Entropy Analysis in MHD Forced Convective Flow through a Circular Channel Filled with Porous Medium in the Presence of Thermal Radiation

Entropy is the basic concept of physics and information sciences. It is a measure to compare different state of an isolated system. Entropy is the thermodynamic variable that serves to define and relate the thermal properties of the matter and equilibrium state, basically known as second law of thermodynamics. According to the second law of thermodynamics, all real processes are irreversible. Entropy generation is a measure of the amount of irreversibility, which associated with the real processes. As entropy generation takes place, the quality of energy decreases Cengel et al. [1]. Bejan [2] introduced that the continuous entropy generation in the fluid flow system is occurred due to the exchange of the heat and momentum within the fluid and at the impermeable boundaries. Bejan [3], [4] discussed that entropy generation for forced convective in viscous fluid flow through the channel is due to heat transfer and viscous friction in the fluid. Entropy generation mostly occurs in the thermal and heat engineering system, therefore in the engineering field it has wide range of applications, such as industrial heat exchanger, nuclear reactor, petroleum equipment’s and many other applications in the mechanical and chemical engineering. Hence lot of research carried out in last two and three decades in this field. The effect of hall current and magnetic field in a oscillatory channel flow through a rotating system was depicted recently by Ahmed and Das [5]. Entropy generation in a laminar fluid flow through a circular pipe is investigated by Ahmet et al. [6]. Yurusoy et al. [7] studied the effects of entropy generation and temperature dependent viscosity on a non-Newtonian fluid flow in annular pipes. Entropy generation for forced convection in a porous saturated circular tube with uniform wall temperature was discussed by Hooman et al. [8]. Pakdemirli et al. [9] analyzed the entropy generation in pipe due to non-Newtonian fluid flow in the case of constant viscosity. They concluded that an increment in the non-Newtonian parameters, reduces the fluid friction in the region close to the pipe wall. Natural convection and entropy generation of nano-fluids in a square cavity was concluded by Bouchoucha and Bessaih [10]. Unsteady heat and mass transfer effects on an impulsively started infinite vertical plate in the presence of porous medium was studied by Loganathan et al. [11]. Ganesan et al. [12] considered the Radiation and mass transfer effects on flow of an incompressible viscous fluid past a moving vertical cylinder. Their work was extended by Suneeta et al. [13] for MHD fluid flow. On the same fluid flow conditions, the entropy generation was analyzed by Mahian et al. [14]. Tshehla et al. [15] premeditated the entropy generation in a variable viscosity fluid flow between two concentric pipes with a convective cooling at the surface. Aldos et al. [16] discussed the mixed convection from a horizontal cylinder in a porous medium. Effect of slip conditions on forced convection and entropy generation in a circular channel occupied by a highly porous medium Darcy extended Brinkmann Forchheimer model was conferred by Chauhan et al. [17]. Kumar Navin et al. [18] extended this work. They concluded the MHD forced convection and entropy generation in a circular channel occupied by hyper porous medium. 

Motivated by the above cited research work, we have made an attempt to analyze the entropy generation on forced convective flow of viscous incompressible electrically conducting fluid through a circular channel filled with a hyper porous medium saturated with a rarefied gas in the presence of transverse magnetic field, thermal radiation and uniform heat flux at the walls of the channel.

We now introduce the following dimensionless quantities as

$z=\frac{\overline{z}}{P e R}, r=\frac{\overline{r}}{R}, u=\frac{\mu \overline{u}}{\mathrm{P} R^{2}}, \mu_{1}=\frac{\mu_{e f f}}{\mu}, D a=\frac{\overline{K}}{R^{2}}, \eta=\sqrt{\frac{1}{D a}}$$F=\frac{\rho C_{i} \mathrm{PR}^{3}}{\mu^{2}}, P e=\frac{\rho C_{p} R \overline{U}_{\operatorname{mean}}}{\kappa}, \alpha=\frac{\overline{\alpha}}{R}, \beta=\frac{\overline{\beta}}{R}, M=\sqrt{\frac{\sigma B_{0}^{2} R^{2}}{\mu}}$(7)

Using (7), the equation (1) is reduced in dimensionless form  

$\frac{d^{2} u}{d r^{2}}+\frac{1}{r} \frac{d u}{d r}-\frac{\eta^{2}}{M} u-\frac{M^{2}}{\mu_{1}} u-\frac{F \eta}{\mu_{1}}+\frac{1}{\mu_{1}}=0$(8)

The corresponding boundary conditions for velocity field became as

at $r=0 : \frac{d u}{d r}=0, r=1 : u=-\alpha \frac{d u}{d r}$(9)

Further defining the mean velocity $\overline{U}_{\text { mean }}$ and bulk mean temperature $\overline{T}_{\text { mean }}$$[\text { Navin et al. }[18]]$

$\overline{U}_{m e c n}=\frac{2}{R^{2}} \int_{0}^{R-} \overline{u} \overline{r} d \overline{r}$ and $\overline{T}_{m e a n}=\frac{2}{R^{2} \overline{U}} \int_{0}^{R} \overline{u} \overline{T} \overline{r} d \overline{r}$(10)

The dimensionless variables are originated as

$\hat{u}=\frac{\overline{u}}{\overline{U}_{\text { mean }}}, T=\frac{\overline{T}-\overline{T}_{w}}{\overline{T}_{\text {mean}}-\overline{T}_{w}}, N u=\frac{2 R q_{w}}{\kappa\left(\overline{T}_{w}-\overline{T}_{\text { mean }}\right)}$(11)

$T$ is the function of radial coordinate $(\overline{r})$only, while the bulk mean temperature $\overline{T}_{\text { mean }}$ is the function of axial coordinate $\overline{(x)}$ only. Now from the first law of thermodynamics, for uniform heat flux on the wall, we have

$\frac{\partial \overline{T}}{\partial \overline{x}}=\frac{2 q_{w}}{\rho C_{p} R \overline{U}_{m e a n}}$(12)

Now using equation (11) and (12), equation (6) is reduced as

$\frac{d^{2} T}{d r^{2}}+\frac{1}{r} \frac{d T}{d r}+(N u) \hat{u} L+\eta^{2} B r \hat{u}^{2} L+B r \mu_{R} L\left(\frac{d \hat{u}}{d r}\right)^{2}+$$+M^{2} B r \hat{u}^{2} L=0$(13)

where $B r=\frac{\mu \overline{U}_{m e a n}^{2}}{\kappa\left(\overline{T}_{m e a n}-\overline{T}_{w}\right)}$is the Brinkmann number, $N=\frac{\kappa \phi}{4 \sigma \overline{T}_{w}^{3}}$ the Radiation parameter [Ganeshan et.al [12]] and $L=\frac{3 N}{3 N+4}$The corresponding boundary/symmetric conditions for temperature field are:

$r=0 : \frac{d T}{d r}=0, r=1 : T=-\beta \frac{d T}{d r}$(14)

The Nusselt number can be found by substituting ${u}$ and $T$ in the compatibility condition

$\int_{0}^{1} {u} T r d r=\frac{1}{2}$(15)

This is an identity from the first law of thermodynamics [Nield et al. [20]].

3.1 Solution of momentum equation

In case of large Darcy number, we write the following asymptotic expression for the velocity distribution with the assumption $\eta<<1$

$u(r)=u_{0}(r)+\eta u_{1}(r)+o\left(\eta^{2}\right)$(16)

Now substituting the equation (16) into (8) and equating the coefficient of like powers of $\eta$, we find the following set of coupled differential equations:

$\frac{d^{2} u_{0}}{d r^{2}}+\frac{1}{r} \frac{d u_{0}}{d r}-\frac{M^{2}}{\mu_{1}} u_{0}+\frac{1}{\mu_{1}}=0$(17)

$\frac{d^{2} u_{1}}{d r^{2}}+\frac{1}{r} \frac{d u_{1}}{d r}+\frac{M^{2}}{\mu_{1}} u_{1}-\frac{F}{\mu_{1}} u_{0}^{2}=0$(18)

and the corresponding boundary conditions are

$r=0 : \quad \frac{d u_{0}}{d r}=0, \quad \frac{d u_{1}}{d r}=0$         $r=1 : u_{0}=-\alpha \frac{d u_{0}}{d r}, u_{1}=-\alpha \frac{d u_{1}}{d r}$(19)

For the diminutive magnetic field, $u_{0}$ and $u_{1}$ can be expanded as

$u_{0}(r)=u_{00}(r)+M^{2} u_{01}(r)+o\left(M^{4}\right)$     $u_{1}(r)=u_{10}(r)+M^{2} u_{11}(r)+o\left(M^{4}\right)$(20)

Substituting the equation (20) in equation (17) and (18) and equating the coefficient of like power of $M^{2}$ and neglecting the higher order of $M^{2}$, we have

$\frac{d^{2} u_{00}}{d r^{2}}+\frac{1}{r} \frac{d u_{00}}{d r}+\frac{1}{\mu_{1}}=0$(21)

$\frac{d^{2} u_{01}}{d r^{2}}+\frac{1}{r} \frac{d u_{01}}{d r}-\frac{u_{00}}{\mu_{1}}=0$(22)

$\frac{d^{2} u_{10}}{d r^{2}}+\frac{1}{r} \frac{d u_{10}}{d r}-\frac{F}{\mu_{1}} u_{00}^{2}=0$(23)

$\frac{d^{2} u_{11}}{d r^{2}}+\frac{1}{r} \frac{d u_{11}}{d r}=\frac{u_{10}}{\mu_{1}}=\frac{2 F}{\mu_{1}} u_{00} u_{01}$(24)

The corresponding boundary conditions are given by

$r=0 : \frac{d u_{00}}{d r}=0, \frac{d u_{01}}{d r}=0, \frac{d u_{10}}{d r}=0, \frac{d u_{11}}{d r}=0$

$r=1 : u_{00}=-\alpha \frac{d u_{00}}{d r}, u_{01}=-\alpha \frac{d u_{01}}{d r}$

$u_{10}=-\alpha \frac{d u_{10}}{d r}, u_{11}=-\alpha \frac{d u_{11}}{d r}$(25)

Equations (21) to (24) are coupled linear differential equations with variable coefficient. These are solvable by straight forward calculation and the solution is given by

$u(r)=A_{23}+r^{2} A_{24}+r^{4} A_{25}+r^{6} A_{26}+r^{8} A_{27}$(26)

where $A_{23}$ to $A_{27}$ are constants.

3.2 Solution of heat equation

Now by using equation (26) in the mean velocity equation (10), we get

$\overline{U}_{m e a n}=\frac{\mathrm{P} R^{2}}{60 \mu}\left(A_{28}+\eta A_{29}\right)$(27)

where $A_{23}$ to $A_{29}$ are constants and not given here due to sake of bravity. Similarly, for the large Darcy number, we write the following asymptotic expression for temperature distribution as

$T(r)=T_{0}(r)+\eta T_{1}(r)+o\left(\eta^{2}\right)$(28)

Using (27), (28) and (10) in equation (13), leaving the higher order than two, we obtained the following set of differential equations

$\frac{d^{2} T_{0}}{d r^{2}}+\frac{1}{r} \frac{d T_{0}}{d r}+\frac{60 N u u_{0}}{A_{28}} L+\frac{3600 B r \mu_{1}}{A_{28}^{2}}\left(\frac{d u_{0}}{d r}\right)^{2} L+$$\frac{3600 B r M^{2} u_{0}^{2}}{A_{28}^{2}} L=0$(29)

$\frac{d^{2} T_{1}}{d r^{2}}+\frac{1}{r} \frac{d T_{1}}{d r}+\frac{60 N u}{A_{28}}\left\{u_{1}-\frac{A_{28}}{A_{29}} u_{0}\right\} L+$$+\frac{7200 B r \mu_{1}}{A_{28}^{2}}\left[\frac{d u_{0}}{d r}\left\{\frac{d u_{1}}{d r}-\frac{A_{29}}{A_{28}}\left(\frac{d u_{0}}{d r}\right)\right\}\right] L+$$+\frac{7200 B r M^{2} u_{0}}{A_{28}}\left\{u_{1}-\frac{A_{29}}{A_{28}} u_{0}\right\} L=0$

The corresponding boundary conditions are:

at $r=0 : \frac{d T_{0}}{d r}=0, \frac{d T_{1}}{d r}=0$

at $r=1 : T_{0}=-\beta \frac{d T_{0}}{d r}, T_{1}=-\beta \frac{d T_{1}}{d r}$(31)

Equation (29) and (30) are ordinary linear differential equations with variable coefficient these are solvable by usual algebric method.        

Finally, solving the equation (15) for determining the Nusselt number, we find

$N u=\frac{A_{28}+\eta A_{29}}{120 A_{106}}-\frac{A_{105}}{A_{106}}$(32)

where $A_{30}$ to $A_{106}$ be constants.

In the present study, an analytical solution is obtained for the velocity and temperature profiles for the fully developed steady MHD forced convective flow through a circular channel filled with a hyper porous medium soaked with a rarefied gas. The entropy generation due to heat transfer, fluid friction and magnetic field is evaluated numerically for different values of governing parameters.

Figure 2 shows the effect of magnetic field and velocity slip parameter on the velocity field. It is depicted that due to the intensity of the magnetic field, the fluid velocity is declining gradually. While slip parameter upsurges velocity field. This is quite visible that the fluid has its high velocity in the centre of the circular tube.

It is observed from Figure 3 that an increase in Forchheimer number F reduces the fluid velocity. But this demotion is comparatively large near the centre line. The dimensionless temperature profiles are plotted against $r$ in Figure from 4 to 7. From Figure 4 it is observed that an increment in the temperature slip parameter $\beta$ supports the magnitude of the temperature while reverse effect is seen due to the enhancement in velocity slip parameter .

Figure 5 describes that when the Brinkmann number $B r$ and intensity of magnetic field M increase, the fluid temperature rises gradually and this effect is more in the centre line of the channel. From Figure 6 it is predicted that with the increasing of Forchheimer number F, the temperature of the fluid is declining. The effect of radiation parameter on temperature fluid can be seen clearly from Figure 7. The fluid temperature is getting increased due to rise in thermal radiation parameter N. This promotion is comparatively higher on the centre line of the circular channel.

Now Figure 8 to 12 illustrate the effect of various physical parameter on entropy generation rate. From Figure 8 we observed that the entropy generation rate is declining due to increase in the Forchheimer number. While Figure 9 to 11 predict that an increase value of temperature slip parameter b, Brinkmann number and Hartmann number M upsurging the entropy generation rate. Figure 12 shows that effect of radiation parameter on entropy generation rate. This is getting increased due to increase of the thermal radiation in the presence of uniform magnetic field.

Figure 13 to 16 reflect the effect of various physical parameter on Bejan number $B e$ . The Bejan number attains its maximum value (i.e., 1) for all value of $B r$. Figure 13 illustrate that Bejan number increases due to an increment in the Brinkmann number. Figure 14 and 15 depict that an increment in the Forchheimer number and Hartmann number raise the value of Bejan number. Figure 16 shows that an increment in the thermal radiation parameter N support the Bejan number.

Table 1 describes that the rate of heat transfers in terms of the Nusselt number rises with an increase of temperature slip parameter b, Brinkmann number $B r$, Forchheimer number F, viscosity ratio m_1, Hartmann number M, and radiation parameter N. While it reduces with the increase value of velocity slip parameter $\alpha$.

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Figure 2. Velocity versus r

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Figure 3. Velocity versus r

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Figure 4. Temperature versus r

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Figure 5. Temperature versus r

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Figure 6. Temperature versus r

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Figure 7. Temperature versus r

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Figure 8. Ns versus r

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Figure 9. Ns versus r

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Figure 10. Ns versus r

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Figure 11. Ns versus r

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Figure 12. Ns versus r

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Figure 13. Be versus r

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Figure 14. Be versus r

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Figure 15. Be versus r

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Figure 16. Be versus r

Table 1. Nusselt number (Nu) at wall when $\eta=0.1$