
using dimensionless quantities in the conventional
continuum approach should be carefully evaluated
since the results might be presented as though they
involve microchannels in MEMS. Kandlikar and
Grande [9], Mehendale et al. [11], Gad-el-Hak [12],
Morini [13], and Barber and Emerson [14] argued the
case extensively and give significant importance to the
physics of the microchannel thermohydraulics.
The model used without physical insight fails to
describe the problem and gives inaccurate and con-
tradictory results. The researcher should be aware of
the range of the problem and the validity of the
model while using non-dimensional forms. In the
present study, the effects of aspect ratio on the en-
tropy generation are investigated during the fluid
flow within parallel-plates channel. The dimensionless
forms are used and the aspect ratio gets smaller, i.e.
the channel becomes narrower. If dimensions are
evaluated at microscales, the analysis of the entropy
generation at the narrow macrochannels becomes
microchannel analysis. That is why the authors would
like to make some explanations with a short intro-
duction to the microdevices in order to attract
attention of the researchers using the dimensionless
Navier–Stokes equations in the field of entropy gen-
eration
The parallel plate geometry is a very simple
geometry but represents a limiting geometry for the
family of rectangular ducts and also for concentric
annular ducts. The engineering function of the par-
allel plates channel is the heat transfer between
plates and the flowing fluid. The wide applications
including the thermodynamic analysis of flows in the
stationary parallel plate macrochannels can be found
in the literature [15–20]. Narusawa [21] examined the
rate of entropy generation both theoretically and
numerically for forced and mixed convection in
rectangular duct heated at the bottom. The station-
ary parallel channel geometry is studied by Nikolay
and Martin [22] who derived a correlation for the
overall Nusselt number for hydro dynamically
730 Heat Mass Transfer (2007) 43:729–739
123
developed laminar flow between stationary parallel
plates from the series solution of the temperature
field.
Experimental and theoretical studies on micro-
channels have been encountered in the literature after
MEMS gained increasing importance in the last dec-
ade. Tunc and Bayazitoglu [23] have investigated
convective heat transfer in a rectangular microchannel
applying the integral transform technique. Adams
et al. [24], Wang and Peng [25] and Peng et al. [26]
have conducted an experimental investigation for a
series of rectangular microchannels to analyze the
influence of liquid velocity, subcooling, property
variations and microchannel geometric configuration
on the thermohydraulic characteristics. Their results
show that all the parameters have significant influence
on the heat transfer performance, cooling character-
istics and liquid flow mode transition. Warrier et al.
[27] have researched the heat transfer and pressure
drop characteristics associated with multiple small
rectangular channels. Gao et al. [28] have measured
the overall friction coefficient and of local Nusselt
numbers and show that the classical laws of hydro-
dynamics and heat transfer are verified for the chan-
nel height e > 0.4 mm by considering the size effects.
The effect of viscous heating in microchannels has
been investigated by Tunc and Bayazitoglu [29]. Koo
and Kleinstreuer [30] have studied the effects of vis-
cous dissipation on the temperature field and the
friction factor in microtubes and microchannels and
concluded that viscous dissipation is a strong function
of the channel aspect ratio, Reynolds number, Eckert
number, Prandtl number and conduit hydraulic
diameter. Agostini et al. [31] have presented the
experimental results for the friction factor and heat
transfer coefficient and put emphasis on metrology
problems.
A velocity boundary layer and temperature profile
develop simultaneously along the inside duct surfaces
when a viscous fluid flows in a parallel-plate macro-
channel having different thermal conditions. The
fundamentals and typical solutions are comprehen-
sively reviewed by Kays and Crawford [32], Kakac
and Yener [33, 34] and Bejan [35]. Typically, simul-
taneously developing flow is observed at the inlet of
the heat exchangers, which results in higher heat
transfer coefficients at the entrance region. Studies on
the first and the second law characteristics of the
simultaneously developing fluid flow and heat trans-
fer in the channel between parallel plates moving
with different velocities are very few in the literature.
The latest study for the effect of the motion of the
bottom plate on the heat transfer has been investi-
gated by Erbay et al. [36]. The time vise variations
and the steady state of the velocity contours, iso-
therms, the effect of Br/W and Re number have been
studied by Erbay et al. [36] for three different cases
of the bottom plate: stationary, moving in parallel
and inverse directions to the flow and for the aspect
ratio D/L = 0.1 by considering the simultaneously
developing flow.
In the literature known to the authors, there has
been no new solution for the effect of aspect ratio both
on the Nusselt number and the entropy generation for
the simultaneously developing transient laminar con-
vection between two parallel plates with moving lower
at both of the macro- and microscales.
In the present study, the attention is drawn to the
effect of the aspect ratio to the characteristics of fluid
flow and heat transfer in the moving parallel-plates
channel under the constant wall temperature bound-
ary conditions to determine the entropy generation
for the contribution of upgrading the system thermal
performance. During the solution steps, the range of
the problem is argued such that a failure at the
validity of the continuum approach is prevented for
obtaining the entropy generation within parallel plate
channels at microscales. The second law analysis is
applied to the simultaneously developing transient
laminar flow between two parallel plates. The irrev-
ersibilities within the channel are presented by the
entropy generation number. The transient solutions of
the set of the governing equations for mass, momen-
tum, energy, and entropy generation for two-dimen-
sional Cartesian coordinates are obtained numerically.
Since the aspect ratio (AR = D/L) is handled as the
chief geometric parameter affecting the thermody-
namic irreversibilities, five different values of aspect
ratio are considered as 0.5, 0.4, 0.3, 0.2, and 0.1. The
study considers the Prandtl numbers of 0.1, 1.0 and 10
and the Reynolds numbers 100, 250 and 500. The
effect of the Brinkman number is also investigated by
using the values 0.1, 1.0, and 10 for Br/W. The entropy
generation is investigated by using three cases at the
bottom plate: stationary plate, moving plate in the
parallel and reverse directions with the flow to com-
prise systems as actuators.
2 Mathematical formulation
The physical system under consideration is shown
schematically in Fig. 1.
The continuity and Navier–Stokes equations in
two-dimensional unsteady and simultaneously devel-
oping laminar flow of viscous fluid with constant
Heat Mass Transfer (2007) 43:729–739 731
123
thermophysical properties between two parallel plates
are used to describe the problem in the absence of
body forces. Non-dimensional governing equations
are obtained by using conventional Navier–Stokes
equations as
@q@sþrq~V ¼ 0 ð1Þ
qD~V
Ds¼ rPþ 1
Rer2~V ð2Þ
DT
Ds¼ 1
RePrr2T ð3Þ
where
Re ¼u ¼
ð4Þ
where the hydraulic diameter D becomes twice the
plate spacing. The entropy generation number, Eq. 5,
is derived using the Navier–Stokes equations and the
Second law of Thermodynamics for incompressible
flows. The dimensionless form of the local entropy
generation equation is obtained by using the same
dimensionless parameters given in Eq. 4
Ns ¼ r2T þ / 2@u
@x
2
þ @v
@y
2" #(
þ @u
@yþ @v
@x
2)
ð5Þ
where
Ns ¼X ¼
ð6Þ
Br is the Brinkman number which determines the rel-
ative importance between dissipation effects and fluid
conduction and cannot be neglected in real flow situ-
ations. The term Br/W is known as the irreversibility
distribution ratio, /. In the literature, Bejan number
can be seen instead of /, which is defined as Be = 1/
(1+/).
The initial, inlet, outlet and the boundary conditions
are given by
u¼ 0; v¼ 0; T¼ 0; ats¼ 0 ð7Þ
u¼ 1; v¼ 0;Pnotknown;T¼ 0; atinlet ð8Þ
@u
@x¼ 0;
@v
@x¼ 0; Pnot known T not known at outlet
ð9Þ
u¼ 0; 1; 1;v¼ 0;P not known;T¼ 1; at top plate
ð10Þ
u ¼ 0; v ¼ 0; Pnotknown;T¼1 at top plate ð11Þ
The Eqs. 1, 2, 3 and 4 with the boundary conditions
given in Eqs. 7, 8, 9, 10 and 11 are valid with the
continuum approach and only if the Knudsen number
is smaller than 0.1.
In the microchannel analysis, the Kn number is the
main characteristic number, which is the ratio of the
mean free path of the molecules of the fluid to the
characteristic macroscopic length of the channel
geometry, but Kn does not appear in the Eqs. 1, 2, 3, 4
and 5, therefore the limit value of the Kn number
cannot be determined from the Navier–Stokes equa-
tions itself. From the molecular analysis, it is found that
the critical limit for no-slip condition is given by
Kn < 0.001 [12, 37–39]. For gases, Kn is calculated
easily by molecular dynamics. For example, air at
standard conditions has the mean free path
k = 0.068 lm, the present solution becomes valid for
channels with the plate-to-plate spacing are greater
than 34 lm. Inevitably, the hydraulic diameter D
(=2H) changes depending on the mean free path of the
gas under consideration. The widths of the flow pas-
sages are often much larger, that is why the channel
can also be assumed as rectangular passage.
A straightforward derivation similar to gases by
molecular dynamics is not available for the liquids.
Then experiments carry more importance for liquids.
The homogeneous velocity profile at the entrance is
expected to become Hagen–Poiseuille parabolic dis-
tribution as the fluid progress through the channel
under the influence of viscosity. The results of the
experiments performed by Celata et al. [40] indicate
that in the laminar flow regime the friction factor is in
good agreement with the Hagen–Poiseuille theory forFig. 1 The schematic of the physical system
732 Heat Mass Transfer (2007) 43:729–739
123
Reynolds number less than 600–800 for R114 and wa-
ter flowing in tubes 130 and 290 lm, respectively, in
diameter. The Reynolds number is restricted by 500 in
the present investigation. Additionally, the limit to the
plate-to-plate spacing is 1 lm for ordinary liquids such
as water can be acceptable for dimensional comments
of the solutions.
Let us consider the compressibility. In general, for
gas flow in microchannels the effects of rarefaction and
compressibility are coupled and tend to conflict with
each other [13]. For gas flows, by using the definitions
of the Reynolds number, the Knudsen number (Kn)
and the Mach number (Ma), the following expression is
obtained via kinetic theory [12]
Re ¼Ma
Kn
ffiffiffiffiffipc2
rð12Þ
The range depicted in the study (Kn < 0.001,
Re = 500) stays in the region of incompressible flow;
therefore, the assumption of incompressibility is valid.
As a last statement of this section it is noted that the
present results to be obtained from the solution of the
governing equations given in Eqs. 1, 2, 3 and 4 with the
properly selected combination of the parameters will
comprise the microchannels.
3 Solution and benchmarking the results
The present combined hydrodynamic and thermal
entry length problem is solved by the finite volume
method with SIMPLE algorithm. A computer pro-
gram was developed and benchmarked with the
velocity and the thermal boundary layers for the
solutions obtained by those of reported by Kakac and
Yener [34] under the case of fixed channel. A grid
sensitivity analysis was carried out. Since the results
of the present study agree with Stephan’s correlation
[34] with a maximum deviation of 3.32%, the grid
structure of 100·40 and the dimensionless time step of
0.005 are used and the rectangular side lengths are
equally divided. This choice was determined from the
values of Nu number for the model of which aspect
ratio is D/L = 0.1.
Total Nusselt number is calculated by using the
average Nusselt numbers for the bottom and the upper
plates, as follows
Nux ¼ @T
@y
y¼0
þ@T
@y
y¼H
!ð13Þ
NuL ¼ 1
L
ZL
0
@T
@y
y¼0
þ@T
@y
y¼H
!dx ð14Þ
4 Results and discussion
The effects of aspect ratio, Reynolds number, Prandtl
number, Brinkman number, and the motion of the
lower plate on the entropy generation were investi-
gated for the parallel-plates microchannel in this study.
The channel aspect ratio is introduced by changing the
channel height getting narrower corresponding aspect
ratios from AR = 0.5–0.1. The Reynolds number varies
as 100, 250, and 500, Prandtl number 0.1, 1.0, and 10,
and Br/W 0.1, 1.0, and 10. The bottom moves in a
parallel direction to flow and the top at rest. The re-
sults would like to be evaluated by considering the
microchannel thermo-hydraulics. Agostini et al. [31]
clearly stated that the results found in literature about
heat transfer in mini-channels for single phase flow are
often contradictory and operating conditions change
from one study to another so that comparisons are
difficult. Therefore, the characteristic quantities defin-
ing the microchannels are examined to avoid going far
from the domain of the continuum approach applied in
this study. The present study considers perfectly the
microscales up to the region of Kn = 0.001.
Total entropy generation values in the non-dimen-
sional sense through the channel are depicted in figures
considering the down stream direction by using the
dimensionless entropy generation number Ns. To dif-
ferentiate the sole effects of the parameters, each one
is treated at certain values of the other parameters for
all motions of the lower plate.
The present work is interested, firstly, in the values
of Nusselt number for constant wall temperature
boundary conditions. In Figs. 2 and 3, Nusselt numbers
are reported for the effect of the motion of the bottom
plate at certain aspect ratios and the effect of aspect
ratios in every motion, respectively. The forward mo-
tion enables the heat transfer to get higher rates as well
as small values of the aspect ratio. The highest values
of the Nusselt number are obtained at the inlet. A
sharp decrease is realized near the entrance and the
decrease continues gradually through the channel. The
lowest values of the Nusselt number are obtained at
the channels with wide gap at the backward motion of
the lower. Obviously, the backward motion causes
some flow instabilities and fluctuations at the very
beginning of the channel, therefore the disturbed
Heat Mass Transfer (2007) 43:729–739 733
123
Nusselt curves are observed especially at the channels
with the smallest aspect ratio.
The irreversibility during the flow is derived in terms
of the average entropy generation numbers by using
Eq. 6. The entropy generation starts on the plate sur-
faces where the friction exists and heat exchanges be-
tween the walls and fluid. The entropy contours
concerning the transient changes and the effect of as-
pect ratio are given in Figs. 4 and 5, respectively. By
addressing both of the figures, it is said that the corners
of the inlet behave as active points for entropy gener-
ation, at which the highest values are obtained. The
formation of the symmetric structure with respect to
mid-plane is only observed at the fixed plate case.
When the lower plate moves in parallel to the flow
direction, the contours and the values of the number of
the entropy generation are slightly higher at the inside
of the upper plate. Figure 6 presents the variation of
the entropy generation number through the channel
length at steady state. The maximum is obtained during
the counter motion of the lower plate and the values of
entropy are about five times greater than that of the
forward motion. At the channel with the smallest as-
pect ratio, the maximum entropy values are obtained at
about 25 times grater than that of the largest channel
since the entropy generation owes its existence to the
irreversibility due to heat transfer and fluid friction
which are higher in narrow gaps.
The influence of the aspect ratio of the channel is
studied and the results are expressed for each of the
motion of the bottom plate, separately. Figure 7 rep-
resents different aspect ratios each comprises the effect
of motion at Re = 250, Pr = 1, and Br/W = 1. The
highest values of entropy generation number are ob-
tained for the smallest aspect ratio. Since the results
are obtained at a fix Reynolds number, the flow be-
comes faster and therefore the entropy generation is
explained by increasing of friction. The increasing
trend of entropy generation does not change at dif-
ferent motions of the lower plate by getting narrow
gaps. Within the motions, the highest effect to increase
the entropy generation belongs to the backward
Nu x
N
u x
Nu x
0
20
40
60
80
100
120
0 2 4 6 8 10
BackwardFixedForward
x
5
10
15
20
25
0 2 4 6 8 10
BackwardFixedForward
x
0
10
20
30
40
0 2 4 6 8 10
BackwardFixedForward
x
(a)
(b)
(c)
Fig. 2 Variations of Nu numbers due to the effect of the motionof the lower plate: a AR = 0.1, b AR = 0.3, c AR = 0.5 at the timestep of s = 50 (Re = 250, Pr = 10)
12
Nu x
0
20
40
60
80
100
120
0 2 4 6 8 10
Ar= 0.1Ar= 0.2Ar= 0.3Ar= 0.4Ar= 0.5
x
Nu x
0
20
40
60
80
100
120
0 2 4 6 8 10
Ar =0.1Ar =0.2Ar= 0.3Ar= 0.4Ar= 0.5
x
Nu x
0
20
40
60
80
100
0
0 2 4 6 8 10
Ar=0.1Ar=0.2Ar=0.3Ar=0.4Ar=0.5
x
(a)
(b)
(c)
Fig. 3 Local Nu numbers considering all aspect ratios for abackward, b stationary and c moving in parallel direction to theflow at the time step of s = 50 (Re = 250, Pr = 1)
734 Heat Mass Transfer (2007) 43:729–739
123
Fig. 4 Transient change in the contours of entropy generation from s = 1 up to 50 for the cases of the motion of the lower plate as abackward, b stationary and c moving in parallel direction to the flow (Re = 500, Pr = 10, Br/W = 1)
Fig. 5 The contours of entropy generation from AR = 0.1 up to AR = 0.5 for the cases of the motion of the lower plate as a backward, bstationary and c moving in parallel direction to the flow (Re = 500, Pr = 10, Br/W = 1)
Heat Mass Transfer (2007) 43:729–739 735
123
motion for certain Re, Pr and Br/W. Since a boundary
layer owes its existence to the difference between the
velocity of the fluid and that of the bottom, the highest
difference between these velocities is obtained at the
backward motion. In case of the forward motion of the
lower, the formation of the boundary layer on the
bottom is eliminated by suppressing the difference
between the velocity of the fluid and that of the bottom
plate. Therefore the forward motion of the lower plate
causes the lowest entropy generation. Entropy gener-
ation occurs at the rest of the channel gap, especially
near the top wall.
The effect of Reynolds number observed on the
entropy generation number is presented in Fig. 8. The
values 100, 250, and 500 are considered for Re number.
The entropy generation has higher levels with
increasing values of Re number. When the results
found at different motions of the moving bottom are
compared, it is seen that the higher values arise at the
backward motion. When the combined effect of Re
number and motion is evaluated, it is seen that the
highest number of entropy generation is at the highest
Re number during the counter motion of the lower
plate. The reason is obviously the increase of the ir-
reversibilities due to the flow friction created by both
of them.
The effect of Prandtl number on the entropy gen-
eration is summarized in Fig. 9. At all motions of the
lower, the smallest entropy generation numbers are
obtained at the smallest Pr value considered in the
present study. Since momentum diffuses quickly rela-
tive to heat for high Pr value, the entropy generation
becomes higher. The values are far from each other
during the backward motion with respect to other
cases. Forward motion suppresses the distinct entropy
generation values.
The Brinkman number determines the relative
importance between dissipation effects and fluid con-
duction. Figure 10 summarizes the effect of Br/W for
each of the lower plate motions on the entropy gen-
eration. It is seen that an increase in the Brinkman
number yields strong increase in the entropy genera-
Ns
1
10
100
0 2 4 6 8 10
Back wardFixedForward
x
Ns
1
10
100
0 2 4 6 8 10
Back wardFixedForward
x
1
10
100
0 2 4 6 8 10
Back wardFixedForward
x
Ns
(a)
(b)
(c)
Fig. 6 The effect of motion on the entropy generation aAR = 0.2, b AR = 0.3 and c AR = 0.5 (Re = 250, Pr = 1, Br/W = 1)
Ns
1
10
100
1000
104
0 2 4 6 8 10
Ar=0.1Ar=0.2Ar=0.3Ar=0.5
x
Ns
1
10
100
1000
104
0 2 4 6 8 10
Ar=0.1Ar=0.2Ar=0.3Ar=0.5
x
Ns
1
10
100
1000
104
0 2 4 6 8 10
Ar=0.1Ar=0.2Ar=0.3Ar=0.5
x
(c)
(b)
(a)
Fig. 7 The effect of aspect ratio on entropy a backward, bstationary and c moving in parallel direction to the flow(Re = 500, Pr = 10, Br/W = 10)
736 Heat Mass Transfer (2007) 43:729–739
123
tion number. When the conduction overcomes the
influence of the friction i.e. Br/W is low, the low values
of entropy generation are realized. If a certain Br/W is
considered, the effect of the motion of the lower plate
is observed obviously. The highest values of the num-
ber of entropy generation exist in the case of the
highest Br/W and backward motion of the lower plate.
Additionally, the effect of the motion of the bottom
plate is differentiated at especially the highest Brink-
man value due to dominated conduction effect in
addition to the friction generated by the lower plate
motion in downstream direction.
5 Conclusion
In the present study, the entropy generation in mi-
crochannels induced by the transient laminar forced
convection in the combined entrance region between
two parallel plates has been investigated. In order to
prepare benchmark values, the results have been
evaluated under the umbrella of the microchannel
thermo-hydraulics, subsequently the present study
considers microscales at region of Kn < 0.001. The
effects of aspect ratio, Reynolds number, Prandtl
number, Brinkman number, and the motion of the
lower plate on the entropy generation have been
investigated.
Based on the results addressing all cases, the fol-
lowing conclusions can be drawn.
At the entrance region simultaneously developing
flow forms and entropy is generated under all para-
metric conditions. Immediately after the sharp entropy
generation at the inlet, the entropy generation con-
tinues to decrease through the downstream. No en-
tropy is generated at the centerline of the duct for all
values of group parameters where as the entropy
generation is largely higher and located on the channel
walls.
The entropy generation has its highest value at the
highest Re, Pr and Br/W values considered in the study.
Ns
0.1
1
10
100
0 2 4 6 8 10
Re=100Re=250Re=500
x
Ns
0.1
1
10
100
0 2 4 6 8 10
Re=100Re=250Re=500
x
Ns
0.1
1
10
100
0 2 4 6 8 10
Re=100Re=250Re=500
x
(a)
(b)
(c)
Fig. 8 The effect of Reynolds number on entropy a backward, bstationary and c moving in parallel direction to the flow(AR = 0.3, Pr = 0.1, Br/W = 0.1)
1
10
100
0 2 4 6 8 10
Pr=0.1Pr=1Pr=10
x
0.5
Ns
1
10
100
0 2 4 6 8 10
Pr=0.1Pr=1Pr=10
x
0.5
Ns
Ns
1
10
100
0 2 4 6 8 10
Pr=0.1Pr=1Pr=10
x
0.5
(a)
(b)
(c)
Fig. 9 The effect of Prandtl number on entropy a backward, bstationary and c moving in parallel direction to the flow(AR = 0.4, Re = 100, Br/W = 1)
Heat Mass Transfer (2007) 43:729–739 737
123
The strong effect of aspect ratio, Reynolds number,
Prandtl and Brinkman number to increase the entropy
generation has been realized especially on the inside
surface of the lower plate moving in counter direction
to the flow.
The maximum entropy generation is obtained at the
most slender channel within the channels considered.
The backward motion causes the higher entropy gen-
eration than that of the fixed and forward motions due
to increased frictional effects. From the engineering
applications viewpoint, this means that the highest
pumping power requirement happens in the case of
backward motion. The lowest entropy generation, i.e.
minimum flow resistance, is realized at the parallel
forward direction at the channel with the largest gap
between upper and lower plates.
As a conclusion, the results comprising the micro-
channels with the properly selected combination of the
parameters have provided significant data and consid-
erable insight and fulfill the lack of systematic inves-
tigation of the entropy generation in microchannels.
Acknowledgments This research is supported by the academicresearch fund, Eskisehir Osmangazi University.
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Ns
0.1
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2 4 6 8 10x
0
Ns
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0 2 4 6 8 10x
Ns
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