Applied Mathematics and Physics, 2013, Vol. 1, No. 3, 78-89 Available online at http://pubs.sciepub.com/amp/1/3/5 © Science and Education Publishing DOI:10.12691/amp-1-3-5

Entropy Generation in MHD Porous Channel Flow Under Constant Pressure Gradient

Sanatan Das1,*, Rabindra Nath Jana2

1Department of Mathematics, University of Gour Banga, Malda, India 2Department of Applied Mathematics, Vidyasagar University, Midnapore, India

*Corresponding author: jana261171@yahoo.co.in

Received September 19, 2013; Revised October 04, 2013; Accepted October 09, 2013

Abstract Effects of magnetic field and suction/injection on the entropy generation in a flow of a viscous incompressible electrically conducting fluid between two infinite horizontal parallel porous plates under a constant pressure gradient have been studied. An exact solution of governing equation has been obtained in closed form. The entropy generation number and the Bejan number are also obtained. The influences of each of the governing parameters on velocity, temperature, entropy generation and Bejan number are discussed with the aid of graphs. It is observed that the magnetic field tends to increase the entropy generation within the channel.

Keywords: MHD flow, porous channel, Reynolds number, entropy generation number, Bejan number Cite This Article: Sanatan Das, and Rabindra Nath Jana, “Entropy Generation in MHD Porous Channel Flow Under Constant Pressure Gradient.” Applied Mathematics and Physics 1, no. 3 (2013): 78-89. doi: 10.12691/amp-1-3-5.

1. Introduction Entropy generation minimization studies are vital for

ensuring optimal thermal systems in contemporary idustrial and technological fields like geothermal systems, electronic cooling, heat exchangers to name a few. All thermal systems confront with entropy generation. Entropy generation is squarely associated with thermodynamic irreversibility. Analysis of the flow of an electrically conducting fluid in a porous channel in the presence of a transverse magnetic field is of special technical significance because of its widespread engineering and industrial applications such as in geothermal reservoirs, nuclear reactor cooling, MHD marine propulsion, electronic packages, micro electronic devices, thermal insulation, and petroleum reservoirs. Some other quite promising applications are in the field of metallurgy such as MHD stirring of molten metal and magnetic-levitation casting. This type of problem also arises in electronic packages and microelectronic devices during their operation. The contemporary trend in the field of heat transfer and thermal design is the second law analysis and its design-related concept of entropy generation minimization. The foundation of knowledge of entropy production goes back to Clausius and Kelvin's studies on the irreversible aspects of the second law of thermodynamics. Since then the theories based on these foundations have rapidly developed. However, the entropy production resulting from combined effects of velocity and temperature gradients has remained untreated by classical thermodynamics, which motivates many researchers to conduct analyses of fundamental and

applied engineering problems based on second law analysis. Entropy generation is associated with thermodynamic irreversibility, which is common in all types of heat transfer processes. Moreover, in thermodynamical analysis of flow and heat transfer processes, one thing of core interest is to improve the thermal systems to avoid the energy losses and fully utilize the energy resources. The magnetohydrodynamic channel flow with heat transfer has attracted the attention of many researchers due to its numerous engineering and industrial applications. Such flows finds applications in thermofluid transport modeling in magnetic geosystems, meteorology, turbo machinery, solidification process in metallurgy and in some astrophysical problems.

One of the methods used for predicting the performance of the engineering processes is the second law analysis. The second law of thermodynamics is applied to investigate the irreversibilities in terms of the entropy generation rate. Since entropy generation is the measure of the destruction of the available work of the system, the determination of the factors responsible for the entropy generation is also important in upgrading the system performances. The method is introduced by Bejan [1,2]. The entropy generation is encountered in many of energy-related applications, such as solar power collectors, geothermal energy systems and the cooling of modern electronic systems. Efficient utilization of energy is the primary objective in the design of any thermodynamic system. This can be achieved by minimizing entropy generation in processes. The theoretical method of entropy generation has been used in the specialized literature to treat external and internal irreversibilities. The irreversibility phenomena, which are expressed by entropy generation in a given system, are related to heat and mass

Applied Mathematics and Physics 79

transfers, viscous dissipation, magnetic field etc. Several researchers have discussed the irreversibility in a system under various flow configurations [3-14]. They showed that the pertinent flow parameters might be chosen in order to minimize entropy generation inside the system. Salas et al. [15] analytically showed a way of applying entropy generation analysis for modelling and optimization of magnetohydrodynamic induction devices. Salas et al. [15] restricted their analysis to only Hartmann model flow in a channel. Thermodynamics analysis of mixed convection in a channel with transverse hydromagnetic effect has been investigated by Mahmud et al. [16]. Flow, thermal and entropy generation characteristic inside a porous channel with viscous dissipation have been investigated by Mahmud [17]. The heat transfer and entropy generation during compressible fluid flow in a channel partially filled with porous medium have been analyzed by Chauhan and Kumar [18]. Tasnim et al. [19] have studied the entropy generation in a porous channel with hydromagnetic effects. Eegunjobi and Makinde [20] have studied the combined effect of buoyancy force and Navier slip on entropy generation in a vertical porous channel. The second law analysis of laminar flow in a channel filled with saturated porous media has been studied by Makinde and Osalusi [21]. Makinde and Maserumule [22] has presented the thermal criticality and entropy analysis for a variable viscosity Couette flow. Makinde and Osalusi [23] have investigated the entropy generation in a liquid film falling along an incline porous heated plate. Cimpean and Pop [24,25] have presented the parametric analysis of entropy generation in a channel. The effect of an external oriented magnetic field on entropy generation in natural convection has been investigated by Jery et al. [26]. Dwivedi et al. [27] have made an analysis on the incompressible viscous laminar flow through a channel filled with porous media. Recently, Makinde1 and Chinyoka [28] have discussed the numerical investigation of buoyancy effects on hydromagnetic unsteady flow through a porous channel with suction/injection.

In the present paper, we study the entropy generation in MHD flow in a porous channel under the constant pressure gradient. Both the channel walls are maintained at constant but different temperatures. A parametric study is carried out to see how the pertinent parameters of the problem affect the flow field, temperature field and the entropy generation.

2. Mathematical Formulation and its Solution

Consider the viscous incompressible electrically conducting fluid bounded by two infinite horizontal parallel porous plates separated by a distance 2h . Choose a cartesian co-ordinate system with x -axis along the centerline of the channel, the y -axis is perpendicular to the planes of the plates y h= ± . A uniform transverse magnetic field 0B is applied perpendicular to the channel plates. The plates are maintained at constant temperatures

0T and hT respectively. Since the magnetic Reynolds number is very small for most fluid used in industrial

applications, we assume that the induced magnetic field is negligible. Since the plates are infinitely long, all physical variables, except pressure, depend on y only. The equation of continuity then gives 0v v= − everywhere in the fluid where 0v is the suction velocity at the plates.

Figure 1. Geometry of the problem

The Ohm's law neglecting Hall currents, the ion-slip and thermoelectric effects as well as the electron pressure gradient for a conducting fluid is

[ ],J E q Bσ= + ×

(1)

where E

is the the electric field vector, J

the current density vector, q the velocity vector, σ the electrical conductivity of the fluid.

The equations of motion along x -direction is

2

00 2

1 ,zBdu p d uv J

dy x dyν

ρ ρ∂

− = − + +∂

(2)

where u is the fluid velocity in the x -direction, ρ is the fluid density, ν the kinematic viscosity and p is the fluid pressure.

The energy equation is

22 2

0 2 ,p p

dT k d T du Jvdy c c dydy

µ λρ ρ σ

− = + +

(3)

where µ is the coefficient of viscosity, k the thermal conductivity, pc the specific heat at constant pressure, the index λ in the equation (3) is set equal to 0 for excluding Jule dissipation and 1 for including Jule dissipation.

The boundary conditions are

00, at ,u T T y h= = = −

0, at ,hu T T y h= = = (4)

where T is the fluid temperature, hT the temperature at upper plate and 0T the temperature at the lower plate.

For steady motion 0E∇× =

which yields 0xdEdy

=

and 0zdEdy

= and hence xE = constant and zE = constant.

On taking 0x zE E= = , we have

80 Applied Mathematics and Physics

00, .x zJ J uBσ= = − (5) On the use of (5), equations (2) and (3) become

220

0 21 ,

Bdu p d uv udy x dy

σν

ρ ρ∂

− = − + −∂

(6)

22

2 20 02 .

p p

dT k d T duv B udy c c dyy

µ λσρ ρ

− = + +

(7)

Introducing the non-dimensional variables

010

, , ,h

T Ty huuh T T

η θν

−= = =

− (8)

equations (6) and (7) become

2

21 112 ,

d u duR M u Pdd ηη

+ − = − (9)

22

2 2112 ,

dud dPe Br M ud ddθ θ λη ηη

− = + + (10)

where 2 2

2 B hM σρν

= is the magnetic parameter,

2

20( )h

Brk T T h

µν=

− the Brinkmann number, 0 p

v h cPe

kρ

=

the Peclet number, 0v h

Rν

= the Reynolds number and

3

2h pP

xρν∂ = − ∂

the non-dimensional pressure gradient.

The boundary conditions for 1( )u η and ( )θ η are

1 10, 0 a 1 and 0, 1 at 1.u t uθ η θ η= = = − = = = (11) The solution of the equation (9) subject to the boundary

conditions (11) is

( ) 21 1 22( ) 1 cosh sinh ,R

Pu a r a r eM

ηη η η

− = − +

(12)

where

12 22

1 2

cosh sinh2 2, = a = .

4 cosh sinh

R RRr M a nd a

r r

= +

(13)

The solution given by the equation (12) exists for both < 0R (corresponding to 0 < 0v for the blowing at the

plates) and > 0R (corresponding to 0 > 0v for the suction at the plates).

On the use of (12), the equation (10) becomes

( )

{ }{ }

2

2

21 24

2 2 2 2 21 2 3 44

2 2 2 2 21 2 3 4

21 2 3 4

2 4 cosh sinh2

( ) ( )2

( ) cosh 2

2( )sinh 2 .

R

R

d dPedd

PBr a r a r eM

PBr a a M a aMa a M a a r

a a M a a r e

η

η

θ θηη

η η

λ

λ η

λ η

−

−

+

= − − +

− − + −

+ + + +

+ +

(14)

Solution of the equation (14) subject to boundary condition (11) is given by

1 2

23 44

7 8 9

( )

22 ( cosh sinh )

2

( cosh 2 sinh 2 ) ,

Pe

R

R

c c e

PBre X r X r

PeM

e X r X r X

η

η

η

θ η

η η η

η η

−

−

−

= +

− − +

+ + +

(15)

where

3 1 2 4 2 11 1, ,2 2

a Ra ra a Ra ra= − = −

2 2 2 2 2 25 1 2 2 3 6 1 2 3 4( ), 2( ),a a a M a a a a a M a aλ λ= + + + = +

1

2

,2 2

,2 2

R RX r r Pe

R RX r r Pe

= − − + = + + −

3 1 21 2 1 2

1 1 1 1 ,X a aX X X X

= + + −

4 1 21 2 1 2

1 1 1 1 ,X a aX X X X

= − + +

( )( )( )( )

5

2

2 2 ,

2 2 ,

X r R r R Pe

X r R r R Pe

= − − +

= + + −

7 5 65 6 5 6

1 1 1 1 1 ,2

X a aX X X X

= + + −

(16)

8 5 65 6 5 6

1 1 1 1 1 ,2

X a aX X X X

= − + +

2 2 2 2 21 2 3 4

9( )

,( )

a a M a aX

R R Peλ− + −

=−

2 3 44

7 8 9

2 2 ( cosh sinh )2

( cosh 2 sinh 2 ) ,

R

R

Pr BrA e X r X rPeM

e X r X r X

= − − +

+ + +

2 3 44

7 8 9

2 2 ( cosh sinh )2

( cosh 2 sinh 2 )

R

R

Pr BrB e X r X rPeM

e X r X r X

−

−

= − +

+ + +

1 2(1 ) 1, = .

2sinh 2sinh

Pe PeB e Ae A Bc cPe Pe

+ − − −= −

3. Results and Discussion To study the effects of magnetic field and Reynolds

number on the velocity field we have presented the non-dimensional velocity 1u against η in Figure 2 and Figure

3 for several values of magnetic parameter 2M and Reynolds number R when 1P = . It is seen from Figure 2 that the fluid velocity 1u decreases with an increase in

Applied Mathematics and Physics 81

magnetic parameter 2M . This can be attributed to the presence of Lorentz force acting as a resistance to the flow as expected. Figure 3 shows that the fluid velocity 1u increases near the lower plate and decreases near the upper plate with an increase in Reynolds number Re . We have plotted the temperature distribution θ against η in Figure 4, Figure 5 and Figure 6 for several values of magnetic parameter 2M , Brinkmann Br and Peclet number Pe . It is seen from Figure 4, Figure 5 and Figure 6 that the fluid temperature θ increases with an increase

in either magnetic parameter 2M or Brinkmann Br or Peclet number Pe . As 2M increases due to increasing magnetic field intensity, the fluid temperature increases within the channel. So, an increase in the fluid temperature is due to the presence of Ohmic heating (or Lorentz heating) which serves as additional heat source to the flow system. The terms linked to the Brinkman number act as strong heat sources in the energy equation. Increases in the Brinkman number hence significantly also increase the fluid temperature shown in Figure 5.

Figure 2. Velocity profiles for different 2M when 0.5R =

Figure 3. Velocity profiles for different R when 2 5M =

82 Applied Mathematics and Physics

Figure 4. Temperature profiles for different 2M when 0.5R = and 2Pe =

Figure 5. Temperature profiles for different Br when 2 5M = and 2Pe =

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Figure 6. Temperature profiles for different Pe when 2 5M = and 0.5R =

The rate of heat transfer at the plates 1η = − and 1η = can be obtained from (15) as

2

3 424 3

47 8

8 7 9

( 1)

( 2 )cosh2( 2 )sinh

2 ( 2 )cosh 2( 2 )sinh 2

' Pe

R

R

c Pee

RX rX re

RX rX rPePBrM X rX r

eX rX r X

θ − = −

− + − − − − − − − −

(17)

3 42

4 34

7 8

8 7 9

2

( 2 ) cosh2

( 2 ) sinh,

2 ( 2 ) cosh 2

( 2 ) sinh 2

(1)R

R

' Pe

RX rX re

PBr RX rX rPe

M X rX re

X rX r X

c Peeθ

−

−

−

−+

− −−

−−

+ − −

= −

(18)

where 3X , 4X , 7X , 8X and 2c are given by (16). The numerical values of the rate of heat transfers ( 1)'θ − and (1)'θ− are entered in the Table 1 and Table 2

for several values of 2M , R , Br and Pe . It is seen from the Table 2 that the rate of heat transfer ( 1)'θ − at the plate 1η = − increases whereas the rate of heat transfer

(1)'θ− at the plate 1η = decreases with an increase in

either magnetic parameter 2M or Reynolds number R . Table 2 shows that the rate of heat transfer ( 1)'θ − increases with an increase in either Peclet number Pe or Brinkman number Br . The rate of heat transfer (1)'θ− increases with an increase in Brinkman number Br while it decreases with an increase in Peclet number Pe . On the other hand, (1) < 0'θ means the heat transfer take places from fluid to the upper plate, because when there is significant heat generation in the fluid due to viscous and Ohmic dissipations, the temperature of the fluid exceeds the plate temperature which causes heat flow from fluid to the plate.

Table 1. Rate of heat transfer at the plates 1η = − and 1η = when 0.2Pe = and 1.4Br =

( 1)'θ − (1)'θ− 2\R M 4 6 8 10 4 6 8 10

0.1 2.27592 2.32133 2.31711 2.30077 1.67845 0.33726 0.21077 0.16233 0.2 2.31156 2.33879 2.32705 2.30715 1.09025 0.29633 0.19370 0.15281 0.3 2.32653 2.34734 2.33274 2.31133 0.80138 0.26776 0.18153 0.14574 0.4 2.33218 2.35185 2.33650 2.31453 0.63165 0.24566 0.17173 0.13981

84 Applied Mathematics and Physics

Table 2. Rate of heat transfer at the plates 0η = and 1η = with 0.5R = and 2 5M =

( 1)'θ − (1)'θ− \Br Pe 2 4 6 8 2 3 4 5

0.2 2.08198 4.06058 6.06530 8.06884 0.01657 0.01650 0.01606 0.01374 0.4 2.12665 4.11982 6.13057 8.13767 0.07046 0.03433 0.03215 0.02748 0.6 2.17132 4.17906 6.19584 8.20651 0.12435 0.05217 0.04825 0.04123 0.8 2.21598 4.23830 6.26111 8.27535 0.17824 0.07001 0.06434 0.05497

4. Entropy Generation Second law analysis in terms of entropy generation rate

is a useful tool for predicting the performance of the engineering processes by investigating the irreversibility arising during the processes. According to Wood [29], the local entropy generation rate is defined as

2 2 2

202

0 00.

'

GBk dT duE u

dy T dy TTσµ λ

= + +

(19)

The first term in equation (20) is the irreversibility due to heat transfer, the second term is the entropy generation due to viscous dissipation and third term is due to magnetic field.

The dimensionless entropy generation number may be defined by the following relationship:

2 2

02

0.

( )G

Sh

T h EN

k T T=

− (20)

In terms of the dimensionless velocity and temperature, the entropy generation number becomes

2 2

2 211 ,S

dud BrN M ud dθ λη η

= + + Ω (21)

where 20

0( )e

'h

uBr

k T Tµ

=−

is the Brinkmann number and

0

0

wT TT−

Ω = is the non-dimensional temperature

difference. The entropy generation number SN can be written as a

summation of the entropy generation due to heat transfer denoted by 1N and the entropy generation due to fluid friction with magnetic field denoted by 2N given as

2 2

2 211 2 1, .

dud BrN N M ud dθ λη η

= = + Ω (22)

The entropy in a system is associated with the presence of irreversibility. We have to notice that the contribution of the heat transfer entropy generation to the overall entropy generation rate is needed in many engineering applications. The Bejan number Be is an alternative irreversibility distribution parameter and it represents the ratio between the heat transfer irreversibility 1N and the total irreversibility due to heat transfer and fluid friction with magnetic field SN . It is defined by

1 1= ,1S

NBeN

=+Φ

(23)

where 21

NN

Φ = is the irreversibility ratio. Heat transfer

dominates for 0 < 1≤ Φ and fluid friction with magnetic effects dominates when > 1Φ . The contribution of both heat transfer and fluid friction to entropy generation are equal when = 1Φ . The Bejan number Be takes the values between 0 and 1 (see Cimpean et al. [30]). The value of 1Be = is the limit at which the heat transfer irreversibility dominates, 0Be = is the opposite limit at which the irreversibility is dominated by the combined effects of fluid friction and magnetic field and 0.5Be = is the case in which the heat transfer and fluid friction with magnetic field entropy production rates are equal. Further, the behavior of the Bejan number is studied for the optimum values of the parameters at which the entropy generation takes its minimum.

The influences of the different governing parameters on entropy generation within the channel are presented in Figures 7-10. It is seen from Figure 7 that the entropy generation number SN decreases with an increase in

magnetic parameter 2M . The effect of magnetic parameter 2M on the entropy generation number is displayed in Figure 7. This figure shows that the entropy generation is slightly increases with magnetic parameter

2M . The magnetic parameter 2M is not too much dominating on entropy generation. A large variation of

2M causes a small variation in the rate of entropy generation. Figure 8 and Figure 9 show that the entropy generation number SN increases near lower plate with an increase in either Reynolds number R or Peclet number Pe . As expected, since the fluid velocity and temperature increase near the lower plate. An increase in Peclet number Pe has no effect on the entropy generation in the central region of the channel as well as near the upper plate.

Figure 10 reveals that the entropy generation number

SN increases with an increase in group parameter 1Br −Ω .

It is seen from Figure 11 that the Bejan number Be decreases near the central region of channel with an increase in magnetic parameter 2M . Figure 12 shows that the Bejan number Be decreases near the central region of channel with an increase in Reynolds number R . Figure 13 and Figure 14 shows that the Bejan number Be decreases near the central region of channel with an increase in either Peclet number Pe or group parameter

1Br −Ω . The Bejan number Be is unaffected at the central point of the channel width for Peclet number Pe and group parameter 1Br −Ω . The group parameter is an important dimensionless number for irreversibility analysis. It determines the relative importance of viscous effects to that of temperature gradient entropy generation.

Applied Mathematics and Physics 85

Figure 7. SN for different 2M when 2Pe = , 1 1Br −Ω = and 0.5R =

Figure 8. SN for different R when 2 5M = , 2Pe = and 1 1Br −Ω =

86 Applied Mathematics and Physics

Figure 9. SN for different Pe when 2 5M = , 2Pe = , 1 1Br −Ω = and 0.5R =

Figure 10. SN for different 1Br −Ω when 2 5M = , 2Pe = , 1 1Br −Ω = and 0.5R =

Applied Mathematics and Physics 87

Figure 11. Bejan number for different 2M when 2Pe = , 1 1Br −Ω = and 0.5R =

Figure 12. Bejan number for different R when 2 5M = , 2Pe = and 1 1Br −Ω =

88 Applied Mathematics and Physics

Figure 13. Bejan number for different Pe when 2 5M = , 1 1Br −Ω = and 0.5R =

Figure 14. Bejan number for different 1Br −Ω , 2 5M = , 2Pe = and 0.5R =

Applied Mathematics and Physics 89

5. Conclusion We investigate the entropy generation in an MHD flow

between two infinite parallel porous plates. The analytical results obtained for the velocity and temperature profiles are used in order to obtain the entropy generation production. The non-dimensional entropy generation number and the Bejan number are calculated for the problem involved. It is found that, the entropy generation decreases with an increase in magnetic parameter. The entropy generation increases with an increase in either Reynolds number or Peclet number or group parameter. The rate of heat transfer at the lower plate increases whereas the rate of heat transfer at the upper plate decreases with an increase in either magnetic parameter or Reynolds number. It is important to note that the fluid velocity, the fluid temperature as well as entropy generation are significantly influenced by Jule dissipation.

Acknowledgement The authors would like to express thanks to the

anonymous referees for their valuable suggestions.

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