# Entropy Generation of Double Diffusive Forced Convection in Porous Channels with Thick Walls and Soret Effect

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## Abstract

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## 1. Introduction

_{2}nanofluid. Similar to the investigation of Mourad et al. [22], Chen et al. [24] also showed that the buoyancy ratio could have influential impact on the total entropy generation rate and optimization of the system from a second law perspective. Most recently, Kefayati [28] extended the solution provided to a case of an open cavity [29] with entropy generation, while considering Soret and Dufour effects. A non-Newtonian power law fluid filled the cavity and a horizontal magnetic field was included in the modeling. Similar to previous investigations [8,13,14], a finite difference lattice Boltzmann method was employed, and both local and total entropy generation rates were illustrated and discussed.

## 2. Problem Statement

## 3. Results and Discussion

#### 3.1. Temperature and Local Entropy Generation Rate

#### 3.2. Total Entropy Generation Rate

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## Nomenclature

${B}_{0}$ | magnetic field, $\mathrm{T}$ |

$\mathrm{Br}$ | Brinkman number ($\mathrm{Pr}\cdot \mathrm{Ec}$) |

$Da$ | Darcy number |

$\mathrm{Ec}$ | Eckert number |

$h$ | convection heat transfer (Case two), $\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-1}$ |

${h}_{3}$ | height of the channel, $\mathrm{m}$ |

${k}_{1}$ | reference thermal conductivity for lower solid material, $\mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1}$ |

${k}_{2}$ | reference thermal conductivity for upper solid material, $\mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1}$ |

${k}_{eff}$ | effective thermal conductivity of porous medium, $\mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1}$ |

${k}_{e1}$ | ratio of porous medium thermal conductivity to lower solid material thermal conductivity |

${k}_{e2}$ | ratio of porous medium thermal conductivity to upper solid material thermal conductivity |

${N}_{s}$ | dimensionless local entropy generation rate |

$Nc$ | dimensionless convection heat transfer (Case two) |

$M$ | Hartmann number |

$\mathrm{Pr}$ | Prandtl number |

${Q}_{1}$ | dimensionless volumetric internal heat generation rate for the lower solid material |

${Q}_{2}$ | dimensionless volumetric internal heat generation rate for the upper solid material |

${Q}_{H}$ | dimensionless heat flux boundary condition (Case two) |

${Q}_{p}$ | dimensionless volumetric internal heat generation rate for the porous medium |

${\dot{q}}_{1}$ | volumetric internal heat generation rate for the lower solid material, $\mathrm{W}\cdot {\mathrm{m}}^{-3}$ |

${\dot{q}}_{2}$ | volumetric internal heat generation rate for the upper solid material, $\mathrm{W}\cdot {\mathrm{m}}^{-3}$ |

${q}_{H}$ | heat flux boundary condition (Case two), $\mathrm{W}\cdot {\mathrm{m}}^{-2}$ |

${\dot{q}}_{p}$ | volumetric internal heat generation rate for the porous medium, $\mathrm{W}\cdot {\mathrm{m}}^{-3}$ |

$Rd$ | dimensionless radiation parameter |

${\dot{S}}^{\u2034}$ | local entropy generation rate, $\mathrm{W}\cdot {\mathrm{m}}^{-3}\cdot {\mathrm{K}}^{-1}$ |

$T$ | temperature, $\mathrm{K}$ |

${T}_{1}$ | temperature of the lower solid material, $\mathrm{K}$ |

${T}_{2}$ | temperature of the upper solid material, $\mathrm{K}$ |

${T}_{C}$ | outer temperature of the upper solid material, $\mathrm{K}$ |

${T}_{H}$ | inner temperature of the lower solid material, $\mathrm{K}$ |

${T}_{p}$ | temperature of the porous medium, $\mathrm{K}$ |

$U$ | dimensionless velocity |

$u$ | velocity of the fluid in porous medium, $\mathrm{m}\cdot {\mathrm{s}}^{-1}$ |

${u}_{r}$ | characteristics velocity |

## Greek symbols

$\kappa $ | permeability, ${\mathrm{m}}^{2}$ |

${\kappa}^{*}$ | Rosseland mean absorption coefficient |

$\epsilon $ | porosity |

${\mu}_{f}$ | dynamic viscosity of the base fluid, $\mathrm{Kg}\cdot {\mathrm{s}}^{-1}\cdot {\mathrm{m}}^{-1}$ |

$\theta $ | dimensionless temperature |

${\theta}_{1}$ | dimensionless temperature of the lower solid material |

${\theta}_{2}$ | dimensionless temperature of the upper solid material |

${\theta}_{p}$ | dimensionless temperature of the porous medium |

${\theta}_{H}$ | dimensionless temperature at outer side of the lower wall |

$\sigma $ | electrical conductivity of fluid, $\mathrm{S}\cdot {\mathrm{m}}^{-1}$ |

${\sigma}^{*}$ | Stefan-Boltzmann constant, $\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-4}$ |

## Appendix A

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**Figure 1.**Configuration of the microchannels filled with a porous material; (

**a**) Case one, (

**b**) Case two.

**Figure 3.**Effect of radiation parameter on the local entropy generation rate in the porous channels.

**Figure 5.**Effect of Damkohler parameter on the local entropy generation rate in the porous channels.

**Figure 6.**Variation of the total entropy generation rate versus radiation parameter with various values for Soret parameters.

**Figure 7.**Variation of the total entropy generation rate versus Damkohler parameter with various values for (

**a**) hot wall temperature, (

**b**) convection coefficient and (

**c**) heat flux.

**Figure 8.**Variation of the total entropy generation rate versus lower wall’s thickness with various values for Damkohler parameter.

**Figure 9.**Variation of the total entropy generation rate versus lower wall’s thickness with various values for ${Y}_{2}$.

**Figure 10.**Variation of the total entropy generation rate versus ${Y}_{2}$ with various values for Damkohler parameter.

**Figure 11.**Variation of the total entropy generation rate versus ${Y}_{2}$ with various values for the lower wall’s thickness.

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**MDPI and ACS Style**

Torabi, M.; Torabi, M.; Peterson, G.P.B.
Entropy Generation of Double Diffusive Forced Convection in Porous Channels with Thick Walls and Soret Effect. *Entropy* **2017**, *19*, 171.
https://doi.org/10.3390/e19040171

**AMA Style**

Torabi M, Torabi M, Peterson GPB.
Entropy Generation of Double Diffusive Forced Convection in Porous Channels with Thick Walls and Soret Effect. *Entropy*. 2017; 19(4):171.
https://doi.org/10.3390/e19040171

**Chicago/Turabian Style**

Torabi, Mohsen, Mehrdad Torabi, and G.P. Bud Peterson.
2017. "Entropy Generation of Double Diffusive Forced Convection in Porous Channels with Thick Walls and Soret Effect" *Entropy* 19, no. 4: 171.
https://doi.org/10.3390/e19040171