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Entropy Generation in MHD Porous Channel Flow Under Constant Pressure Gradient

Sanatan Das*, Rabindra Nath Jana

Department of Mathematics, University of Gour Banga, Malda, India

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

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

Citation: Sanatan Das, Rabindra Nath Jana, Entropy Generation in MHD Porous Channel Flow Under Constant Pressure Gradient. Applied Mathematics and Physics, 1 (3). 78-89. January 2013. 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 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 . Choose a cartesian co-ordinate system with -axis along the centerline of the channel, the -axis is perpendicular to the planes of the plates. A uniform transverse magnetic field is applied perpendicular to the channel plates. The plates are maintained at constant temperatures and 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 only. The equation of continuity then gives everywhere in the fluid where 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

[picture] | (1)

where is the the electric field vector, the current density vector, the velocity vector, the electrical conductivity of the fluid.

The equations of motion along -direction is

[picture] | (2)

where is the fluid velocity in the -direction, is the fluid density, the kinematic viscosity and is the fluid pressure.

The energy equation is

[picture] | (3)

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

The boundary conditions are

[picture] | (4)

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

For steady motion which yields and and hence constant and constant. On taking , we have

[picture] | (5)

On the use of (5), equations (2) and (3) become

[picture] | (6)

[picture] | (7)

Introducing the non-dimensional variables

[picture] | (8)

equations (6) and (7) become

[picture] | (9)

[picture] | (10)

where is the magnetic parameter, the Brinkmann number, the Peclet number, the Reynolds number and the non-dimensional pressure gradient.

The boundary conditions for and are

[picture] | (11)

The solution of the equation (9) subject to the boundary conditions (11) is

[picture] | (12)

where

[picture] | (13)

The solution given by the equation (12) exists for both (corresponding to for the blowing at the plates) and (corresponding to for the suction at the plates).

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

[picture] | (14)

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

[picture] | (15)

where

[picture] | (16)

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 against in Figure 2 and Figure 3 for several values of magnetic parameter and Reynolds number when . It is seen from Figure 2 that the fluid velocity decreases with an increase in magnetic parameter . 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 increases near the lower plate and decreases near the upper plate with an increase in Reynolds number . We have plotted the temperature distribution against in Figure 4, Figure 5 and Figure 6 for several values of magnetic parameter , Brinkmann and Peclet number . It is seen from Figure 4, Figure 5 and Figure 6 that the fluid temperature increases with an increase in either magnetic parameter or Brinkmann or Peclet number . As 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 M² when R=0.5

Figure 3. Velocity profiles for different R when M²=5

Figure 4. Temperature profiles for different M² when R=0.5 and Pe=2

Figure 5. Temperature profiles for different Br when M²=5 and Pe=2

Figure 6. Temperature profiles for different Pe when M²=5 and R=0.5

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

[picture] | (17)

[picture] | (18)

where , , , and are given by (16).

The numerical values of the rate of heat transfers and are entered in the Table 1 and Table 2 for several values of , , and . It is seen from the Table 2 that the rate of heat transfer at the plate increases whereas the rate of heat transfer at the plate decreases with an increase in either magnetic parameter or Reynolds number . Table 2 shows that the rate of heat transfer increases with an increase in either Peclet number or Brinkman number . The rate of heat transfer increases with an increase in Brinkman number while it decreases with an increase in Peclet number . On the other hand, 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 η=-1 and η=1 when Pe=0.2 and Br=1.4

Table 2. Rate of heat transfer at the plates η=0 and η=1 with R=0.5 and M²=5

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

[picture] | (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:

[picture] | (20)

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

[picture] | (21)

where is the Brinkmann number and is the non-dimensional temperature difference.

The entropy generation number can be written as a summation of the entropy generation due to heat transfer denoted by and the entropy generation due to fluid friction with magnetic field denoted by given as

[picture] | (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 is an alternative irreversibility distribution parameter and it represents the ratio between the heat transfer irreversibility and the total irreversibility due to heat transfer and fluid friction with magnetic field . It is defined by

[picture] | (23)

where is the irreversibility ratio. Heat transfer dominates for and fluid friction with magnetic effects dominates when . The contribution of both heat transfer and fluid friction to entropy generation are equal when . The Bejan number takes the values between 0 and 1 (see Cimpean et al. [30]). The value of is the limit at which the heat transfer irreversibility dominates, is the opposite limit at which the irreversibility is dominated by the combined effects of fluid friction and magnetic field and 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 decreases with an increase in magnetic parameter . The effect of magnetic parameter on the entropy generation number is displayed in Figure 7. This figure shows that the entropy generation is slightly increases with magnetic parameter . The magnetic parameter is not too much dominating on entropy generation. A large variation of causes a small variation in the rate of entropy generation. Figure 8 and Figure 9 show that the entropy generation number increases near lower plate with an increase in either Reynolds number or Peclet number . As expected, since the fluid velocity and temperature increase near the lower plate. An increase in Peclet number 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 increases with an increase in group parameter . It is seen from Figure 11 that the Bejan number decreases near the central region of channel with an increase in magnetic parameter . Figure 12 shows that the Bejan number decreases near the central region of channel with an increase in Reynolds number . Figure 13 and Figure 14 shows that the Bejan number decreases near the central region of channel with an increase in either Peclet number or group parameter . The Bejan number is unaffected at the central point of the channel width for Peclet number and group parameter . 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.

Figure 7. Ns for different M² when Pe=2, BrΩ⁻¹=1 and R=0.5

Figure 8. Ns for different R when M²=5, Pe=2 and BrΩ⁻¹=1

Figure 9. Ns for different Pe when M²=5, Pe=2, BrΩ⁻¹=1 and R=0.5

Figure 10. Ns for different BrΩ⁻¹ when M²=5, Pe=2, BrΩ⁻¹=1 and R=0.5

Figure 11. Bejan number for different M² when Pe=2, BrΩ⁻¹=1 and R=0.5

Figure 12. Bejan number for different R when M²=5, Pe=2 and BrΩ⁻¹=1

Figure 13. Bejan number for different Pe when M²=5 BrΩ⁻¹=1 and R=0.5

Figure 14. Bejan number for different BrΩ⁻¹, M²=5, Pe=2 and R=0.5

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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