# Effect of an External Oriented Magnetic Field on Entropy Generation in Natural Convection

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}of the channel which corresponds to minimum entropy generation, and it is given by:

## 2. Governing Equations

_{x}, B

_{y}, 0) is applied and the magnetic Reynolds number (R

_{em}= μ

_{e}σ

_{e}V

_{0}L

_{0}<< 1, where σ

_{e}, μ

_{e}, V

_{0}and L

_{0}are the magnetic permeability, the viscosity, the characteristic velocity and length, respectively) is assumed to be small so that the induced magnetic field is negligible in comparison to the applied magnetic field. Since there is no applied or polarization voltage imposed on the flow field, the electric field E = 0.

_{c}and T

_{h}, respectively, and are such that T

_{c}< T

_{h}, horizontal walls are adiabatic as illustrated in Figure 1. The cavity is permeated by the uniform oriented magnetic field B = B

_{x}e

_{x}+ B

_{y}e

_{y}(where B

_{x}and B

_{y}are space independent) of constant magnitude B

_{0}= $\sqrt{{B}_{x}^{2}\text{}+\text{}{B}_{y}^{2}}$ and e

_{x}and e

_{y}are unit vectors in Cartesian coordinate system. The orientation of the magnetic field forms an angle α with horizontal axis, such that tan α = B

_{y}/B

_{x}.

_{P}is the isobaric specific heat of the fluid. Thus, the governing conservative equations of the mass, the momentum and the energy in their dimensionless variables take the following form:

- continuity equation:$$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$$
- momentum equation in X direction:$$\frac{\partial U}{\partial \tau}+div\left(Uv-gradU\right)=-\frac{\partial P}{\partial X}+H{a}^{2}\left(V\mathrm{sin}\alpha \mathrm{cos}\alpha -U{\mathrm{sin}}^{2}\alpha \right)$$
- momentum equation in Y direction:$$\frac{\partial V}{\partial \tau}+div\left(Vv-gradV\right)=-\frac{\partial P}{\partial Y}+G{r}_{T}\theta +H{a}^{2}\left(U\mathrm{sin}\alpha \mathrm{cos}\alpha -V{\mathrm{cos}}^{2}\alpha \right)$$
- energy equation:$$\frac{\partial \theta}{\partial \tau}+div\left(\theta v-\frac{1}{\mathrm{Pr}}grad\theta \right)=0$$

## 3. Second Law Formulation

_{0}. According to Bejan [24], the characteristic entropy transfer rate is given by:

_{0}and ΔT are the thermal conductivity, the characteristic length of the enclosure, a reference temperature and a reference temperature difference, respectively.

## 4. Numerical Scheme

_{T}= 10

^{3}, 10

^{4}and 10

^{5}respectively, is found sufficiently enough to achieve the imposed global and local convergence criteria given respectively by:

^{−3}and 10

^{−5}for Pr = 0.71 and 0.02, respectively.

## 5. Results and Discussions

_{T}, ranging between 10

^{3}and 10

^{5}, the irreversibility coefficient related to fluid friction χ

_{1}, ranging between 10

^{−}

^{4}and 10

^{−}

^{2}, the Hartmann number Ha, ranging between 0 and 100 and the inclination angle of the magnetic field α, ranging between 0° and 90°. Obviously, the irreversibility coefficient related to the magnetic field χ

_{2}, depends directly on χ

_{1}and Ha. To be realistic, the numerical values of Prandtl number are chosen to be Pr = 0.71 and Pr = 0.02, which, correspond to air and liquid gallium, respectively. Liquid gallium is widely used in integrated circuits with optoelectronic devices, to dope semiconductors and produce solid-state devices like transistors, as a component in low-melting alloys and in some high temperature thermometers.

_{T}= 10

^{4}, heat transfer decreases as Hartmann number increases. It is important to notice that heat transfer is reduced to pure conduction mode as Hartmann number tends towards the value 100. Consequently, the magnetic field tends to suppress the convection and retards the fluid motion via the Lorentz force (magnetic force).

_{1}= 10

^{−}

^{3}) at different values of thermal Grashof and Hartmann numbers as illustrated in Figure 3 and Figure 4. At Pr = 0.71 and for Gr

_{T}≤ 10

^{3}, Figure 3a shows that entropy generation magnitude increases from the value equal to unity at initial time towards an asymptotic value as time proceeds. Asymptotic value (which describes a stationary state) is rapidly achieved and considerably decreased in magnitude as Hartmann number increases. As a consequence, increasing Hartmann number induces the decrease of entropy generation. On increasing thermal Grashof number (Gr

_{T}= 10

^{4}), same results are obtained as the previous case for higher values of Hartmann number (i.e., 25 ≤ Ha ≤ 100) and inversely, an oscillatory behaviour of entropy generation is observed for lower values of Hartmann number (0 ≤ Ha ≤ 10) before achieving the stationary state (see Figure 3b). For relatively considerable thermal Grashof number values (Gr

_{T}≥ 10

^{5}), Figure 3c shows an oscillatory behaviour of entropy generation at the very beginning of transient state for lower values of Hartmann number (0 ≤ Ha ≤ 35). A critical Hartmann number Ha

_{c}= 40 is obtained from which the oscillatory behaviour vanishes and entropy generation amplitude quickly reaches a maximum value before decreasing asymptotically towards the stationary state. As a consequence, amplitude and oscillation numbers of entropy generation are important as thermal Grashof number increases and Hartmann number decreases. Figure 4 depicts transient entropy generation for Gr

_{T}= 10

^{4}and 10

^{5}at different Hartmann number values for Pr = 0.02. Figures 4a and 4b show that at fixed Hartmann number, entropy generation increases with time towards a constant value. As Hartmann number increases, asymptotic behaviour of entropy generation is quickly achieved with decreased amplitude of irreversibility. As a consequence, increasing Hartmann number induces the decrease of entropy generation. This is due to the fact that the magnetic field causes the fluid velocity deceleration and the decrease in heat transfer.

_{T}= 10

^{5}and Ha = 0 as an illustrative example as shown in Figure 5.

**Figure 3.**Total dimensionless entropy generation (σ

_{t}) versus dimensionless time for Pr = 0.71, α = 0° and χ

_{1}= 10

^{−}

^{3}at different Hartmann number values: (a) Gr

_{T}= 10

^{3}, (b) Gr

_{T}= 10

^{4}, (c) Gr

_{T}= 10

^{5}.

**Figure 4.**Total dimensionless entropy generation (σ

_{t}) versus dimensionless time for Pr = 0.02, α = 0°, and χ

_{1}= 10

^{−}

^{3}at different Hartmann number values: (a) Gr

_{T}= 10

^{4}, (b) Gr

_{T}= 10

^{5}.

**Figure 5.**Nusselt number versus time at Gr

_{T}= 10

^{5}, χ

_{1}= 10

^{−}

^{3}and Ha = 0 for Pr = 0.02 and 0.71.

_{T}≥ 10

^{4}for small Hartmann number values (Figures 3b-3c). This result is consistent with the findings of Magherbi et al. [51] when Gr

_{T}≥ 10

^{4}in absence of magnetic field. Magherbi et al. [51] showed that steady state is relatively far from equilibrium state, thus the system takes a spiral approach towards the stationary state corresponding to an oscillation of entropy generation over time. In this case the Prigogine’s theorem [52] is not verified and the system evolves in the non-linear branch of thermodynamics for irreversible processes. The asymptotic behaviour of entropy generation observed for Gr

_{T}≤ 10

^{3}(at any Hartmann number), for 10

^{3}< Gr

_{T}≤ 10

^{4}(when Ha ≥ 10) and for Gr

_{T}= 10

^{5}(when Ha ≥ 40), shows that the system evolves in the linear branch of thermodynamics for irreversible processes, where the famous Onsager reciprocity relations are applicable. In these cases, the stationary state is sufficiently close to the equilibrium state, as a consequence, entropy generation, quickly reaches a maximum value, then tends towards a constant value. In order to have clear idea, Figure 6 shows critical values of Hartmann number versus the inclination angle of the magnetic field for a fixed thermal Grashof number (Gr

_{T}= 10

^{5}). For Ha > Ha

_{c}, the system evolves in the linear branch of thermodynamics of irreversible processes. Further, for 0° ≤ α ≤ 90°, it was found that maximum Ha

_{c}(= 45) is obtained at α = 30° (see Figure 6), this indicates that the stationary regime is far from the equilibrium state as compared to other inclination angle values of the magnetic field. In this case, Ha

_{c}= 30, 35 and 40 for α = 90°, 60° and 0°, respectively. As an important conclusion, increasing the inclination angle of the magnetic field (30° ≤ α ≤ 90°) tends to decrease critical Hartmann number, and consequently to decrease transient oscillatory behaviour of entropy generation.

**Figure 6.**Hartmann number versus inclination angle of the magnetic field for Pr = 0.71, Gr

_{T}= 10

^{5}and χ

_{1}= 10

^{−}

^{3}.

**Figure 7.**Transient entropy generation for Gr

_{T}= 10

^{5}, χ

_{1}= 10

^{−}

^{3}and Ha = 0: (a) viscous irreversibility, (b) thermal irreversibility.

_{T}≥ 10

^{4}, critical thermal Grashof number versus Hartmann number was studied for fixed values of the inclination angle of the magnetic field and irreversibility coefficient χ

_{1}(i.e., α = 0° and χ

_{1}= 10

^{−3}). Curve plotted in Figure 8 depicts critical values of thermal Grashof number obtained at each Hartmann number value (0 ≤ Ha ≤ 50), thus for Gr

_{T}> Gr

_{c}, the stationary regime is far from the equilibrium state. Figure 8 is divided into two regions: (I) represents the region corresponding to higher values of thermal Grashof number for which the system evolves in the non linear branch of thermodynamics of irreversible processes. In this case, entropy generation exhibits oscillatory behaviour corresponding to a non equilibrium state. (II) represents the region corresponding to thermal Grashof number values for which the stationary regime is close to the equilibrium state. In this case, the oscillatory behaviour of entropy generation vanishes and the system evolves in the linear branch of thermodynamics of irreversible processes. It’s important to notice that as Hartmann number value increases, the system tends towards stationary state because critical thermal Grashof number from which the system evolves in non equilibrium state is obtained at higher values. As an illustrative example, Gr

_{c}≈ 10

^{4}at Ha = 10 and Gr

_{c}≈ 1.2 × 10

^{5}at Ha = 50.

_{T}= 10

^{4}and Ha > 50 for Gr

_{T}= 10

^{5}(see Figure 3 and Figure 4). As mentioned above, for Pr = 0.02, heat transfer by conduction mode is more pronounced than that for Pr = 0.71. This is proved by the plot of Nusselt number versus Hartmann number for Gr

_{T}= 10

^{4}and 10

^{5}as illustrated in Figure 9. As Hartmann number increases, Nusselt number tends towards unity and then isothermal lines become practically parallel to the active wall and the fluid velocity decreases.

**Figure 8.**Critical thermal Grashof number versus Hartmann number for Pr = 0.71, α = 0° and χ

_{1}= 10

^{−}

^{3}.

**Figure 9.**Nusselt number versus Hartmann at different thermal Grashof numbers for Pr = 0.71 and 0.02 at α = 0°.

_{T}= 10

^{4}, α = 0° and χ

_{1}= 10

^{−4}, 10

^{−3}and 10

^{−2}, respectively. As can be seen from Figure 10, entropy generation decreases as Hartmann number increases, and inversely, it increases with the increase of irreversibility coefficient value. Thus, the presence of the magnetic field (described by Hartmann number), tends to reduce fluid flowing inside the cavity, as Hartmann number increases, the major part of the fluid becomes practically immobile and the flow is simply described by approximately conduction mode for higher values of Hartmann number.

**Figure 10.**Total dimensionless entropy generation versus Hartmann number at α = 0°, Gr

_{T}= 10

^{4}at χ

_{1}= 10

^{−2}, 10

^{−3}, 10

^{−4}: (a) Pr = 0.71, (b) Pr = 0.02.

_{1}and thermal Grashof number Gr

_{T}are fixed and set to be 10

^{−}

^{2}and 10

^{5}, respectively. As can be seen from Figure 11, the study of irreversibility versus inclination angle of the magnetic field was performed for lower (respectively higher) Hartmann number. For lower Hartmann number (Ha = 10), Figures 11a, 11c show that irreversibility due to fluid friction is the major contribution of total entropy generation.

**Figure 11.**Total and local irreversibilities versus α at Gr

_{T}= 10

^{5}and χ

_{1}= 10

^{−}

^{2}: (a) Pr = 0.71 and Ha = 10, (b) Pr = 0.71 and Ha = 100, (c) Pr = 0.02 and Ha = 10, (d) Pr = 0.02 and Ha = 100.

_{2}= χ

_{1}Ha

^{2}= 10

^{−}

^{2}× 100

^{2}= 100, further, local irreversibility due to magnetic field is given by σ

_{l,a,Mag}= χ

_{2}(Usina – Vcosa)

^{2}, as a consequence total entropy generation is entirely dependent on magnetic irreversibility and convective currents are suppressed. For Pr = 0.71, Maximum value of irreversibilities due to magnetic field and viscous effects as well as total entropy generation is obtained at α = 90° (in this case σ

_{l,a,Mag}= 100(U)

^{2}), minimum values of total, magnetic and viscous irreversibilities are obtained at α ≈ 135°. For Pr = 0.02 (Figure 11d), maximum value of total entropy generation as well as that related to magnetic irreversibility is found at α = 60°. Minimum amplitude of magnetic as well as total irreversibilities is found at α = 90°, maximum viscous irreversibility is found also at α = 90°. From the previous figures, irreversibility due to thermal gradients is practically absent, this is due to the fact that the magnetic field induces a quasi equilibrium of isothermal lines through the cavity.

_{T}= 10

^{5}in presence of a magnetic field (Ha = 50) as illustrated in Figure 12 and Figure 13. It can be seen from Figure 12 that, as the direction of the external magnetic field changes from horizontal (α = 0°) to vertical (α = 90°), the flow increases up to angle α = 60° and 30° for Pr = 0.02 and 0.71, respectively. A higher value of the buoyancy force (Gr

_{T}= 10

^{5}) means that the effect of the magnetic field direction is so significant on the flow structure. Figure 13 depicts the corresponding isothermal lines. For liquid Gallium (Pr = 0.02), isothermal lines are practically parallel to the active walls, which means that heat transfer is conduced by conduction mode. For air, thermal boundary layer thickness decreases as the inclination angle increases inducing an increase in heat transfer from horizontal to vertical direction of the magnetic field.

^{3}≤ Gr

_{T}≤ 10

^{5}at χ

_{1}= 10

^{−3}. For Pr = 0.71, entropy generation lines increase in magnitude with thermal Grashof number and are confined through the active walls (isothermal walls), namely on bottom of the warmed wall and on top of the cooled wall due to the presence of strong thermal and velocity gradients in these regions. For liquid gallium, Figure 14b shows distribution of irreversibility through the whole cavity except its center. Magnitude of entropy generation lines increases with buoyancy effect. For small Prandtl number value, as thermal Grashof number increases, velocity gradients considerably increase causing the augmentation of local entropy generation. In presence of the magnetic field (i.e., for Ha = 50) and for Gr

_{T}= 10

^{5}and χ

_{1}= 10

^{−}

^{3}, Figure 15 shows the influence of the inclination angle of the magnetic field on entropy generation lines. Entropy generation is lower in magnitude around the center of the cavity. Entropy generates at a higher magnitude near the cavity walls. At α = 0°, both of the active walls act as strong concentrators of irreversibility due to higher value of magnetic irreversibility as shown above.

**Figure 14.**Entropy generation maps at Ha = 0 and χ

_{1}= 10

^{−}

^{3}: (a) Pr = 0.71, (b) Pr = 0.02, (1) Gr

_{T}= 10

^{3}, (2) Gr

_{T}= 10

^{4}, (3) Gr

_{T}= 10

^{5}.

**Figure 15.**Entropy generation maps at Gr

_{T}= 10

^{5}, χ

_{1}= 10

^{−}

^{3}, Ha = 50: (a) Pr = 0.71, (b) Pr = 0.02.

## 6. Conclusions

- For fixed value of the inclination angle of the magnetic field, transient entropy generation exhibits oscillatory behaviour for air when Gr
_{T}≥ 10^{4}at small values of Hartmann number (magnetic field). Asymptotic behaviour is obtained for considerable values of Hartmann number. Transient irreversibility always exhibits asymptotic behaviour for liquid gallium. Magnetic field induces the decrease of entropy generation magnitude. - For air, increasing the inclination angle of the magnetic field (30° ≤ α ≤ 90°), tends to decrease critical Hartmann number, and consequently to decrease transient oscillatory behaviour of entropy generation.
- For air and at fixed inclination angle of the magnetic field, increasing Hartmann number tends to increase critical Grashof number from which the system evolves in the non-linear branch of thermodynamics for irreversible processes concerning air.
- In steady state, for lower Hartmann number value (Ha = 10) and for relatively higher thermal Grashof number (Gr
_{T}= 10^{5}), maximum value of entropy generation is found at an inclination angle of the magnetic field, α = 90° and 60° for air and liquid gallium, respectively. For both fluids, irreversibility due to viscous effects is the major contribution of entropy generation. - In steady state, for higher Hartmann number (Ha = 100) and for relatively higher thermal Grashof number (Gr
_{T}= 10^{5}), for both studied fluids, entropy generation increases via the increase of magnetic irreversibility. Maximum value of irreversibility is also obtained at α = 90° and 60° for air and liquid gallium, respectively. - Increasing Hartmann number (Ha ≥ 50), induces the decrease of entropy generation magnitude for lower Prandtl number values.
- Heat transfer rate is always described by pure conduction mode for liquid gallium, whereas it presents oscillatory behaviour for air when Gr
_{T}≥ 10^{4}. - At local level and for relatively higher thermal Grashof number (Gr
_{T}= 10^{5}), entropy generation distribution is strongly dependent on magnetic field direction, magnitude of irreversibility lines increases up to 30°, then gradually decreases. No entropy is generated in the cavity center.

## Nomenclature

a | thermal diffusivity (m^{2}·s^{−1}) |

B | magnetic field (T) |

E | electric field (V m^{−1}) |

E_{M} | electromagnetic force (N) |

g | gravitational acceleration (m·s^{−}^{2}) |

h | heat transfer coefficient (W·m^{−2}·K^{−1}) |

Gr_{T} | thermal Grashof number |

Ha | Hartmann number |

Ha_{c} | critical Hartmann number |

J | current density (A·m^{−2}) |

k | thermal Conductivity (J·m^{−1}·s^{−1}·K^{−1}) |

L | length of the cavity (m) |

P | dimensionless pressure |

p | pressure (N·m^{−2}) |

Pr | Prandtl number |

S_{gen} | rate of entropy generation per unit volume (J·s^{−1}·K^{−1}·m^{−3}) |

T | temperature (K) |

T_{0} | mean Temperature (T_{0} = (T_{h} + T_{c})/2) (K) |

t | time (s) |

u, v | velocity components in x and y directions respectively (m·s^{−1}) |

U, V | dimensionless velocity components in X and Y directions respectively |

υ | dimensionless velocity vector |

w | dimensional velocity vector (m·s^{−1}) |

x, y | Cartesian coordinates (m) |

X, Y | dimensionless Cartesian coordinates |

## Greek letters

α | inclination angle of the magnetic field (°) |

β | thermal expansion coefficient (K^{−1}) |

ΔT | temperature difference (ΔT = T_{h} – T_{c} ) (K) |

χ_{i} | irreversibility coefficient (i=1, 2) |

μ | dynamic viscosity (kg·m^{−1}·s^{−1}) |

v | kinematic viscosity (m^{2}·s^{−1}) |

ρ | mass density (kg·m^{−3}) |

σ | dimensionless entropy generation |

σ_{e} | electric conductivity (Ω^{−1}·m^{−1}) |

θ | dimensionless temperature |

τ | dimensionless time |

Ω | volume of the system |

## Subscripts

a | dimensionless |

c | cold wall |

h | hot wall |

Mag | magnetic |

Th | thermal |

t | total |

Vis | viscous |

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**MDPI and ACS Style**

El Jery, A.; Hidouri, N.; Magherbi, M.; Brahim, A.B.
Effect of an External Oriented Magnetic Field on Entropy Generation in Natural Convection. *Entropy* **2010**, *12*, 1391-1417.
https://doi.org/10.3390/e12061391

**AMA Style**

El Jery A, Hidouri N, Magherbi M, Brahim AB.
Effect of an External Oriented Magnetic Field on Entropy Generation in Natural Convection. *Entropy*. 2010; 12(6):1391-1417.
https://doi.org/10.3390/e12061391

**Chicago/Turabian Style**

El Jery, Atef, Nejib Hidouri, Mourad Magherbi, and Ammar Ben Brahim.
2010. "Effect of an External Oriented Magnetic Field on Entropy Generation in Natural Convection" *Entropy* 12, no. 6: 1391-1417.
https://doi.org/10.3390/e12061391