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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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