Entropy generation in parallel plate microchannels

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


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


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


x

0

10

20

30

40

0 2 4 6 8 10


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)


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)


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


x

1

10

100

0 2 4 6 8 10


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)


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)


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.

References

1. Bejan A (1980) Second law analysis in heat transfer. EnergyInt J 5:721–732

2. Bejan A (1996) Entropy generation minimization. CRCPress, Boca Raton

3. Bejan A (1994) Entropy generation through heat and fluidflow. Wiley, Canada

4. Krane RJ (1987) A Second law analysis of the optimumdesign and operation of thermal energy storage systems. Int JHeat Mass Transfer 30:43–57

5. Arpacı VS (1993) Radiative entropy production—lost heatinto entropy. Int J Heat Mass Transfer 36:4193–4197

6. Tsatsaronis G (1995) Design optimization of thermal sys-tems using exergy—based techniques. In: Proceedings ofSecond Law Analysis: Towards the 21st Century, Roma,183–191

7. Erbay LB, Altac Z, Sulus B (2004) Entropy generation in asquare enclosure with partial heating from a vertical lateralwall. Heat Mass Transfer 40:909–918

8. Gad-el-Hak M (2001) Flow physics in MEMS. Mec Ind2:313–341

9. Kandlikar SG, Grande W J (2003) Evaluation of micro-channel flow passages—thermohydraulic performance andfabrication technology. Heat Transfer Eng 24:3–17

10. Morini GL (2004) Single-phase convective heat transfer inmicrochannels: a review of experimental results. Int J ThermSci 43:631–651

11. Mehendale SS, Jacobi AM, Shah RK (1999) Heat exchang-ers at micro and meso-scales, in compact heat exchangersand enhancement technology for the process industries.Bergell House, NY, pp 55–74

12. Gad-el-Hak M (2003) Comments on critical view on newresults in micro-fluid mechanics. Int J Heat Mass Transfer46:3941–3945

13. Morini GL, Spiga M, Tartarini P (2004) The rarefaction ef-fect on the friction factor of gas flow in microchannels. Su-perlattices Microstruct 35(3–6):587–599

14. Barber RW, Emerson DR (2002) The influence of Knudsennumber on the hydrodynamic development length withinparallel plate micro-channels. In: Rahman M, Verhoeven R,Brebbia CA (eds) Advances in fluid mechanics, vol IV. WITPress, Southampton, pp 207–216

15. Nag PK, Kumar N (1989) Second law optimization of con-vective heat transfer through a duct with constant heat flux.Int J Energy Res 13:537–543

16. S ahin AZ (1998) Irreversibilities in various duct geometrieswith constant wall heat flux and laminar flow. Int J Energy23:465–473

17. S ahin AZ (2000) Entropy generation in turbulent liquid flowthrough a smooth duct subjected to constant wall tempera-ture. Int J Heat Mass Transfer 43:1469–1478

18. S ahin AZ (2002) Entropy generation and pumping power ina turbulent fluid flow through a smooth pipe subjected toconstant heat flux. Int J Exergy 2:314–321

19. Mahmud S, Fraser RA (2002) Thermodynamic analysis offlow and heat transfer inside channel with two parallel plates.Int J Exergy 2:140–146

20. Mahmud S, Fraser RA (2003) The second law analysis infundamental convective heat transfer problems. Int J ThermSci 42(2):177–186

Ns

0.1

1

10

100

1000

2 4 6 8 10x

0

Ns

0.1

1

10

100

1000

0 2 4 6 8 10x

Ns

0.1

1

10

100

1000

0 2 4 6 8 10

Br/Ω =0.1 Br/Ω =1Br/Ω =10

Br/Ω =0.1Br/Ω =1Br/Ω =10

Br/Ω =0.1Br/Ω =1Br/Ω =10

x

(a)

(b)

(c)

Fig. 10 The effect of Br/W number on entropy a backward, bstationary and c moving in parallel direction to the flow(AR = 0.5, Re = 250, Pr = 1)


123

21. Narusawa U (2001) The second law analysis of mixed con-vection in rectangular ducts. Heat Mass Transfer 37:197–203

22. Nikolay M, Martin H (2002) Improved approximation forthe Nusselt number for hydrodynamically developed laminarflow between parallel plates. Int J Heat Mass Transfer45:3263–3266

23. Tunc G, Bayazitoglu Y (2002) Heat transfer in rectangularchannel. Int J Heat Mass Transfer 45:765–773

24. Adams TM, Halik S, Jeter SM, Qureshis ZH (1998) Anexperimental investigation of single-phase forced convectionin microchannels. Int J Heat Mass Transfer 41:851–857

25. Wang BX, Peng XF (1994) Experimental investigation onliquid forced-convection heat transfer through microchan-nels. Int J Heat Mass Transfer 37:73–82

26. Peng XF, Wang BX, Peterson GP, Ma HB (1995) Experi-mental investigation of heat transfer in flat plates with rect-angular microchannels. Int J Heat Mass Transfer 38:127–137

27. Warrier GP, Dhir VK, Momoda LA (2002) Heat transferand pressure drop in narrow rectangular channels. ExpTherm Fluid Sci 26:53–64

28. Gao P, Person SLP, Marinet MF (2002) Scale effects onhydrodynamics and heat transfer in two-dimensional miniand microchannels. Int J Therm Sci 41:1017–1027

29. Tunc G, Bayazitoglu Y (2001) Heat transfer in microtubeswith viscous dissipation. Int J Heat Mass Transfer 44:2395–2403

30. Koo J, Kleinstreuer C (2004) Viscous dissipation effects inmicrotubes and microchannels. Int J Heat Mass Transfer47:3159–3169

31. Agostini B, Watel B, Bontemps A, Thonon B (2004) Liquidflow friction factor and heat transfer coefficient in smallchannels: an experimental investigation. Exp Therm FluidSci 28:97–103

32. Kays WM, Crawford ME (1980) Convective heat and masstransfer, 2nd edn. McGraw-Hill, New York

33. Kakac S, Yener Y (1995) Convective heat transfer, 2nd edn.CRC Press, Boca Raton

34. Kakac S, Yener Y (1983) Laminer force convection in thecombined entrance region of ducts. In: Kakac S, Bergles AE,Shah RK (eds) Low Reynolds number flow heat exchangers.Hemisphere, New York

35. Bejan A (1995) Convective heat transfer, 2nd edn. Wiley,New York

36. Erbay LB, Ercan MS , Sulus B, Yalcın MM (2003) Entropygeneration during fluid flow between two parallel plates withmoving bottom plate. Entropy Int Interdiscip J 5:506–518

37. Yu S, Amel TA (2001) Slip-flow heat transfer in rectangularmicrochannels. Int J Heat Mass Transfer 44:4225–4234

38. Arkilic EB, Schmidt MA, Breuer KS (1997) Gaseous slipflow in long microchannels. J Microelectromech Syst 6:167–178

39. Xue H, Ji HM, Shu C (2001) Analysis of micro-Coutte flowusing the Burnett equations. Int J Heat Mass Transfer44:4139–4146

40. Celata GP, Cumo M, Zummo G (2004) Thermal-hydrauliccharacteristics of single-phase flow in capillary pipes. ExpTherm Fluid Sci 28:87–95


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Entropy generation in parallel plate microchannels

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