# Effects of Radiation Heat Transfer on Entropy Generation at Thermosolutal Convection in a Square Cavity Subjected to a Magnetic Field

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{T}= 10

^{4}. Entropy generation increases with the buoyancy forces that induce an augmentation of exchanged heat between the flow and the walls. Hidouri et al. [15] studied the influence of Soret effect on entropy generation in double diffusive convection for the case of a binary perfect gas mixture for both aiding and opposing buoyancy forces through a square cavity. They showed that entropy generation takes a minimum value for the case of opposing buoyancy forces with equal intensity (i.e., N = −1) when thermal Grashof values Gr

_{T}≥ 10

^{5}. In a similar way, the influence of Dufour effect on entropy generation in convective heat and mass transfer for a binary perfect gas mixture was numerically studied by Magherbi et al. [16]. They showed that entropy generation takes also a minimum value when N = −1 for values of thermal Grashof number Gr

_{T}≥ 10

^{4}.

## 2. Problem Statement

_{h}, C’

_{h}) and (T’

_{c}, C’

_{c}), while the two horizontal walls are insulated and adiabatic. Radiatively, fluid nonparticipating medium and all surfaces are gray; the radiative heat flux in y direction is negligible as compared to that in the x direction. The fluid, consists of a binary mixture of air and a diffusing species and considered as a Newtonian, Boussinesq incompressible fluid whose properties are described by its kinematic viscosity ν, its thermal and solutal diffusivities, α

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

_{T}and β

_{c}. The mass density of the fluid varies linearly with temperature and concentration, such that:

_{o}[1 − β

_{T}(T’ − T’

_{o}) − β

_{c}(C’ − C’

_{o})]

## 3. Analysis

#### 3.1. Governing Equations

#### 3.2. Initial and Boundary Conditions

#### 3.3. Entropy Generation

#### 3.4. Numerical Solution

^{3},10

^{4}and 10

^{5}are found sufficiently enough to achieve convergence criterion given by equation of continuity such that:

^{−4}for all considered Thermal Rayleigh 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 [20] | Nithyadevi et al. [21] | 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 |

Grid Size | Ra | Nu | Er (%) |
---|---|---|---|

21 × 21 | 1,000 | 1.1024 | - |

31 × 31 | 1.0992 | 0.29 | |

41 × 41 | 1.0979 | 0.408 | |

31 × 31 | 10,000 | 2.3201 | - |

41 × 41 | 2.2956 | 1.055 | |

51 × 51 | 2.2832 | 1.59 | |

41 × 41 | 100,000 | 4.7097 | - |

51 × 51 | 4.6641 | 0.968 | |

61 × 61 | 4.6593 | 1.07 |

## 4. Results and Discussions

^{3}≤ Ra ≤ 10

^{5}; 10

^{−7}≤ ${\lambda}_{1}$ ≤ 10

^{−4}, 10

^{−1}≤ ${\lambda}_{2}$ ≤ 0.5; 10

^{−5}≤ ${\lambda}_{3}$ ≤ 10

^{−2}and 0° ≤ α ≤ 180°, respectively. ${\lambda}_{2}$ and ${\lambda}_{3}$ are equal to 0 in natural convection regime.

^{3}≤ Ra ≤ 10

^{4}, entropy generation quickly decreases from a maximum value at the beginning of the transient state), then asymptotically decreases towards a constant value in steady state, showing that the system evolves in the linear branch of thermodynamics for irreversible processes. On increasing the thermal Rayleigh number (i.e., Ra ≥ 10

^{5}), transient entropy generation exhibits an oscillatory behavior showing that the system is in a spiral approach corresponding to non linear branch of thermodynamics for irreversible processes. As time proceeds, entropy generation reaches a maximum then asymptotically decreases towards a constant value in steady state.

^{5}is illustrated in Figure 3a and Figure 3b. As it can be seen, magnetic and radiation parameters induce the decrease of heat transfer expressed by Nusselt number towards unity value. Thus radiation effect reduces Nusselt number and homogenizes the temperature inside the cavity by reducing the temperature gap between the two insulated walls. It enhances the heat transfer inside the cavity. Magnetic effect by Lorentz force suppresses the flow. The increase of heat transfer inside the enclosure induces the increase of entropy generation as illustrated in Figure 3b. It is important to notice that the increase of Hartmann number considerably induces the decrease of entropy generation regardless the radiation parameter.

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

^{5}; N = 0.

**Figure 4.**Dimensionless entropy generation versus radiation parameter for N = −1 and Ha = 25 (a) and versus buoyancy ratios for different radiation parameter values at Ha = 25 (b) Ra = 10

^{5}; α = 0°.

**Figure 5.**Average Nusselt number (a) Average Sherwood number (b) versus buoyancy ratio for different radiation parameters (c) Nusselt and Sherwood numbers versus Hartmann number for different radiation parameters: Ra = 10

^{5}; α = 0°.

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

^{5}; α = 0°; Ha = 25; N = 0.

**Figure 7.**Midsection y-component temperature (a) and concentration (b) at y = 0.5 for different Radiation parameters Ra = 10

^{5}; α = 0°; Ha = 25; N = 0.

**Figure 8.**Average Nusselt number (a) Average Sherwood number (b) and Total entropy generation (c) versus inclination angle of magnetic field for different radiation parameters: Ra = 10

^{5}; α = 0°; Ha = 25 and N = 0.

^{5}). Figure 11 shows the case where thermal and concentration gradients have the same amplitude and act in opposite way (i.e., N = −1). In this case, isothermal lines are mainly parallel to active walls showing a quasi conduction mode with a unicellular structure and increasing two cells structure of entropy generation as radiation parameter enhances both heat and mass transfer inside the enclosure.

**Figure 9.**Isothermal lines, Stream lines and Isentropic lines for Ha = 0; N = 0 and Ra = 10

^{5}with different radiation parameter at horizontal magnetic field.

**Figure 10.**Isothermal lines, Stream lines and Isentropic lines for Ha = 25; N = 0 and Ra = 10

^{5}with different radiation parameter at horizontal magnetic field.

**Figure 11.**Isothermal lines, Stream lines and Isentropic lines for Ha = 25; N = −1 and Ra = 10

^{5}with different radiation parameter at horizontal magnetic field.

## 5. Conclusions

## Nomenclature

A | aspect Ratio of the cavity |

C | dimensionless concentration |

C’ | concentration (mol∙m ^{−3}) |

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

C_{p} | isobaric specific heat (J∙Kg ^{−1}∙K^{−1}) |

D | species 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 | dimensionless diffusion flux |

k * | mean absorption coefficient (m ^{−1}) |

Le | Lewis number |

N | Buoyancy ratio |

Nu | Nusselt number |

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

S | dimensionless total entropy generation |

Nr | Radiation parameter |

P | pressure (N∙m ^{−2}) |

Pr | Prandtl number |

$R{a}_{T}$ | Rayleigh number |

Sc | Schmidt number |

Sh | Sherwood number |

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

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

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

T | dimensionless temperature |

T’ | temperature (K) |

t | dimensionless time |

t’ | time (s) |

T_{c}’ | hot side temperature (K) |

T’_{f} | cold side temperature (K) |

T’_{o} | bulk temperature (K) |

u, v | dimensionless velocity components |

V | dimensionless velocity vector |

U^{*} | characteristic velocity (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 (°) |

α_{T} | thermal diffusivity (m ^{2}∙s^{−1}) |

Β | inclination angle of the cavity (°) |

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

β_{c} | compositional expansion coefficient (mol ^{−1}∙m^{3}) |

λ_{i} | irreversibility distribution ratio, (i = 1, 2, 3, 4) |

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

ρ | fluid density (kg∙m ^{−3}) |

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

σ_{0} | Stephan-Boltzmann constant |

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

ΔT’ | temperature difference (K) |

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

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**MDPI and ACS Style**

Hidouri, N.; Bouabid, M.; Magherbi, M.; Brahim, A.B.
Effects of Radiation Heat Transfer on Entropy Generation at Thermosolutal Convection in a Square Cavity Subjected to a Magnetic Field. *Entropy* **2011**, *13*, 1992-2012.
https://doi.org/10.3390/e13121992

**AMA Style**

Hidouri N, Bouabid M, Magherbi M, Brahim AB.
Effects of Radiation Heat Transfer on Entropy Generation at Thermosolutal Convection in a Square Cavity Subjected to a Magnetic Field. *Entropy*. 2011; 13(12):1992-2012.
https://doi.org/10.3390/e13121992

**Chicago/Turabian Style**

Hidouri, Nejib, Mounir Bouabid, Mourad Magherbi, and Ammar Ben Brahim.
2011. "Effects of Radiation Heat Transfer on Entropy Generation at Thermosolutal Convection in a Square Cavity Subjected to a Magnetic Field" *Entropy* 13, no. 12: 1992-2012.
https://doi.org/10.3390/e13121992