#
Dynamics of Colloidal Mixture of Cu-Al_{2}O_{3}/Water in an Inclined Porous Channel Due to Mixed Convection: Significance of Entropy Generation

## Abstract

**:**

_{2}O

_{3}/water). A symmetrical uniform heat flux was considered at the walls and a constant flow rate was given through the channel. The mathematical model, consisting of a system of equations with given boundary conditions, was transformed in terms of dimensionless variables and the proposed analytical solution was found to be valid for all the cases of the inclined channel. The solution was validated by comparison with previously published results. The behavior of the velocity and temperature of the hybrid nanofluid were studied together with the entropy generation inside the channel by considering the influence of different important parameters, such as the nanoparticle volume fraction, the mixed-convection parameter and the inclination angle of the channel from horizontal. The results were focused to prevent the dissipation of energy by calculating the maximum thermal advantage at a minimum entropy generation in the system.

## 1. Introduction

_{2}O

_{3}) and hybrid nanofluids (Al

_{2}O

_{3}-Cu) and observed that hybrid nanofluids have a higher coefficient of convection heat transfer. The neoteric class of nanofluids is given by hybrid nanofluids, which contain a small amount of metal nanoparticles and non-metallic nanoparticles. The addition of metallic nanoparticles in a base fluid, such as Cu, Zn, Al, has reported high thermal conductivities, but their use has restrictions such as stability or reactivity. In contrast, nonmetallic nanoparticles, such as Al

_{2}O

_{3}, Fe

_{3}O

_{4}or CuO, give lower thermal conductivities compared to metallic nanoparticles, but they have important properties such as stability or chemical inertness. Thus, as reported by Suresh et al. [43] and Tayebi and Chamkha [44], it was observed that by adding a small amount of Cu nanoparticle volume fraction to an Al

_{2}O

_{3}/based nanofluid, the thermophysical characteristics of the resulting hybrid nanofluid could be increased without reducing the stability. In addition, Devi and Devi [45] examined a Cu-Al

_{2}O

_{3}/water hybrid nanofluid using a mathematical approach. They compared their results with experimental results and found an excellent agreement.

_{2}O

_{3}/water hybrid nanofluid and a nondimensional analysis was used. The analytical (exact) solution was found by considering a new approach. The validity of this new solution was confirmed by a comparison with previously published results. Moreover, two hybrid nanofluid models were applied for the presented solutions and the results are in very good agreement. Considering the literature review and the authors knowledge, no study with these findings has been reported before in the literature.

- Is it a thermal advantage to use a Cu-Al
_{2}O_{3}/water hybrid nanofluid instead of using a regular Al_{2}O_{3}/water nanofluid in a porous inclined channel? - Is the obtained analytical solution the most appropriate solution to calculate the entropy generation rate in the channel?
- Does the addition of the Cu nanoparticle volume fraction in the Al
_{2}O_{3}/water nanofluid enhance the heat transfer in all the cases of the inclination angle of the channel? - Is it significant to change the influence of free convection over forced convection into the channel to improve the thermal performance of the system?
- Is it relevant to change the main parameters, such as the nanoparticle volume fraction, the mixed-convection parameter and the inclination angle of the channel from horizontal, to obtain the maximum thermal advantage at a minimum of the entropy generation rate in the system?

## 2. Mathematical Model

_{0}at the channel entrance and the walls are heated by a uniform heat flux q

_{w}. The flow is assumed to be fully developed and steady. The Boussinesq approximation was employed and the homogeneity and local thermal equilibrium in the porous medium were considered. The hybrid nanofluid mixture consisted of nanoparticles of Al

_{2}O

_{3}which were the first added to the base fluid (water) with 0.1 vol. solid volume fraction, and this was fixed throughout the problem. Then, Cu nanoparticles were added with various volume fractions to obtain the hybrid nanofluid mixture, namely Cu-Al

_{2}O

_{3}/water. The hybrid nanofluid was assumed to saturate the solid matrix and both are in thermodynamic equilibrium.

#### 2.1. Basic Equations and Boundary Conditions

#### 2.2. Thermophysical Models of Hybrid Nanofluids

- -
- Hybrid nanofluid density$${\rho}_{hnf}={\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}{\rho}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}+{\phi}_{\mathrm{Cu}}{\rho}_{\mathrm{Cu}}+\left(1-{\phi}_{\mathrm{Cu}}-{\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\right){\rho}_{f}$$
- -
- Hybrid nanofluid buoyancy coefficient$${\left(\rho \beta \right)}_{hnf}={\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}{\left(\rho \beta \right)}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}+{\phi}_{\mathrm{Cu}}{\left(\rho \beta \right)}_{\mathrm{Cu}}+\left(1-{\phi}_{\mathrm{Cu}}-{\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\right){\left(\rho \beta \right)}_{f}$$
- -
- Hybrid nanofluid heat capacitance$${\left(\rho c\right)}_{hnf}={\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}{\left(\rho c\right)}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}+{\phi}_{\mathrm{Cu}}{\left(\rho c\right)}_{\mathrm{Cu}}+\left(1-{\phi}_{\mathrm{Cu}}-{\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\right){\left(\rho c\right)}_{f}$$
- -
- Hybrid nanofluid thermal conductivity$$\frac{{k}_{hnf}}{{k}_{f}}=\left\{\frac{{\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}{k}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}+{\phi}_{\mathrm{Cu}}{k}_{\mathrm{Cu}}}{{\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}+{\phi}_{\mathrm{Cu}}}+2{k}_{f}+2\left({\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}{k}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}+{\phi}_{\mathrm{Cu}}{k}_{\mathrm{Cu}}\right)-2\left({\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}+{\phi}_{\mathrm{Cu}}\right){k}_{f}\right\}\phantom{\rule{0ex}{0ex}}\times {\left\{\frac{{\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}{k}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}+{\phi}_{\mathrm{Cu}}{k}_{\mathrm{Cu}}}{{\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}+{\phi}_{\mathrm{Cu}}}+2{k}_{f}-\left({\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}{k}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}+{\phi}_{\mathrm{Cu}}{k}_{\mathrm{Cu}}\right)+\left({\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}+{\phi}_{\mathrm{Cu}}\right){k}_{f}\right\}}^{-1}$$
- -
- Hybrid nanofluid viscosity$${\mu}_{hnf}={\mu}_{f}{\left(1-{\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}-{\phi}_{\mathrm{Cu}}\right)}^{-2.5}$$

- -
- Hybrid nanofluid density$${\rho}_{hnf}=\left(1-{\phi}_{\mathrm{Cu}}\right)\left[\left(1-{\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\right){\rho}_{f}+{\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}{\rho}_{{p}_{1}}\right]+{\phi}_{\mathrm{Cu}}{\rho}_{{p}_{2}}$$
- -
- Hybrid nanofluid buoyancy coefficient$${\left(\rho \beta \right)}_{hnf}=\left(1-{\phi}_{\mathrm{Cu}}\right)\left[\left(1-{\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\right){\left(\rho \beta \right)}_{f}+{\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}{\left(\rho \beta \right)}_{p}{}_{1}\right]+{\phi}_{\mathrm{Cu}}{\left(\rho \beta \right)}_{p}{}_{2}$$
- -
- Hybrid nanofluid heat capacitance$${\left(\rho c\right)}_{hnf}=\left(1-{\phi}_{\mathrm{Cu}}\right)\left[\left(1-{\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\right){\left(\rho c\right)}_{f}+{\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}{\left(\rho c\right)}_{p}{}_{1}\right]+{\phi}_{\mathrm{Cu}}{\left(\rho c\right)}_{p}{}_{2}$$
- -
- Hybrid nanofluid thermal conductivity$$\frac{{k}_{hnf}}{{k}_{bf}}=\frac{{k}_{{p}_{2}}+\left(n-1\right){k}_{bf}-\left(n-1\right){\phi}_{\mathrm{Cu}}\left({k}_{bf}-{k}_{{p}_{2}}\right)}{{k}_{{p}_{2}}+\left(n-1\right){k}_{bf}+{\phi}_{\mathrm{Cu}}\left({k}_{bf}-{k}_{{p}_{2}}\right)},\phantom{\rule{0ex}{0ex}}\mathrm{where}\frac{{k}_{bf}}{{k}_{f}}=\frac{{k}_{{p}_{1}}+\left(n-1\right){k}_{f}-\left(n-1\right){\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\left({k}_{f}-{k}_{{p}_{1}}\right)}{{k}_{{p}_{1}}+\left(n-1\right){k}_{f}+{\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\left({k}_{f}-{k}_{{p}_{1}}\right)}$$
- -
- Hybrid nanofluid viscosity$${\mu}_{hnf}={\mu}_{f}{\left[\left(1-{\phi}_{\mathrm{Cu}}\right)\left(1-{\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\right)\right]}^{-2.5}$$

_{2}O

_{3}was considered to be the first added to the base fluid (water) with a 0.1 vol. solid volume fraction (i.e., ${\phi}_{1}={\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}=0.1$), which was fixed throughout the problem. Then, Cu nanoparticles were added with various volume fractions (${\phi}_{2}={\phi}_{\mathrm{Cu}}=0.02-0.06)$ to form the hybrid nanofluid, namely Cu-Al

_{2}O

_{3}/water.

_{2}O

_{3}and Cu nanoparticles are given in the Table 1.

#### 2.3. Nondimensionalization Method

_{1}–A

_{3}are dependent on the nanoparticle volume fractions and have the following forms, corresponding to the considered hybrid nanofluid models:

## 3. Analytical Solution

#### 3.1. The Solution for the General Case of $a>0$

#### 3.2. The Solution for $a=0$

#### 3.2.1. The Solution for a Horizontal Channel ($\gamma =0$)

#### 3.2.2. The Solution for the Forced Convection Limit ($\lambda =0)$

## 4. Entropy Generation

## 5. Results and Discussion

_{2}O

_{3}/water hybrid nanofluid was considered for investigation. For the problem of mixed-convection hybrid nanofluid flow in a porous inclined channel, the analytical solutions were obtained and plotted. The behavior of the velocity and temperature for different relevant parameters were obtained and discussed. In the presented results, the inclination angle of the channel $\gamma \in \left[0,\frac{\pi}{2}\right]$ was considered. The entropy generation of the system was also calculated for the relevant parameters in order to find its minimum and to obtain the best energy performance.

_{2}O

_{3}/water with a concentration of alumina oxide nanoparticles ${\phi}_{1}={\phi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}=0.1$, the second nanoparticle volume fraction (cooper) was added gradually with different concentrations ${\phi}_{2}={\phi}_{\mathrm{Cu}}=0.02,0.04,0.06$. The influence of the second nanofluid volume fraction addition is shown in Figure 4a–d for the mixed-convection parameter $\lambda =10,50$ and the inclination of the channel $\gamma =\pi /6$. The velocity profiles did not show a significant change when the hybrid nanofluid was considered (Figure 4a,b) but the temperature reported important changes. The velocity U(Y) reported changes for an increased mixed-convection parameter from 10 (Figure 4a) to 50 (Figure 4b). This was due to the increasing effect of free convection over forced convection at the lower wall of the channel. U(Y) took negative values from the middle of the channel (Y = 0.5) to the upper wall (Y = 1), reporting a region of reversed flow for all the values of the parameters. This behavior confirmed the influence of free convection obtained by the heated walls of the channel.

_{0}) and compressed as it reached the depths of the reservoir (initial cold medium).

#### Entropy Generation for the Hybrid Nanofluid Flow

## 6. Conclusions

_{2}O

_{3}/water hybrid nanofluid in an inclined infinitely long two-dimensional porous channel bounded by impermeable parallel plane walls was examined in this paper. The system of the governing equations of the hybrid nanofluid was written following the model of Tiwari and Das [15]. The system of partial differential equations, together with the boundary conditions, was changed to a system of linear ordinary differential equations by using non-dimensional transformations. Two hybrid nanofluid models were used to observe the behavior of the solution in the channel and the results were found in very good agreement.

- The obtained analytical solution of the problem includes for the first time, all the cases: the inclined, the horizontal and the vertical channel, respectively. This new solution is the most appropriate for an accurate calculating of entropy generation since it is an analytical (not an approximate) solution. Moreover, this exact solution was used to observe the thermal advantage of the hybrid nanofluid for mixed-convective flow in a porous channel.
- The thermal properties of the fluid were enhanced considerably by adding small concentrations of the Cu nanoparticle volume fraction in the regular nanofluid Al
_{2}O_{3}/water, but the velocity was not significantly affected by this change. This behavior was only relevant for values of the mixed-convection parameter λ > 25. - The use of a Cu-Al
_{2}O_{3}/water hybrid nanofluid instead of a regular Al_{2}O_{3}/water nanofluid in the porous inclined channel was not always a thermal advantage. For smaller values of the mixed-convection parameter (λ < 25), a simple nanofluid model has increased thermal properties at a minimum entropy generation in the system. This result could be useful to improve the systems dedicated to solar power collectors. - The inclination angle of the channel from horizontal has an important role on the behavior of the hybrid nanofluid flow inside the channel. Reversed flow was reported for balanced conditions of the heat transfer by fluid motion over the heat transfer by thermal conductivity, Peclet number Pe = 1, for all the values of the inclination angle of the channel. In addition, the temperature increased with a decrease in the inclination angle of the channel. The cumulated results could be used in thermal transmission applications such as heat pipes, etc.
- In the case of the horizontal channel, the hybrid nanofluid flow decreased the thermal performance of the system compared to a regular nanofluid and the entropy generation had minimum values for a higher mixed-convection parameter (λ < 25). This case could be suitable for cooling energetic systems, for example, electronic equipment.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

D | channel width (m) |

g | acceleration due to gravity (m s^{−2}) |

K | specific permeability (m^{2}) |

k | thermal conductivity (W·m^{−1}·K^{−1}) |

Pe | Péclet number |

Ra | Rayleigh number |

q_{w} | heat flux (W·m^{−2}) |

c | specific heat capacity (kJ·kg^{−1}·K^{−1}) |

p | pressure (Pa) |

T | hybrid nanofluid temperature (K) |

F | dimensionless temperature |

U | dimensionless velocity |

$x$ | coordinate along the channel (m) |

$y$ | coordinate normal to the wall (m) |

U_{0} | velocity at the channel entrance (m·s^{−1}) |

$u$ | velocity component along x-axis (m·s^{−1}) |

$v$ | velocity component along y-axis (m·s^{−1}) |

${T}_{0}$ | uniform fluid temperature at the inflow (K) |

Greek symbols | |

α | thermal diffusivity (m^{2}·s^{−1}) |

$\beta $ | thermal expansion coefficient (K^{−1}) |

γ | inclination angle of the channel (°) |

τ | dimensionless temperature |

φ | nanoparticles volume fraction |

λ | mixed-convection parameter |

ρ | density (kg·m^{−3}) |

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

Subscripts | |

f | base fluid |

hnf | hybrid nanofluid |

nf | nanofluid |

p | nanoparticle |

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**Figure 2.**Comparison of temperature profiles of the previous results reported by Cimpean and Pop [52] (

**a**) and the present results (

**b**).

**Figure 4.**Velocity and temperature profiles for $Pe=1,\gamma =\frac{\pi}{6}$ for $\lambda =10$ (

**a**,

**c**) and $\lambda =50$ (

**b**,

**d**) for different concentrations of Cu-nanoparticle volume fraction.

**Figure 5.**Velocity (

**a**) and temperature (

**b**) profiles for horizontal channel, for $Pe=1,\lambda =50$ and ${\phi}_{1}=0.1$ and for different concentrations of Cu-nanoparticle volume fraction ${\phi}_{2}$.

**Figure 6.**Velocity (

**a**) and temperature (

**b**) profiles, for $Pe=1,\gamma =\frac{\pi}{6}$ for different values of the mixed-convection parameter $\lambda $, compared between simple nanofluid (${\phi}_{2}=0.0$) plotted with a straight line and hybrid nanofluid (${\phi}_{2}=0.04)$ plotted with a dotted line.

**Figure 7.**Velocity (

**a**,

**b**) and temperature (

**c**,

**d**) profiles, for $Pe=1,{\phi}_{1}=0.1,{\phi}_{2}=0.02$, for $\lambda =10$ (

**a**,

**c**) and $\lambda =50$ (

**b**,

**d**) for different values of the inclination angle of the channel.

**Figure 8.**Entropy generation number for $Pe=1$ (

**a**) and for $Pe=10$ (

**b**), for $\lambda =1,10,50$ and $\gamma =\frac{\pi}{4},$ depicted with a straight line for ${\phi}_{1}=0.1,{\phi}_{2}=0.0$ (regular nanofluid) and with a broken line for ${\phi}_{1}=0.1,{\phi}_{2}=0.02$ (hybrid nanofluid).

**Figure 9.**Entropy generation number for vertical channel (

**a**) and for horizontal channel (

**b**), for $Pe=1,$ depicted with a straight line for ${\phi}_{1}=0.1,{\phi}_{2}=0.0$ (regular nanofluid) and with a broken line for ${\phi}_{1}=0.1,{\phi}_{2}=0.02$ (hybrid nanofluid).

**Figure 10.**Entropy generation number for different inclinations of the channel, $\gamma =\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3},$ for $Pe=1,\lambda =50$ depicted with a straight line for ${\phi}_{1}=0.1,{\phi}_{2}=0.0$ (regular nanofluid) and with a broken line for ${\phi}_{1}=0.1,{\phi}_{2}=0.02$ (hybrid nanofluid).

Physical Characteristics | Host Liquid (Water) | Al_{2}O_{3} | Cu |
---|---|---|---|

c (J·kg^{−1}·K^{−1}) | 4179 | 765 | 385 |

ρ (kg·m^{−3}) | 997.1 | 3970 | 8933 |

k (W·m^{−1}·K^{−1}) | 0.613 | 40 | 400 |

β × 10^{−5} (K^{−1}) | 21.0 | 0.85 | 1.67 |

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**MDPI and ACS Style**

Cimpean, D.S.
Dynamics of Colloidal Mixture of Cu-Al_{2}O_{3}/Water in an Inclined Porous Channel Due to Mixed Convection: Significance of Entropy Generation. *Coatings* **2022**, *12*, 1347.
https://doi.org/10.3390/coatings12091347

**AMA Style**

Cimpean DS.
Dynamics of Colloidal Mixture of Cu-Al_{2}O_{3}/Water in an Inclined Porous Channel Due to Mixed Convection: Significance of Entropy Generation. *Coatings*. 2022; 12(9):1347.
https://doi.org/10.3390/coatings12091347

**Chicago/Turabian Style**

Cimpean, Dalia Sabina.
2022. "Dynamics of Colloidal Mixture of Cu-Al_{2}O_{3}/Water in an Inclined Porous Channel Due to Mixed Convection: Significance of Entropy Generation" *Coatings* 12, no. 9: 1347.
https://doi.org/10.3390/coatings12091347