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ORIGINAL

Entropy generation during ﬂuid ﬂow in a channel under the effect

of transverse magnetic ﬁeld

R. A. Damseh ÆM. Q. Al-Odat ÆM. A. Al-Nimr

Received: 22 December 2005 / Accepted: 28 August 2007 / Published online: 14 September 2007

Springer-Verlag 2007

Abstract Entropy generation due to ﬂuid ﬂow and heat

transfer inside a horizontal channel made of two parallel

plates under the effect of transverse magnetic ﬁeld is

numerically investigated. The ﬂow is assumed to be steady,

laminar, hydro-dynamically and thermally fully developed

of electrically conducting ﬂuid. Both horizontal walls are

maintained at constant temperatures higher than that of the

ﬂuid. The governing equations in Cartesian coordinate are

solved by an implicit ﬁnite difference technique. After the

ﬂow ﬁeld and the temperature distributions are obtained,

the entropy generation proﬁles are computed and presented

graphically. The factors, which were found to affect the

problem under consideration are the magnetic parameter,

Eckert number, Prandtl number, and the temperature

parameter (h

?

). It was found that, entropy generation

increased as all parameters involved in the present problem

increased.

List of symbols

B

o

magnetic ﬁeld ﬂux density

C

p

speciﬁc heat of at constant pressure (kJ/kg K)

Ec Eckert number

Ha Hartmann number ﬃﬃﬃﬃﬃ

M

p

ggravitational acceleration (m/s

2

)

kthermal conductivity (W/m K)

Mmagnetic parameter

Ppressure

Pe Peckelt number

Pr Prandtl number

Ttemperature (K)

T

?

free stream temperature (K)

T

w

wall temperature (K)

S

¢¢¢

dimensionless entropy generation

s

¢¢¢

entropy generation per unit volume (W/m

3

K)

S

1,2,3

dimensionless entropy generation due to ﬂuid

friction, heat transfer, magnetic effect, respectively

Wchannel width

Udimensionless volumetrically averaged axial

velocity

uvolumetrically averaged axial velocity (m/s)

vvolumetrically averaged lateral velocity (m/s)

Vdimensionless volumetrically averaged lateral

velocity

x,ycoordinates along and normal to the channel,

respectively (m)

X,Ydimensionless coordinates along and normal to the

channel, respectively

Greek symbols

aeffective thermal diffusivity (m

2

/s)

qﬂuid density (kg/m

3

)

rﬂuid electrical conductivity

hdimensionless ﬂuid temperature

h

?

free stream temperature parameter

ldynamic viscosity

mkinematic viscosity (m

2

/s)

Subscripts

wwall

?free stream condition

R. A. Damseh (&)M. Q. Al-Odat

Mechanical Engineering Department,

Al-Huson University College, Al-Balqa Applied University,

P.O.B (50), Irbid, Jordan

e-mail: Rdamseh@yahool.com

M. A. Al-Nimr

Mechanical Engineering Department,

Jordan University of Science and Technology,

Irbid, Jordan

123

Heat Mass Transfer (2008) 44:897–904

DOI 10.1007/s00231-007-0342-8

1 Introduction

Forced convection ﬂow of an electrically conducting ﬂuid

in a channel in the presence of a transverse magnetic ﬁeld

is of special technical signiﬁcance because of its frequent

occurrence in many industrial and engineering applications

such as geothermal reservoirs, cooling of nuclear reactors,

thermal insulation, and petroleum reservoirs. This type of

problem also arises in electronic packages; micro elec-

tronic devices during their operations. In the absence of

magnetic ﬁeld, references [1–3] will give some ideas about

ﬂuid ﬂow and thermal characteristics inside a vertical

channel.

The optimal design of thermal systems can be achieved

by minimizing entropy generation in these systems. This

issue has been the topic of great importance in many

engineering ﬁeld such as heat exchangers, cooling of

nuclear reactors, MHD power generators, geophysical ﬂuid

dynamics energy storage systems and cooling of electronic

devices, etc. Entropy generation is associated with ther-

modynamics irreversibility, which is common in all types

of heat transfer processes. Different sources of irrevers-

ibility are responsible for entropy generation such as heat

transfer across ﬁnite temperature gradient, characteristics

of convective heat transfer, magnetic ﬁeld effect, viscous

dissipation effect etc. Bejan [1,2] showed that the entropy

generation for forced convective heat transfer is due to

temperature gradient and viscosity effect in the ﬂuid.

Entropy generation in thermal engineering systems

destroys system available work and thus reduces its efﬁ-

ciency. Abu-Hijleh et al. [3] studied the entropy generation

due to laminar mixed convection from an isothermal

rotating cylinder. Tasnim et al. [4] presented an analytical

work to study the First and Second Laws (of thermody-

namics) characteristics of ﬂow and heat transfer inside a

vertical channel made of two parallel plates embedded in a

porous medium and under the action of transverse mag-

netic ﬁeld. Mahmud and Fraser [5] investigated

analytically the effects of radiation heat transfer on mixed

convection through a vertical channel in the presence of

transverse magnetic ﬁeld, applying both First and Second

Laws of thermodynamics to analyze the problem. Arpaci

and Selamet [6] investigated the entropy production in

boundary layers Khalkhali [7] developed a thermodynamic

model of conventional cylindrical heat pipes based on the

second law of thermodynamics. Abu-Hijleh [8] computed

entropy generation due to laminar mixed heat convection

from an isothermal heated cylinder in an air cross ﬂow for

different values of the Reynolds number, buoyancy

parameter, and cylinder diameter. Mahmud and Fraser [9]

analyzed second law characteristics of heat transfer and

ﬂuid ﬂow due to forced convection of steady-laminar ﬂow

of incompressible ﬂuid inside channel with circular cross-

section and channel made of two parallel plates. Haddad

et al. [10] focused on the local entropy generation of steady

two-dimensional symmetric ﬂow past a parabolic cylinder

in a uniform stream parallel to its axis.

Soundalgekar [11] analyzed the two-dimensional ﬂow

on an incompressible, viscous ﬂuid over an inﬁnite porous

vertical plate with uniform suction velocity normal to the

plate, the difference between the temperature of the plate

and the free stream moderately large causing the natural

convection currents. Raptis and Kafoussias [12] studied the

ﬂow and heat transfer characteristics in the presence of

porous medium and magnetic ﬁeld. Chamkha [13] studied

the problem of steady, laminar, free convection ﬂows over

vertical porous surface in the presence of magnetic ﬁeld

and heat generation or absorption. Elbashbeshy [14]

investigated heat transfer over a stretching surface with

variable and uniform heat ﬂux subjected to suction.

Recently, Odat et al. [15] discussed the entropy generation

of an electrically conducting ﬂuid past a horizontal ﬂat

plate in the presence of transverse magnetic ﬁeld.

The main objective of this paper is to study the local

entropy generation due to steady fully developed laminar

forced convection channel ﬂow in the presence of a

transverse magnetic ﬁeld. In the present work the fully

developed forced convection equations are solved using

ﬁnite difference method. The entropy generation rates due

to forced convection are computed for different values of

magnetic parameter, Eckert number and Prandtl number.

2 Problem formulation

Consider fully developed, steady, laminar forced convec-

tion ﬂow of an electrically conducting, incompressible,

Newtonian ﬂuid in a channel under the effect of a trans-

verse magnetic ﬁeld B

0

applied normal to the ﬂow

direction. A schematic diagram of the problem under

consideration is shown in Fig. 1. The ﬂuid is assumed to be

incompressible with constant properties. The magnetic

Reynolds number is assumed to be small, so that the

y

x

W

Tw

Tw

B0

Flow

Τ∞

Fig. 1 The geometry and the coordinate system

898 Heat Mass Transfer (2008) 44:897–904

123

induced magnetic ﬁeld is neglected and the Hall- effect of

magnetohydrodynamics is assumed to be negligible.

The governing equations for steady state fully developed

laminar force convection in a channel is given by

ld2u

dy2þdp

dx rB2

0u¼0ð1Þ

ko2T

oy2qcu oT

oxþlou

oy

2

þrB2

0u2¼0ð2Þ

It is more convenient for the subsequence analysis to write

the governing equations in dimensionless form by

introducing the following parameters:

uo¼dp

dx

W2

l;M¼rB2

0W2

l;DT¼TwT1;U¼u

uo

;

Y¼y

W;X¼x

W;h¼TT1

DT;Pr ¼t

a;

Pe¼uoL

a;a¼k

qCP

;Ec¼CP

u2

o

cDT;h1¼T1

DT

9

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

;

ð3Þ

Using the dimensionless parameter (3) the non-dimensional

form of the governing equations 1 and 2 can be written as

d2U

dY2MU þ1¼0ð4Þ

o2h

oY2PeU oh

oXþPrEc oU

oY

2

þMEcPr U2¼0ð5Þ

where is X,Yare the axial and transverse distances,

respectively, Uis the dimensionless axial velocity, his the

dimensionless temperature, Wis the channel width, tis the

kinematic viscosity, ais the thermal diffusivity, Pe,Pr, and

Ec are Peckelt, Prandtl and Eckert numbers, respectively,

and Mis magnetic parameter.

The dimensionless boundary conditions are:

at X¼0 (Channel entrance) h¼0

at Y¼0U¼0;h¼1

at Y¼1U¼0;h¼19

=

;ð6Þ

3 Second law analysis

The basis of facts of entropy generation goes back to

Clausius and Kelvin’s studies on the irreversible aspect of

the second law of thermodynamics. Convective heat

transfer in a channel is essentially irreversible. The entropy

generation due to temperature differences has remained

untreated by classical thermodynamics, which motivates

many workers to perform analysis of fundamental and

applied engineering problems based on second law analy-

sis. A comprehensive review of such problems is reported

by Bejan [16]. A continuous entropy generation is caused

due to the exchange of energy and momentum, within the

ﬂuid and at the solid boundaries. The ﬁrst part of this

entropy generation is due to heat transfer in the direction of

ﬁnite temperature gradient. The second part of this entropy

generation takes place due to the ﬂuid friction. The mag-

netic effect brings in an additional work due to the

magnetic ﬁeld and magnetization. The volumetric rate of

local entropy generation in 2D Cartesian coordinates can

by written as [17]:

s000 ¼l

T

du

dy

2

þk

T2

oT

ox

2

þoT

oy

2

"#

þrB2

0

u2

Tð7Þ

The dimensionless volumetric entropy generation is

deﬁned as S000 ¼s000=s000

o;where s000

o¼lu2

o=DTL2;(h

?

)is

the free stream temperature parameter. Thus Eq. (6)

becomes:

S000 ¼1

hþh1

dU

dY

2

þ1

PrEc hþh1

ðÞ

2

oh

oX

2

þoh

oY

2

"#

þM

hþh1

U2ð8Þ

The dimensionless entropy generation equation consists of

three parts. The ﬁrst part is the dimensionless entropy

generation due to contribution of ﬂuid friction (S

1

) and the

second part is the dimensionless entropy generation due to

ﬁnite temperature gradient (S

2

), this part is due to con-

duction heat transfer, while the third represents the

dimensionless entropy generation due to magnetic ﬁeld

effect (S

3

). The overall dimensionless entropy generation,

for a particular problem, is an internal combination

between S

1

,S

2

and S

3

. Dimensionless entropy generation is

computed after the numerical solution of the dimensionless

velocity and temperature distributions are obtained.

4 Solution methodology

The governing differential equations (4)–(5) along with the

boundary conditions were solved numerically using a ﬁnite

difference method that was described by Patanker [18].

Applying central differences for spatial derivatives in the

governing equations, a non-linear system of equations is

generated over a non-uniform grid, to accommodate the

step velocity and temperature at the wall. The resulting

system of algebraic equations is solved by using the

Gauss–Seidel iterative procedure.

A grid independence study was carried out (see Fig. 2)

with 41 ·41, 61 ·61, 81 ·81 mesh size. The results

obtained using a ﬁner grid of 81 ·81 do not reveal

Heat Mass Transfer (2008) 44:897–904 899

123

discernible changes in the predicted heat transfer and ﬂow

ﬁeld. Thus, due to computational cost and accuracy con-

siderations a 61 ·61-mesh size was used in this

investigation.

Using algebraic package (MAPLE), the analytical

solution of Eq. (4) is given as

UðYÞ¼ 1

Ha2

1coshðHaYðÞÞþcoshðHaÞ1ÞsinhðHaYÞ

sinhðHaÞ

;ð9Þ

where Ha ¼ﬃﬃﬃﬃﬃ

M

pis Hartmann number. Without magnetic

ﬁeld effect (i.e. M= 0) the solution of Eq. (4) can be

simply written as

UYðÞ¼

Y

21YðÞ ð10Þ

The results obtained in this study were validated by com-

parison with the analytical solution as given in Eqs. (9) and

(10). The presented results show excellent agreement with

the analytical solution (as shown in Fig. 3), this will

establish conﬁdence in the reported results.

5 Results and discussion

Figure 3shows the dimensionless velocity distributions at

different values of magnetic parameter M. As expected, the

velocity proﬁles are symmetrical a bout the centerline

(Y= 0.5) of the channel. It is clear that, increasing the

value of Mhave a tendency to slow down the ﬂuid motion.

This is because of the presence of the transverse magnetic

ﬁeld, which creates a resistive force similar to the drag

force that acts in the opposite direction of the ﬂuid motion,

thus causing the velocity of the ﬂuid to decrease. Figure 4

shows the effect of magnetic ﬁeld on the ﬂuid temperature

distortions. As expected, increasing Mcauses the ﬂuid to

become warmer and therefore increases its temperature

(due to decrease in the heat transfer rate). This behavior is

U

Mesh Size

0.2 0.4 0.6 0.8

0.10800

0.10900

0.11000

0.11100

0.11200

0.11300

0.11400

Y

Mesh Size

Mesh Size

61 × 61

41 × 41

81 × 81

Fig. 2 Grid independence study

Present numerical method

Analytical solution

U

0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.04

0.08

0.12

0.16

Y

M = 0, 1, 2, 4

Pr = 7

Pe = 2 E+7

Ec = 0.01

Fig. 3 Comparison between dimensionless velocity distribution

obtained in this study and that of analytical solutions at different

values of magnetic parameter (M)on

0.0 0.2 0.4 0.6 0.8 1.0

0.6

0.7

0.8

0.9

1.0

Y

M = 0, 1, 2, 4

Pr = 7

Pe = 2 E+7

Ec = 0.01

Fig. 4 Effect of magnetic parameter (M) on dimensionless temper-

ature distribution

900 Heat Mass Transfer (2008) 44:897–904

123

attributed to decrease the ﬂuid velocity temperature due to

the magnetic ﬁeld as shown in Fig. 3.

The effect of Prandtl number on entropy generation due

to ﬂuid friction, heat transfer irreversibility and the overall

entropy generation is shown in Figs. 5,6and 7, respec-

tively. These ﬁgures show that as Prandtl number increases,

the local entropy generation near the wall due to friction

irreversibility increases. However, far from the wall the

effect of Prandtl number is signiﬁcant. Actually, the local

entropy generation increases due to three factors: (1) The

increase in local velocity gradient. (2) The increase in local

temperature gradient. (3) The decrease in temperature. As

Prandtl number increases, the velocity gradient increases.

This will cause an increase in the local entropy generation,

Fig. 5. However, the increase in Prandtl number causes a

decrease in the temperature gradient. This in turns causes a

decrease in the local entropy generation, Fig. 6. The net

effect increasing Prandtl number to decrease the total

entropy generation, since the decrease in entropy genera-

tion due to heat transfer is dominating the increase in

entropy due to ﬂuid friction, as shown in Fig. 7. Far from

the wall, the effect of the decrease in local temperature and

the local velocity is almost insigniﬁcant. For all values of

Prandtl number, the entropy generation rate decreases in

the transverse direction from the lower wall towards the

channel centerline and gradually increases towards the

upper wall. This clearly implies that viscous dissipation has

no effect on the entropy generation rate at the centerline of

a channel subjected to transverse magnetic ﬁeld.

The effect of Eckert number on entropy generation due

to ﬂuid friction, heat transfer and is shown in Figs. 8and 9,

respectively. It is clear that the increase in Eckert number

causes an increase in the velocity gradient at the wall.

Accordingly, the ﬂow velocity decreases near the center of

the channel in order to satisfy the continuity equation. The

increase in the velocity gradient with the Eckert number

results in an increase in the entropy generation due to

viscous effect as shown in Fig. 8. This behavior is simply

explained by recalling Eq. (8). Figure 9shows that entropy

generated due to heat transfer decreases by increasing the

Eckert number with minimum values at the channel

Pr = 10.0

Pr = 7.0

Pr = 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

Y

M = 2

Pe = 2 E+7

Ec = 0.01

= 2.0

S

S

′′′

1

Pr = 14.0

Fig. 5 Effect of Prandtl number (Pr) on entropy generation due to

ﬂuid friction

Pr = 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.4

0.6

0.8

1.0

Pr = 10.0

Pr = 7.0

Y

M = 2

Pe = 2 E+7

Ec = 0.01

= 2.0

S

S

′′′

2

Pr = 14.0

Fig. 6 Effect of Prandtl number (Pr) on entropy generation due to

heat transfer

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Y

Pr = 7

Pe = 2 E+7

Ec = 0.01

= 2.0

S′′′

Pr = 1.0

Pr = 10.0

Pr = 7.0

Pr = 14.0

Fig. 7 Effect of Prandtl number (Pr) on total entropy generation

Heat Mass Transfer (2008) 44:897–904 901

123

centerline. However, Fig. 10 shows that as Eckert number

increases the total entropy generation decreases. This can

be attributed to the substantial decrease in temperature

gradient as Eckert number increases. This effect is math-

ematically obvious in Eqs. (5) and (8).

Figures 11,12,13 and 14 illustrate the effect of the

magnetic parameter on the dimensionless volumetric

entropy generation spatial distributions. It is clear that the

magnetic parameter have a signiﬁcant effect on entropy

generation due to the decrease in the ﬂow velocity and the

increase in local ﬂuid temperature. From Fig. 11 it can be

seen that the local entropy generation due to ﬂuid friction

decreases with M. This behavior may be explained by the

increase in energy loss (decrease in velocity) with the

magnetic parameter, see Fig. 3. Moreover, increasing

the magnetic parameter will increase the temperature of the

ﬂuid at various locations during the channel width, which

in turns reduces the temperature gradient. Therefore, the

entropy generation due to magnetic parameter will

decrease, Fig. 12. The inﬂuence of magnetic parameter on

entropy generation due to presence of magnetic ﬁeld is

plotted in Fig. 13. Increasing Mtends to increase the

entropy generation, this effect has its maximum value at

the centerline of the heated channel (Y= 0.5). The effect of

magnetic parameter on total entropy generation is pre-

sented in Fig. 14. It is clear that, the total entropy decreases

with an increase in magnetic parameter.

It is worth noting that, entropy generation proﬁles are

asymmetric about the centerline of the channel due to the

asymmetric temperature distribution. For all factors affec-

ted the problem under consideration, each wall acts as a

strong concentrator of entropy generation because of the

high near-wall gradients of velocity and temperature. Fluid

friction entropy generation is zero at channel centerline

(Y= 0.5) due to zero velocity gradient. Furthermore,

entropy generation is independent of all factors at Y= 0.5.

Therefore, the magnitude of entropy generation is same at

centerline of the channel for all factors. Minimum entropy

generation ratio occurs very near where the temperature

gradient is zero. Generally, it is observed that an increase in

the dimensionless parameters strengthens the effect of ﬂuid

Ec =0.01

0.00.20.40.60.81.0

0.0

0.2

0.4

0.6

0.8

Y

Pr = 7

Pe =2E+7

M=2.0

=2.0

S

S

′′′

1

Ec =0.05

Ec =0.08

Ec =0.1

Fig. 8 Effect of Eckert number (Ec) on entropy generation due to

ﬂuid friction

Ec = 0.1

Ec = 0.08

Ec = 0.05

Ec = 0.01

0.0 0.2 0.4 0.6 0. 8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Y

Pr = 7

Pe = 2 E+7

M = 2.0

= 2.0

S

S

′′′

2

Fig. 9 Effect of Eckert number (Ec) on entropy generation due to

heat transfer

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

Ec = 0.05, 0.08, 0.1

Ec =0.01

Y

Pr = 7

Pe =2E+7

M=2.0

=2.0

S′′′

Fig. 10 Effect of Eckert number (Ec) on total entropy generation

902 Heat Mass Transfer (2008) 44:897–904

123

friction irreversibility, but heat transfer entropy generation

dominates over ﬂuid friction entropy generation.

6 Concluding remarks

This study is focused on the inﬂuence of transverse mag-

netic ﬁeld effect on the local entropy generation of steady

fully developed 2D-laminar forced convection ﬂow elec-

trically conducting ﬂuid in a horizontal channel under the

inﬂuence of a transverse magnetic ﬁeld. The velocity and

temperature proﬁles are obtained and use to compute the

entropy generation. The effect of Eckert number, Prandtl

number and the magnetic parameter on the entropy gen-

eration is analyzed. It was found that, total entropy

generation decreases as Prandtl number, Eckert number

and the magnetic parameter increases. Moreover, this study

shows that the entropy generation due to heat transfer

dominates over ﬂuid friction irreversibility and viscous

dissipation has no effect on the entropy generation rate at

the centerline of the channel.

0.00.20.40.60.8 1.0

0.00

0.05

0.10

0.15

0.20

0.25

Y

M= 0, 1, 2, 4

Pr = 7

Pe =2E+7

Ec =0.01

=2.0

S

S

′′′

1

Fig. 11 Effect of magnetic parameter (M) on entropy generation due

to ﬂuid friction

0.00.20.40.60.81.0

0.25

0.50

0.75

1.00

Y

M= 0, 1, 2, 4

Pr = 7

Pe =2E+7

Ec =0.01

=2.0

S

S

′′′

2

Fig. 12 Effect of magnetic parameter (M) on entropy generation due

to heat transfer

0.0 0.2 0.4 0.6 0. 8 1.0

-0.2

0.0

0.2

0.4

0.6

0.8

Y

M = 1, 2, 4

Pr = 7

Pe = 2 E+7

Ec = 0.01

= 2.0

S

S

′′′

3

M = 0

Fig. 13 Effect of magnetic parameter (M) on entropy generation due

magnetic effect

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Y

M = 0, 1, 2, 4

Pr = 7

Pe = 2 E+7

Ec = 0.01

θ∞ = 2.0

S′′′

Fig. 14 Effect of magnetic parameter (M) on total entropy generation

Heat Mass Transfer (2008) 44:897–904 903

123

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