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Sustainable Heat Transfer Management: Modeling of Entropy Generation Minimization and Nusselt Number Development in Internal Flows with Various Shapes of Cross-Sections Using Water and Al_{2}O_{3}/Water Nanofluid

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Setup

^{2}, and it is exerted on a 154 cm length.

#### 2.2. Nanofluid Preparation

## 3. Governing Equations

#### 3.1. Thermophysical Properties of Nanofluid

#### 3.2. Energy Analysis Equation

#### 3.3. Energy Analysis Equation

## 4. Uncertainty Analysis

## 5. Data Collection and Validation

#### Validation of the Experimental Setup

## 6. Results and Discussion

#### 6.1. Energy Analysis

#### 6.1.1. Circular Cross-Section

#### 6.1.2. Square Cross-Section

#### 6.1.3. Rectangular Cross-Section

#### 6.1.4. Average Nusselt Number

#### 6.2. Entropy Generation Analysis

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$A$ | The area of heat transfer (${\mathrm{m}}^{2}$) |

${C}_{p}$ | Specific heat of the fluid ($\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$) |

d & D | Diameter ($\mathrm{m}$) |

$f$ | Friction factor |

$g$ | Gravitational constant ($\mathrm{m}{\mathrm{s}}^{-2}$) |

I | Current (A) |

$k$ | Conductivity ($\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$) |

$M$ | Molar concentration |

$\dot{m}$ | Mass flux ($\mathrm{k}\mathrm{g}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$) |

Nu | Nusselt number |

$P$ | Peripheral (${\mathrm{m}}^{2}$) |

${q}^{\u201d}$ | Heat flux ($\mathrm{W}{\mathrm{m}}^{-2}$) |

Re | Reynolds number |

$S$ | Entropy |

$T$ | Temperature ($K$) |

$v$ | Velocity ($\mathrm{m}{\mathrm{s}}^{-1}$) |

$V$ | Voltage (V) |

$X$ | Entrance length (m) |

Greek letters | |

$\delta $ | Uncertainty |

$\mu $ | Viscosity ($\mathrm{P}\mathrm{a}.\mathrm{s}$) |

$\rho $ | Density ($\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$) |

$\phi $ | Nanoparticle concentration |

Subscripts | |

$act$ | Actual |

$ave$ | Average |

$el$ | Electrical |

$gen$ | Generation |

$loss$ | Loss |

$T$ | Total |

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**Figure 3.**Local Nu of water with different cross-sections in (

**a**) Re = 850, (

**b**) Re = 1250, (

**c**) Re = 1600, and (

**d**) Re = 2100.

**Figure 4.**Local $Nu$ of 1% nanofluid with different cross-sections in (

**a**) $Re=850$, (

**b**) $Re=1250$, (

**c**) $Re=1600$, and (

**d**) $Re=2100$.

**Figure 5.**Local Nu of 2% nanofluid with different cross-sections in (

**a**) Re = 850, (

**b**) Re = 1250, (

**c**) Re = 1600, and (

**d**) Re = 2100.

**Figure 6.**Local $Nu$ of 3% nanofluid with different cross-sections in (

**a**) $Re=850$, (

**b**) $Re=1250$, (

**c**) $Re=1600$, and (

**d**) $Re=2100$.

**Figure 7.**Local $Nu$ of 4% nanofluid with different cross-sections in (

**a**) $Re=850$, (

**b**) $Re=1250$, (

**c**) $Re=1600$, and (

**d**) $Re=2100$.

**Figure 8.**The results of local Nu for circular cross-section with different nanoparticle concentrations in (

**a**) Re = 850, (

**b**) Re = 1250, (

**c**) Re = 1600, and (

**d**) Re = 2100.

**Figure 9.**The results of local Nu for square cross-section with different nanoparticle concentrations in (

**a**) Re = 850, (

**b**) Re = 1250, (

**c**) Re = 1600, and (

**d**) Re = 2100.

**Figure 10.**The results of local Nu for rectangular cross-section with different nanoparticle concentrations in (

**a**) Re = 850, (

**b**) Re = 1250, (

**c**) Re = 1600, and (

**d**) Re = 2100.

**Figure 11.**The average Nu number for all cross-sections in different fluid flows in (

**a**) Re = 850, (

**b**) Re = 1250, (

**c**) Re = 1600, and (

**d**) Re = 2100.

**Figure 12.**The average Nu number in all cross-sections within a range of Re numbers for (

**a**) water, (

**b**) nanofluid with 1% concentration, (

**c**) nanofluid with 2% concentration, (

**d**) nanofluid with 3% concentration, and (

**e**) nanofluid with 4% concentration.

**Figure 13.**Total entropy generation of all cross-sections in a range of Re numbers for (

**a**) water, (

**b**) nanofluid with 1% concentration, (

**c**) nanofluid with 2% concentration, (

**d**) nanofluid with 3% concentration, and (

**e**) nanofluid with 4% concentration.

**Figure 14.**Total entropy generation in all cross-sections for different fluids in (

**a**) Re = 850, (

**b**) Re = 1250, (

**c**) Re = 1600, and (

**d**) Re = 2100.

Cross-Section | 2a | 2b | 2b/2a | Dh |
---|---|---|---|---|

Circular | - | - | - | 0.0154 |

Square | 0.0154 | 0.0154 | 1 | 0.0154 |

Rectangular | 0.0231 | 0.01155 | 0.5 | 0.0154 |

${\mathit{\rho}}_{\mathit{p}}$$\left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^{3}\right)$ | $\mathit{C}{\mathit{p}}_{\mathit{p}}\left(\mathbf{j}/\mathbf{k}\mathbf{g}\mathbf{K}\right)$ | ${\mathit{K}}_{\mathit{p}}\left(\mathbf{w}/\mathbf{m}\mathbf{K}\right)$ | ${\mathit{d}}_{\mathit{p}}\left(\mathbf{n}\mathbf{m}\right)$ |
---|---|---|---|

3690 | 880 | 18 | 15 |

Equipment | Measurement Range | Minimum Measuring Value | The Studied Range in the Present Study | Uncertainty Percentage |
---|---|---|---|---|

K-Type thermocouple | 0–120 ($\xb0\mathrm{C}$) | 0.1 | 24.5–38.5 | 0.260 |

RTD-Pt100 thermocouple | 0–200 ($\xb0\mathrm{C}$) | 0.1 | 25.5–34.5 | 0.290 |

Voltmeter | 0–100 (V) | 0.01 | 24–48 | 0.021 |

Ampere meter | 0–10 (A) | 0.001 | 0.85–1.2 | 0.083 |

Ohmmeter | 0–100 ($\mathsf{\Omega}$) | 0.001 | 27.4–54.5 | 0.002 |

Pressure transducer | 0–100 (mbar) | 0.1 | 8.5–45 | 0.222 |

Flow meter | 0–70 (L/min) | 1 | 10–60 | 1.667 |

Geometrical dimensions | 1–20 (mm) | 0.1 | 1–20 | 0.500 |

Physical properties | - | - | - | 0.100 |

Parameter | Uncertainty Percentage |
---|---|

$q$ | 0.086 |

$h$ | 0.091 |

$Nu$ | 0.518 |

$Re$ | 1.746 |

$f$ | 1.827 |

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**MDPI and ACS Style**

Jery, A.E.; Satishkumar, P.; Abdul Jaleel Maktoof, M.; Suplata, M.; Dudic, B.; Spalevic, V. Sustainable Heat Transfer Management: Modeling of Entropy Generation Minimization and Nusselt Number Development in Internal Flows with Various Shapes of Cross-Sections Using Water and Al_{2}O_{3}/Water Nanofluid. *Water* **2023**, *15*, 89.
https://doi.org/10.3390/w15010089

**AMA Style**

Jery AE, Satishkumar P, Abdul Jaleel Maktoof M, Suplata M, Dudic B, Spalevic V. Sustainable Heat Transfer Management: Modeling of Entropy Generation Minimization and Nusselt Number Development in Internal Flows with Various Shapes of Cross-Sections Using Water and Al_{2}O_{3}/Water Nanofluid. *Water*. 2023; 15(1):89.
https://doi.org/10.3390/w15010089

**Chicago/Turabian Style**

Jery, Atef El, P. Satishkumar, Mohammed Abdul Jaleel Maktoof, Marian Suplata, Branislav Dudic, and Velibor Spalevic. 2023. "Sustainable Heat Transfer Management: Modeling of Entropy Generation Minimization and Nusselt Number Development in Internal Flows with Various Shapes of Cross-Sections Using Water and Al_{2}O_{3}/Water Nanofluid" *Water* 15, no. 1: 89.
https://doi.org/10.3390/w15010089