# Analysis of the Magnetic Field Effect on Entropy Generation at Thermosolutal Convection in a Square Cavity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−2}power law. DeVahl Davis [2] studied the two-dimensional natural convection in a square cavity with differentially heated side walls and has suggested a bench mark solution. Valencia and Frederick [3] investigated the natural convection of air in square cavities with half-active and half-insulated vertical walls numerically for various Rayleigh numbers. They observed that the heat transfer rates could be controlled, to a certain extent, by varying the relative positions of the hot and cold elements. Saravanan and Kandaswamy [4] analyzed the convection in a low Prandtl number fluid driven by the combined mechanism of buoyancy and surface tension in the presence of a uniform vertical magnetic field. They showed that the heat transfer across the cavity from the hot wall to cold wall becomes poor for a decrease in thermal conductivity in the presence of a vertical magnetic field. Deng et al. [5] studied numerically a two-dimensional, steady and laminar natural convection in a rectangular enclosure with discrete heat sources on walls. They remarked that the heat source on the floor increases the thermal instability and acts as a proportional effect on convection, while the heat source on the side wall increases the thermal stability and acts as a reverse effect on convection. Nithyadevi et al. [6] investigated the effect of aspect ratio on the natural convection of a fluid contained in a rectangular cavity with partially thermally active side walls. They found that heat transfer rate increases with increase in the aspect ratio and when the cooling location is at the top of the enclosure.

^{5}the parabolic profile is destroyed. For Rayleigh number, Ra = 10

^{3}, and small Hartmann numbers, the flow and heat transfer are characterized by a parallel flow structure in the central region of the cavity. The conduction is the dominant mode of heat transfer and vertical velocity profiles and temperatures are almost parabolic. The number of convection rolls depends on the Rayleigh and the Hartmann numbers. The resulting Lorentz forces and their braking effect explain the seemingly strange shape of the velocity profiles with excessive intensities in regions near the corners. Unsteady double-diffusive magneto convection of water in an enclosure with Soret and Dufour effects around the density maximum has been numerically investigated by Nithyadevi and Yang [20]. They observed that the density inversion leaves strong effects on fluid flow, heat and mass transfer due to the formation of bi-cellular structure. The formation of dual cell structure and strength of each cell is always dependent on the density inversion parameter, thermal Rayleigh and Hartmann numbers. The heat and mass transfer rates decrease with an increase of Hartmann number. The heat and mass transfer rates are found to increase with increasing thermal Rayleigh number. The heat transfer rate increases and mass transfer rate decreases when the density inversion parameter increases in the presence of Soret and the absence of Dufour parameters. Hakan et al. [21] found in their numerical analysis that heat transfer increases with increasing amplitude of sinusoidal function and decreases with increasing of Hartmann number. However, heat transfer also increases with the increasing of Rayleigh number. Hartmann number can be a control parameter for heat transfer and flow field.

^{2}) and irreversibility coefficient (i.e., φ = 10

^{−4}), the heat transfer irreversibility is strongly dominant and the total entropy generation increases with the increase of the aspect ratio. For cavities with high Ra values (i.e., Ra = 10

^{5}) and φ = 10

^{−4}, fluid friction irreversibility is dominant and total entropy generation increases with the aspect ratio, reaches a maximum, then decreases. A peak point for the maximum total entropy generation exists. For irreversibility coefficient, φ = 10

^{−2}, the magnitudes of fluid friction irreversibilities are considerably greater than for those for φ = 10

^{−4}. Hence, the fluid friction entropy generation for a cavity with φ = 10

^{−2}is more dominant compared to that for φ = 10

^{−4}. Similar peak points for the total entropy generation are also observed for a cavity with φ = 10

^{−2}. The average Bejan number is a proper criterion to predict the domination of heat transfer or fluid friction irreversibilities for the entire domain. The total entropy generation in a cavity increases with Rayleigh number, however, the rate of increase depends on the aspect ratio. For the same Rayleigh number, the total entropy generation for a tall cavity may be less than that for a shorter one. The dominant contribution to entropy generation came from heat transfer irreversibilities, with fluid friction accounting for only a small fraction of the total entropy generation. Entropy generation in convective heat and mass transfer through an inclined enclosure is numerically investigated by Magherbi et al. [25]. They showed that entropy generation increases with the thermal Grashof number and the buoyancy ratio for moderate Lewis number. At local level, irreversibility due to heat and mass transfer are nearly identical and are localized in the bottom heated and the top cooled walls of the enclosure. The inclination angle of the cavity has more effect on entropy generation for thermal Grashof number Gr

_{t}≥ 10

^{4}. In this case, irreversibility increases towards a maximum value obtained for α = 45°, then decreases and tends towards unity value for inclination angle α ≈ 180°. The influence of an oriented magnetic field on entropy generation in natural convection flow for air and liquid gallium is numerically studied by Eljery et al. [26], they showed that transient entropy generation exhibits oscillatory behavior for air when Gr

_{t}≥ 10

^{4}at small values of Hartmann number. Asymptotic behavior is obtained for considerable values of Hartmann number. A transient irreversibility always exhibits asymptotic behavior for liquid gallium. They also showed that heat transfer rate is always described by pure conduction mode for liquid gallium, whereas it presents oscillatory behavior for air Gr

_{t}≥ 10

^{4}. Local irreversibility is strongly dependent on magnetic field direction, magnitude of irreversibility lines increases up to 30°, and then gradually decreases.

^{7}.They obtained that the flow is primarily dominated by thermal buoyancy effects, whereas for N > 1, the flow is mainly dominated by compositional buoyancy effects. The average Nusselt number Nu and the average Sherwood number Sh at the inner wall both are monotonic increasing functions of aspect ratio for N > 1. Compositional convection in a rotating annulus with opposing temperature and concentration gradients is investigated by [28]. Results show that the flow will keep steady even with very high frequency rotation. But for Ra

_{T}= 10

^{6}, the flow will become unsteady either when the ratio of buoyancy forces approaches to unity or when rotation frequency increases. A new LB-based entropy generation analysis algorithm is developed for complex systems [29,30]. Firstly, important features of entropy generation in a vertically concentric annular space are revealed: when Pr and curvature ratio are equal to unity, the time-averaged total entropy generation number is a monotonic increasing function of Ra, and there is an approximate linear relationship between the logarithm of S

_{total}and Ra. But the time-volume-averaged Bejan number has an inverse trend. The entropy intensely generates within two layers along the vertical walls. The differences between these two layers become being erased with curvature ratio increasing and they are point symmetric with respect to the geometric center of the cavity when this parameter is sufficiently big. The variations of total entropy generation and time-volume-averaged Bejan numbers with curvature ratio are slight, and they approach asymptotically to the values of their planar counterparts. The maximum of entropy generation number will jump from the inner wall to the outer wall with Ra and curvature ratio increasing. The total entropy generation number and average Bejan number both increase monotonously with Pr. Entropy generation inside disk driven rotating convectional flow is secondly investigated. The situation becomes complex when the influences of rotating of the disk and the buoyancy force both should be considered. The variations of the total entropy generation number and Bejan number depend on the detailed combination of Reynolds and Grashof numbers. Entropy generation due to heat/mass transfer of turbulent natural convection due to internal heat generation in a cavity is studied by [31]. It was found that the time-volume averaged Bejan number almost equals one and then decreases quickly against Ra increasing. Though the maximum of entropy generation number increases quickly with Ra, the time-averaged total entropy generation number changes in the opposite trend.

_{total}, the inlet Reynolds number Re and the equivalence ratio can be approximated as a linear increasing function. For the second time, entropy generation in hydrogen-enriched ultra-lean counter-flow methane-air non-premixed combustion confined is analyzed. Entropy generation in this kind of combustion is different from that reported in previous case: despite the fact that the whole domain can be divided into two parts similar to premixed case, the area of Zone I is expanded with equivalence ratio increasing, which is contrary to its premixed counterpart. In this kind of combustion, with more fuel mixture being injected into the combustion, the share of irreversibility due to mixing is reduced while that due to heat transfer increases, which is fully contrary to its premixed counterpart. However, with more H

_{2}being added into the fuel blends of this kind of non-premixed combustion, the share of irreversibility due to heat transfer is reduced while that, due to mixing increases, which is consistent with its premixed counterpart and is quite different from the co-flow non-premixed combustion.

_{total}) increases with Ra, and the relative total entropy generation rates are nearly insensitive to Ra when Ra ≤ 10

^{9}; Since N > 1, S

_{total}increases quickly and linearly with N and the relative total entropy generation rate due to diffusive irreversibility becomes the dominant irreversibility; S

_{total}increases nearly linearly with aspect ratio. The relative total entropy generation rate due to diffusive and thermal irreversibilities both are monotonic decreasing functions against aspect ratio while that due to viscous irreversibility is a monotonic increasing function with aspect ratio.

## 2. Governing Equations

_{h}, C’

_{h}) and (T’

_{c}, C’

_{c}), respectively while the two other walls are insulated and adiabatic. The fluid, is considered mixture (air and a gas diffusing species) as a Newtonian, Boussinesq incompressible fluid, their properties are described by its kinematic viscosity ν, its thermal and solutal diffusivities, α

_{T}and D, respectively and its thermal and solutal volumetric expansion coefficients β

_{T}and β

_{c}respectively. The mass density of the fluid is considered to vary linearly with temperature and concentration such as:

_{o}[1 − β

_{T}(T − T

_{o}) − β

_{c}(C − C

_{o})]

## 3. Formulation

_{0}) and temperature (T’

_{0}) in the denominator of Equation (14) with a single diffusing species, Equation (14) can therefore be written as follows:

_{th}), fluid friction (S

_{vis}), mass transfer (S

_{diff}) by pure concentrations gradients, mass transfer by mixed product of concentration and thermal gradients and magnetic field (S

_{mag}), respectively.

## 4. Numerical Procedure

## 5. Results and Discussion

^{3}≤ Gr

_{t}≤ 10

^{5}, 10

^{−7}≤ ${\lambda}_{1}$ ≤ 10

^{−4}, 10

^{−1}≤ ${\lambda}_{2}$ ≤ 0.5, 10

^{−5}≤ ${\lambda}_{3}$ ≤ 10

^{−2}and 0° ≤ α ≤ 180°. ${\lambda}_{2}$ and ${\lambda}_{3}$ are equal to 0 in natural convection. Grids of sizes of 31 × 31, 41 × 41 and 51 × 51 nodal points are used for Grt = 103, 104 and 105, respectively. A step time, Δt = 10

^{−4}for is used for all the studied thermal Grashof numbers.

**Table 1.**Comparison of average Nusselt number for different values of Rayleigh number in a square cavity with Pr = 0.71, N = 0, Ha = 0.

Ra = Gr × Pr | Davis [2] | Nithyadevi et al. [6] | Present Study |
---|---|---|---|

10^{3} | 1.118 | 1.123 | 1.099 |

10^{4} | 2.243 | 2.304 | 2.295 |

10^{5} | 4.519 | 4.899 | 4.664 |

_{t}= 10

^{5}). In absence of the magnetic field (i.e., Ha = 0), entropy generation quickly passes from a minimum value at the very beginning of the transient state towards a maximum value, then exhibits an oscillatory behaviour before reaching a constant value in steady state.

**Figure 2.**Dimensionless total entropy generation versus time for: N = 0; β = 90°; α = 0°; Ra = 10

^{5}.

^{5}). Maximum value of Nusselt number is obtained at about α = 90° for which entropy generation is also maximum for both studied Hartmann number values (see Figure 3b). This is due to the increased value of thermal and velocity gradients at this point. Increasing Hartmann number induces the decrease of heat transfer and consequently the dissipated energy expressed by entropy generation. It is important to notice that minimum of entropy generation is obtained for α ≈ 140°.

**Figure 3.**(a) Average Nusselt number and (b) Dimensionless total entropy generation versus inclination angle for different Hartmann numbers: Ra = 10

^{5}; β = 90°; N = 0.

**Figure 4.**(a) Dimensionless total entropy generation versus Hartmann number for different buoyancy ratios and (b) Thermal entropy generation versus buoyancy ratio for different Hartmann numbers: Ra = 10

^{5}; β = 90°; α = 0°.

**Figure 5.**(a) Magnetic and Viscous entropy generation and (b) Diffusion entropy generation versus buoyancy ratio: Ra = 10

^{5}; β = 90°; α = 0°.

_{cr}. Both Nu and Sh tend to decrease with increasing values of N for N < N

_{cr}and to increase with increasing values of N for N > N

_{cr}. Both average Nu and Sh increase and decrease when the buoyancy forces assist or oppose, respectively those from thermal origin. The profiles of Nu and Sh with N tended to be symmetric about the value of N

_{cr}. For a horizontal magnetic field, results show that average Nu and Sh increase in the beginning, reach maximums values and then decrease towards the pure conduction regime.

**Figure 6.**(a) Midsection x-component velocity at y = 0.5 and (b) midsection y-component velocity at x = 0.5 for different Hartmann numbers: Ra = 10

^{5}; β = 90°; α = 0°; N = 0.

**Figure 7.**Midsection y-component temperature at y = 0.5 for different Hartmann numbers: Ra = 10

^{5}; β = 90°; α = 0°; N = 0.

**Figure 8.**Midsection y-component temperature (a) and concentration (b) at y = 0.5 for different buoyancy ratios Ra = 10

^{5}; Ha = 25; β = 90°; α = 0°.

**Figure 9.**Average Nusselt number (a) and Total entropy generation (b) versus Hartmann number for different buoyancy ratios and Rayleigh numbers, β = 90°; α = 0°.

**Figure 10.**Isotherms, Streamlines and Isentropic lines for N = 0 with different values of Hartmann number for α = 0°; Gr = 10

^{4}.

## 6. Conclusions

_{t}= 10

^{4}. Contributions of thermal, diffusive, friction and magnetic terms on entropy generation are investigated. The more effect was due to heat transfer and then to mass transfer. Results showed that magnetic effect is more pronounced than friction one. The magnetic field parameter suppresses the flow in the cavity and this lead to a decrease of entropy generation, temperature and concentration decrease with increasing value of the magnetic field parameter. At local level, results show that entropy generation lines are localized on lower heated and upper cooled regions of the active walls.

## Nomenclature

B | magnetic field (T) |

C | dimensionless concentration |

C’ | concentration (mol∙m ^{−3}) |

${C}_{h}^{\prime}$ | hot side concentration (mol∙m ^{−3}) |

${C}_{c}^{\prime}$ | cold side concentration (mol∙m ^{−3}) |

${C}_{0}^{\prime}$ | bulk concentration (mol∙m ^{−3}) |

C_{p} | specific heat (J∙Kg∙K−1) |

D | mass diffusivity (m ^{2}∙s^{−1}) |

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

Gr_{t} | thermal Grashof number |

Gr_{c} | solutal Grashof number |

H (L) | height (length) of the cavity (m) |

Ha | Hartmann number |

J_{k} | diffusion Flux (k = u, v, T, C) |

Le | Lewis number |

N | Buoyancy ratio |

Nu | Nusselt number |

${\mathrm{{\rm N}}}_{s,\ell}$ | dimensionless local entropy generation |

$S$ | dimensionless total entropy generation |

P | pressure (kg∙m ^{−1}∙s^{−2}) |

Pr | Prandtl number |

Ra_{T} | Rayleigh number |

Sc | Schmidt number |

Sh | Sherwood number |

$\stackrel{*}{{S}_{gen}}$ | local Volumetric entropy generation (J∙m ^{−3}∙s^{−1}∙K^{−1}) |

T | dimensionless temperature |

T’ | temperature (K) |

t’ | time (s) |

t | dimensionless Time |

T_{h}’ | hot side temperature (K) |

T’_{c} | cold side temperature (K) |

T’_{o} | bulk temperature (K) |

u, v | dimensionless velocity components |

U^{*} | characteristic Velocity (m∙s ^{−1}) |

V | velocity vector (m∙s ^{−1}) |

u’, v’ | velocity components along x’, y’ respectively (m∙s ^{−1}) |

x, y, z | dimensionless Coordinates |

x’, y’, z’ | cartesian coordinates (m) |

Greek Symbols | |

α | magnetic field’s angle with horizontal direction (°) |

${\alpha}_{T}$ | thermal diffusivity (m ^{2}∙s^{−1}) |

β | inclination angle of the cavity (°) |

β_{T} | thermal expansion coefficient |

β_{c} | compositional expansion coefficient |

${\lambda}_{i}$ | irreversibility distribution ratios, (I = 1, 2, 3, 4) |

μ | dynamic viscosity of the fluid (kg∙m ^{−1}∙s^{−1}) |

ρ | fluid density (kg∙m ^{−3}) |

σ_{e} | electrical conductivity (Ω ^{−1}∙m^{−1}) |

υ | kinematics’ viscosity (m ^{2}∙s^{−1}) |

ΔT’ | temperature difference (K) |

ΔC’ | concentration difference (mol∙m ^{−3}) |

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**MDPI and ACS Style**

Bouabid, M.; Hidouri, N.; Magherbi, M.; Brahim, A.B.
Analysis of the Magnetic Field Effect on Entropy Generation at Thermosolutal Convection in a Square Cavity. *Entropy* **2011**, *13*, 1034-1054.
https://doi.org/10.3390/e13051034

**AMA Style**

Bouabid M, Hidouri N, Magherbi M, Brahim AB.
Analysis of the Magnetic Field Effect on Entropy Generation at Thermosolutal Convection in a Square Cavity. *Entropy*. 2011; 13(5):1034-1054.
https://doi.org/10.3390/e13051034

**Chicago/Turabian Style**

Bouabid, Mounir, Nejib Hidouri, Mourad Magherbi, and Ammar Ben Brahim.
2011. "Analysis of the Magnetic Field Effect on Entropy Generation at Thermosolutal Convection in a Square Cavity" *Entropy* 13, no. 5: 1034-1054.
https://doi.org/10.3390/e13051034