# Entropy Generation and MHD Convection within an Inclined Trapezoidal Heated by Triangular Fin and Filled by a Variable Porous Media

^{1}

^{2}

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## Abstract

**:**

_{2}O

_{3}-Cu hybrid nanofluids. The porosity is varied exponentially with the smallest distance to the nearest wall and the permeability is depending on the particle diameter. Because of using the two energy equations model (LTNEM), sources of the entropy are entropy due to the transfer of heat of the fluid phase, entropy due to the fluid friction and entropy due to the porous phase transfer of heat. A computational domain with new coordinates (ξ,η) is created and Finite Volume Method (FVM) in case of the non-orthogonal grids is used to solve the resulting system. Various simulations for different values of the inclination angle, Hartmann number and alumina-copper concentration are carried out and the outcomes are presented in terms of streamlines, temperature, fluid friction entropy and Bejan number. It is remarkable that the increase in the inclination angle causes a diminishing of the heat transfer rate. Additionally, the irreversibility due to the temperature gradients is dominant near the heated fins, regardless of the values of the Hartmann number.

## 1. Introduction

- Using an irregular flow domain heated by irregular fin that was not presented before.
- Properties of the porous medium are considered variables and this assumption makes the work more attractive to readers.
- Physically, the local thermal non-equilibrium state (two-energy equations model) is more realistic than the local thermal equilibrium case.
- Most of the published works in this field consider the magnetic field as inclined and the geometry as non-inclined, and hence, the formulation of the governing system when the magnetic force is horizontal and the geometry is inclined is unusual and novel.
- Analyses of the second law of thermodynamics for such kinds of complex geometries (triangular fin within a trapezoidal enclosure) have not been presented before.
- The numerical methodology which depends on mapping between the real–irregular domain and regular–rectangular computational domain has received less focus.
- Finally, various practical applications for the current simulations can be found, e.g., air conditioning systems in buildings, furnace and home heating, electronic equipment cooling, drying foods and double pane windows.

## 2. Formulation of the Problem

#### 2.1. Correlations of the Hybrid Nanofluids

#### 2.2. Heat Transfer Coefficient

#### 2.3. Entropy Generation Analysis

## 3. Numerical Treatments

## 4. Discussion of Results

_{f}decreases as the $\gamma $ increases. The local entropy generation for $\gamma $ equals zero appears to have the maximum values near the side walls. As the $\gamma $ increases, this maximum value takes place at the upper wall of the cavity. Moreover, the graph shows that for $\gamma $, $Be$ less than 0.5 contours occupies most of the domain of the cavity; this is mainly due to the friction. Additionally, Be contours greater than 0.5 are observed at or near the sidewalls of the cavity as well as the fin walls. For these cases where $Be$ is greater than 0.5, the entropy generation is mainly due to the transfer of heat.

_{f}decreases as $Ha$ increases. Moreover, it is noted from $Be$ contours and for all cases of $Ha$ that the local entropy generation is dominated by the thermal component near the fin walls and at the corners of the cavity; while it is dominated by the friction at a small portion of the top wall of the cavity.

_{T}increases. Moreover, the graph shows that as the $\varphi $ increases, S

_{T}increases as well. Higher volume fraction is associated with better conductivity and hence a higher entropy generation due to heat is achieved.

_{total}increases as Ra increases. Furthermore, the graph shows for the cases of $Ra={10}^{4}$ and $Ra={10}^{5}$, as $\varphi $ increases, the S

_{total}increases. Finally, the graph shows that for the case where $Ra={10}^{6}$, as the $\varphi $ increases, the S

_{total}increases. This is mainly because the entropy generation due to the heat increases with the volume fraction while the entropy generation due to friction decreases.

_{TS}increases as $Ra$ increases. Additionally, the graph shows that as the $\varphi $ increases, the S

_{TS}decreases. Physically, the increase in the concentration of the nanoparticles (${\varphi}_{Al},{\varphi}_{Cu}$) enhances the temperature gradients for the fluid phase which results in a diminishing in the temperature gradients of the solid phase, and hence ${S}_{Ts}$ is reduced.

_{f}in the entire domain for the variation of the Ra and nanoparticle volume fraction ${\varphi}_{Al}={\varphi}_{Cu}$ at$\text{}Ha=15,\text{}H=0.1,\text{}\gamma =\pi /6,\text{}\mathsf{\Phi}=\frac{\pi}{3}.$ The graph shows that as $Ra$ increases, t S

_{f}increases. Moreover, for low Ra values, the entropy generation is almost constant with the $\varphi $; while this value decreases as the $\varphi $ increases at Ra= 10

^{6}. In fact, high values of $Ra$ ($Ra={10}^{6})$ make the velocity fields more affected by variations of $\varphi $. In addition, an increase in $\varphi $ augments the overall viscosity which in turn decreases the gradients of the velocity, and hence, the total entropy is reduced.

## 5. Conclusions

- 1
- The irreversibility due to the transfer of heat is dominant along the fin boundary in the case of a horizontal domain, while in the case of a vertical geometry, it is dominant near the boundaries of the trapezoidal.
- 2
- Activity of the hybrid nanofluid as well as the fluid friction entropy are diminished as the $Ha$ grows.
- 3
- The transfer of heat entropy is augmented as the concentration of the nanoparticles is boosted.The increase in the $\gamma $ reduces the Nusselt number while it increases as the Rayleigh number or the volume fraction parameter are growing.
- 4
- From the obtained results, it is recommended to use a non-inclined irregular domain to enhance the heat transfer rate. Additionally, the variable-properties porous medium is more realistic than the porous medium with constant properties.
- 5
- It is recommended to use hybrid nanofluids for enhancement of the heat transfer instead of mono nanofluids.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviation

Nomenclature | |

$A$ | Aspect ratio |

$Be$ | Bejan number |

${c}_{p}$ | Specific heat capacity $\left({\mathrm{J}\text{}\mathrm{kg}}^{-1\text{}}{\text{}\mathrm{K}}^{-1}\right)$ |

${C}_{f}$ | Inertial coefficient |

$Da$ | Darcy number |

${d}_{p}$ | Solid particles diameter (m) |

$\mathrm{g}$ | Gravity acceleration $\left({\mathrm{m}\text{}\mathrm{s}}^{-2}\right)$ |

$h$ | Heat-transfer coefficient [${\mathrm{W}\text{}\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$] |

$Ha$ | Hartmann number |

$k$ | Thermal conductivity $\left({\mathrm{W}\text{}\mathrm{m}}^{-1}{\text{}\mathrm{K}}^{-1}\right)$ |

$K$ | Porous medium permeability $\left({\mathrm{m}}^{2}\right)$ |

$Kr$ | Thermal conductivity ratio |

$L$ | Bottom wall length $\left(\mathrm{m}\right)$ |

$Nu$ | Nusselt number |

$P$ | Pressure $\left({\mathrm{N}\text{}\mathrm{m}}^{-2}\right)$ |

$Pr$ | Prandtl number |

$Ra$ | Rayleigh number |

$ST$ | Entropy generation due to the heat transfer |

$Sf$ | Entropy generation due to the fluid friction |

${S}_{total}$ | Total entropy |

$t$ | Time $\left(\mathrm{s}\right)$ |

$T$ | Temperature$\left(\mathrm{K}\right)$ |

$\left({u}_{p},{v}_{p}\right)$ | Dimensional velocity component (${\mathrm{ms}}^{-1}$) |

$\left({U}_{p},{V}_{p}\right)$ | Dimensionless velocity component |

$\left(x,y\right)$ | Cartesian coordinates (m) |

$\left(X,Y\right)$ | Dimensionless Cartesian coordinates |

Greek Symbols | |

$\alpha $ | Thermal diffusivity $\left({\mathrm{m}}^{2}\text{}{\mathrm{s}}^{-1}\right)$ |

$\beta $ | Coefficient of thermal expansion $\left({\mathrm{K}}^{-1}\right)$ |

$\gamma $ | Inclination angle of the cavity vector |

$\tau $ | Dimensionless time |

$\theta $ | Dimensionless temperature |

$\mu $ | Dynamic viscosity $\left({\mathrm{kg}\text{}\mathrm{m}}^{-1\text{}}{\text{}\mathrm{s}}^{-1}\right)$ |

$\nu $ | Kinematic viscosity $\left({\mathrm{m}}^{2}\text{}{\mathrm{s}}^{-1}\right)$ |

$\rho $ | Density $\left({\mathrm{kg}\text{}\mathrm{m}}^{-3\text{}}\right)$ |

$\varphi $ | Solid volume fraction |

$\Phi $ | Trapezoidal angle |

$\sigma $ | Electrical conductivity |

$\epsilon $ | Porosity |

($\xi ,\eta )$ | Coordinates of the rectangular domain |

Subscripts | |

eff | Effective |

$f$ | Fluid |

$\mathrm{P}$ | Porous medium |

$hnf$ | Hybrid Nanofluid |

$h$ | Hot |

$fp$ | Fluid phase |

$p$ | Porous phase |

$C$ | Cold |

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**Figure 3.**Comparisons of the streamlines, isotherms, local Bejan number and local entropy generation due to heat transfer with those of Ilis et al. [47] $Pr=0.71,Ra={10}^{5},{\phi}_{1}={10}^{-4}$.

**Figure 4.**Features of the streamlines and temperature distributions for the fluid phase and porous phase for the variation of the inclination angle $\gamma $ ($\gamma =0,\frac{\pi}{6},\frac{\pi}{3},\frac{\pi}{2}$) at $Ra={10}^{6},Ha=15,H=0.1,{\varphi}_{Al}={\varphi}_{Cu}=1\%,\mathsf{\Phi}=\frac{\pi}{3}$.

**Figure 5.**Features of the local entropy generation due to the fluid friction and local Bejan number for the variation of the inclination angle $\gamma $ ($\gamma =0,\frac{\pi}{6},\frac{\pi}{3},\frac{\pi}{2}$) at $Ra={10}^{6},Ha=15,H=0.1,{\varphi}_{Al}={\varphi}_{Cu}=1\%,\mathsf{\Phi}=\frac{\pi}{3}.$

**Figure 6.**Features of the streamlines and temperature distribution for the fluid phase and porous phase for the variation of the Hartmann number$Ha$ ($Ha=0,15,25,50,100$) at $Ra={10}^{5},H=0.1,{\varphi}_{Al}={\varphi}_{Cu}=1\%,\mathsf{\Phi}=\frac{\pi}{3},\gamma =0$.

**Figure 7.**Features of the local entropy generation due to the fluid friction and local Bejan number for the variation of the Hartmann number$Ha$ ($Ha=0,15,25,50,100$) at $Ra={10}^{5},H=0.1,{\varphi}_{Al}={\varphi}_{Cu}=1\%,\mathsf{\Phi}=\frac{\pi}{3},\gamma =0$.

**Figure 8.**Profiles of the Nu

_{f}for the variation of the inclination angle $\gamma $ ($\gamma =0,\frac{\pi}{6},\frac{\pi}{3},\frac{\pi}{2}$) at $Ra={10}^{6},\text{}Ha=15,\text{}H=0.1,\text{}{\varphi}_{Al}={\varphi}_{Cu}=1\%,\text{}\mathsf{\Phi}=\frac{\pi}{3}$.

**Figure 9.**Profiles of the Nu

_{f}for the variation of the Rayleigh number and nanoparticle volume fraction ${\varphi}_{Al}={\varphi}_{Cu}$ at$Ha=15,H=0.1,\gamma =\pi /6,\mathsf{\Phi}=\frac{\pi}{3}$.

**Figure 10.**Profiles of the S

_{T}of the nanofluid phase in the entire domain for the variation of the Rayleigh number and nanoparticle volume fraction ${\varphi}_{Al}={\varphi}_{Cu}$ at$Ha=15,H=0.1,\gamma =\pi /6,\mathsf{\Phi}=\frac{\pi}{3}$.

**Figure 11.**Profiles of the S

_{total}in the entire domain for the variation of the Rayleigh number and nanoparticle volume fraction ${\varphi}_{Al}={\varphi}_{Cu}$ at$Ha=15,H=0.1,\gamma =\pi /6,\mathsf{\Phi}=\frac{\pi}{3}$.

**Figure 12.**Profiles of the S

_{TS}of the solid phase in the entire domain for the variation of the Rayleigh number and nanoparticle volume fraction ${\varphi}_{Al}={\varphi}_{Cu}$ at$Ha=15,H=0.1,\gamma =\pi /6,\mathsf{\Phi}=\frac{\pi}{3}$.

**Figure 13.**Profiles of the S

_{f}in the entire domain for the variation of the Rayleigh number and nanoparticle volume fraction ${\varphi}_{Al}={\varphi}_{Cu}$ at$Ha=15,H=0.1,\gamma =\pi /6,\mathsf{\Phi}=\frac{\pi}{3}$.

Property | H_{2}O | Cu | Al_{2}O_{3} |
---|---|---|---|

ρ | 997.1 | 8933 | 3970 |

C_{p} | 4179 | 385 | 765 |

k | 0.613 | 401 | 40 |

β | 21 × 10^{-5} | 1.67 × 10^{-5} | 0.85 × 10^{-5} |

σ | 0.05 | 5.96 × 10^{7} | 1 × 10^{-10} |

**Table 2.**Grid independency test at $Ra={10}^{5},\text{}H=0.1,\text{}Ha=10,\text{}{\varphi}_{Al}={\varphi}_{Cu}=1\%,\text{}\mathsf{\Phi}=\frac{\pi}{3},\gamma =0$.

Grid Size | ${\left(\mathit{N}{\mathit{u}}_{\mathit{f}}\right)}_{\mathit{a}\mathit{v}}$ | ${\left(\mathit{N}{\mathit{u}}_{\mathit{s}}\right)}_{\mathit{a}\mathit{v}}$ |
---|---|---|

$31\times 31$ | 1.396400 | 2.751032 |

$41\times 41$ | 1.297873 | 2.877911 |

$61\times 61$ | 1.935634 | 3.034790 |

$81\times 81$ | 1.964531 | 3.125698 |

$101\times 101$ | 1.993429 | 3.196916 |

$121\times 121$ | 2.021942 | 3.259058 |

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**MDPI and ACS Style**

Almuhtady, A.; Alhazmi, M.; Al-Kouz, W.; Raizah, Z.A.S.; Ahmed, S.E.
Entropy Generation and MHD Convection within an Inclined Trapezoidal Heated by Triangular Fin and Filled by a Variable Porous Media. *Appl. Sci.* **2021**, *11*, 1951.
https://doi.org/10.3390/app11041951

**AMA Style**

Almuhtady A, Alhazmi M, Al-Kouz W, Raizah ZAS, Ahmed SE.
Entropy Generation and MHD Convection within an Inclined Trapezoidal Heated by Triangular Fin and Filled by a Variable Porous Media. *Applied Sciences*. 2021; 11(4):1951.
https://doi.org/10.3390/app11041951

**Chicago/Turabian Style**

Almuhtady, Ahmad, Muflih Alhazmi, Wael Al-Kouz, Zehba A. S. Raizah, and Sameh E. Ahmed.
2021. "Entropy Generation and MHD Convection within an Inclined Trapezoidal Heated by Triangular Fin and Filled by a Variable Porous Media" *Applied Sciences* 11, no. 4: 1951.
https://doi.org/10.3390/app11041951