Entropy Generation in MHD Porous Channel Flow Under Constant Pressure Gradient

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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

(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

(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

(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

(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

(5)

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

(6)

(7)

Introducing the non-dimensional variables

(8)

equations (6) and (7) become

(9)

(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

(11)

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

(12)

where

(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

(14)

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

(15)

where

(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

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

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

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

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

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

(17)

(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

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

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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

(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:

(20)

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

(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

(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

(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

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

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

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

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

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

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

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

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