#
Free Convection Heat Transfer and Entropy Generation in an Odd-Shaped Cavity Filled with a Cu-Al_{2}O_{3} Hybrid Nanofluid

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^{2}

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^{5}

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## Abstract

**:**

_{2}O

_{3}hybrid nanosuspension in an odd-shaped cavity. The side walls are adiabatic, and the internal and external borders of the enclosure are isothermally kept at high and low temperatures of T

_{h}and T

_{c}, respectively. The control equations based on conservation laws are formulated in dimensionless form and worked out employing the Galerkin finite element technique. The outcomes are demonstrated using streamlines, isothermal lines, heatlines, isolines of Bejan number, as well as the rate of generated entropy and the Nusselt number. Impacts of the Rayleigh number, the hybrid nanoparticles concentration (ϕ

_{hnf}), the volume fraction of the Cu nanoparticles to ϕ

_{hnf}ratio (ϕ

_{r}), width ratio (WR) have been surveyed and discussed. The results show that, for all magnitudes of Rayleigh numbers, increasing nanoparticles concentration intensifies the rate of entropy generation. Moreover, for high Rayleigh numbers, increasing WR enhances the rate of heat transport.

## 1. Introduction

_{2}O nanosuspension, which concluded horizontal and vertical parts. Furthermore, the entropy production resulting from energy transference, streamlines, and isotherms is shown for different Ra and particle concentrations.

_{2}O

_{3}hybrid nanofluid, the rate of heat transfer could be enhanced up to around 13.5%. Ismael et al. [17] numerically surveyed the irreversibility and heat transfer of Cu–Al

_{2}O

_{3}hybrid nanofluid in a lid-driven cavity. They reported that using hybrid nanofluids is promising for the improvement of heat transfer and reducing the size of heat transfer devices. Roslan et al. [20] investigated the mixed convection heat transfer of Al

_{2}O

_{3}-Cu-water hybrid nanofluids in a rectangular cavity. The top lid of the cavity was moving with a constant velocity, inducing a mixed convection flow. The results of Roslan et al. [20], in agreement with the outcomes of Roslan et al. [20], showed that the presence of hybrid nanoparticles improved the heat transfer. Following [20], Ali et al. [24] analyzed the mixed convection of the same hybrid nanofluid for a double lid-driven cavity, where both top and bottom lids were moving. This research also confirmed the heat transfer enhancement of using hybrid nanofluids.

## 2. Problem Physics

_{2}O hybrid nanoliquid in the region formed between a hot block and its chamber is schematically shown in Figure 1.

_{h}, and the external walls, depicted in blue, are at the temperature of T

_{c}. The remaining borders are assumed to be adequately insulated. It is assumed that the circulation is steady, incompressible, laminar, and the radiation has no significant effect on the process.

#### 2.1. Governing Equations

#### 2.2. Thermophysical Characteristics

_{hnf}, (ρc)

_{hnf}, β

_{hnf}and μ

_{hnf}denote the density, thermal capacitance, heat expansion coefficient and dynamic viscosity of the nanoliquid which could be calculated from the below relations [28]:

_{2}O

_{3}refer to the base fluid, Cu, and Al

_{2}O

_{3}nanoparticles’ properties, respectively. The physical characteristics of the hybrid nano-sized particles and water (as the host fluid) are gathered in Table 1.

#### 2.3. Non-Dimensional Form of the Governing Equations

#### 2.4. Rate of Heat Transfer

_{avg}as:

## 3. Entropy Generation

## 4. Heat Function

_{,x}, j

_{,y}) and for the hybrid nanofluid read as [32]:

## 5. Results and Discussion

^{3}, 10

^{4}and 10

^{5}), width ratio (WR = W/L = 0.2, 0.3 and 0.4), the concentration of the hybrid nanoparticles (0.0 ≤ ϕ

_{hnf}≤ 0.05) and the concentration of Cu to ϕ

_{hnf}ratio (0.0 ≤ ϕ

_{r}= ϕ

_{Cu}/ϕ

_{hnf}≤ 1.0) have been surveyed and discussed. Moreover, the Prandtl number Pr = 6.2 is considered to be constant in all simulations.

^{3}. The layered arrangements of the isotherms confirm that conduction energy transference is the dominant mode of heat transport in the chamber. It should be noted that the isotherms signify the strength of the conduction heat transfer; for instance, it can be observed from Figure 2 that the conduction energy transference has larger values near the walls (especially the hot walls). Conversely, the heat lines at Ra = 10

^{3}and WR = 0.2 show the perpendicular lines toward isotherms, and heat lines in this figure represent the routes of the energy transference from the hot border to the cold border with the conduction mechanism. Furthermore, the first row of Figure 2c demonstrates a vortex zone in the vertical part of the chamber, which represents a slight fluid velocity in this region. By increasing the WR from 0.2 to 0.3 at the same Ra (second row), the isotherms tend to separate as a result of conduction energy transference reduction. In this new state, the maximum value of the heat line is reduced from 9 to 5.5, and this reduction shows the lower energy transport strength in the cavity. Therefore, it can be concluded that the rate of energy transport from the hot border to the cold border decreases if the wall width increases when the conduction is the dominant phenomenon.

^{5}at the same WR = 0.3 (third row), especially in the horizontal zone of the chamber. It depicts that the conduction energy transference is mainly restricted to the walls, and Ra increment leads to increasing buoyancy force. Moreover, the streamlines show different vortices at Ra = 10

^{5}and WR = 0.3, and these vortices are started to evolve in a horizontal direction, and the routes of heat transfer are mainly similar to streamlines. Therefore, it can be concluded; the free convective energy transference now is the dominant mode in the cavity. In the last line of Figure 2, the effect of WR is depicted at the high value of Ra. By increasing the WR from 0.3 to 0.4 at the Ra = 10

^{5}, the conduction effect on heat transfer is weakened, even in the cavity’s vertical part. Furthermore, the stronger vortexes and more extensive heat lines distribution, especially in the horizontal part, show the dominance convection heat transfer, which is boosted by increasing the wall width. It can be found from Figure 2 that the maximum value of the heat lines is raised in the high Ra when the wall width is increased. Therefore, increasing wall width has a direct influence on energy transference at the high Ra. Figure 2d shows the isentropic lines as the Be number is changed in the range between 0 and 1. It can be observed from the figures that the lines, with larger values, are close to walls. It shows in these areas the thermal entropy has mainly the same values as total entropy. In addition, some lines with lower values of Be are emerged in the middle of vertical cavity by increasing Ra and WR that show the effect of viscous entropy in these areas. Furthermore, these lines extend in horizontal part when the convective energy transport is the dominant mode in the chamber. Figure 3 shows the alteration of average Nu values against the total amount of nanoparticle volume fractions, with an equal proportion of Cu and Al

_{2}O

_{3}in the water-based fluid. It can be obtained from Figure 3a, that the Nu

_{avg}has a more considerable value in the higher Ra as a result of natural convection increment, which was discussed in Figure 2. On the other hand, Figure 3a shows that the Nu

_{avg}rises when a higher concentration of solid particles presents in the medium. The investigated nanoparticles in this research have a high heat transfer conduction coefficient, and the presence of these materials can enhance the conduction phenomena. For this reason, the same slops are observed in Figure 3a, and this represents that the nanoparticles have the same and constant effect in different Ra values. Figure 3b shows that the Be number goes down when the Ra number is increased. For high Ra, the energy transport is enhanced. Therefore, the generated entropy due to energy transference increases, but it can be found from Figure 3b that the generated entropy caused by friction of the hybrid nanofluid has a higher value increment compared to S

_{th}that leads to Be number reduction. However, nanoparticle volume fractions have no influence on the Be number in any values of the Ra (see Figure 3b). After increasing the nanoparticle volume fractions, heat transfer entropy and overall entropy both increased, so Be remained almost constant.

_{r}) on the Nu

_{avg}and Be numbers in different values of wall width. Figure 4a reveals that the percentage of Cu nanoparticle volume fractions has no considerable effect on the Nuavg. Both Cu and Al

_{2}O

_{3}have a similar influence on hybrid nanofluid at Ra = 10

^{3}. It can be seen from this Figure that the Nu

_{avg}reduces considerably when the wall width is increased from 0.2 to 0.4. As previously described at Ra = 10

^{3}, the conduction energy transference is the dominant regime; therefore, it is evident that the Nu

_{avg}decreases according to the Fourier law when the WR increases. This reduction in heat transfer rate leads to entropy decrement, generated by heat transfer, and the Be reduction in Figure 4b is caused by this S

_{th}alterations. Similar to the Nu

_{avg}, the percentage of the Cu nanoparticle volume fractions has no considerable effect on Be number, because it does not affect the rate of energy transference at the low Ra; therefore, no entropy is generated by heat transfer and nanofluid frictions; then, Be remains constant at different values of ϕ

_{r}.

^{5}, in opposite results, as obtained from Figure 4a. This phenomenon can be attributed to the larger conduction heat transfer coefficient of Cu compared to Al

_{2}O

_{3}nanoparticles. By increasing the wall width from 0.2 to 0.3, Nu

_{avg}is increased, as shown in Figure 5a. At WR = 0.3, the nanofluid can be circulated more easily in the cavity as a result of buoyancy force at Ra = 10

^{5}. However, energy transference strength decreases considerably when WR increases to 0.4. Figure 5b shows similar results in Figure 4b, but for different reasons. The generated entropy caused by frictions increases when the wall width changes from 0.2 to 0.3 because of the nanofluid movement in the cavity. As for the previous results, Be number more depends on S

_{viscous}than S

_{th}at high Ra numbers. Therefore, a considerable reduction can be observed on Be number in Figure 5b when WR reaches 0.3. However, by increasing WR to 0.4, the convection effect is decreased (see Figure 5a); therefore, the nanofluid velocity decreases that leads to entropy reduction (S

_{viscous}). In this state, the S

_{viscous}tends to S

_{th}value, and because of this, the Be does not change noticeably at WR = 0.4 compared to WR = 0.3. Figure 5b shows that the Cu nanoparticle volume fractions in hybrid nanofluid have no substantial effect on Be number as a result of convection mechanisms dominance at high Ra numbers. However, Figure 5b shows that Be slightly decreases with ϕ

_{r}increment at WR = 0.2. These slight changes can be attributed to Cu density that has a larger value compared to Al

_{2}O

_{3}. The greater density at high Ra with a narrow wall width leads to increase entropy caused by friction factors and decrease Be number.

_{viscous}overtakes its counterpart, S

_{th}and the entropy generation induced by the friction of the hybrid nanofluid completely prevails. On the other hand, increasing the volume fraction of the hybrid nanoparticles, according to Equations (6) and (8) leads to the increment of the thermal conductivity and also the viscosity of the suspension. Hence, it slightly enhances the S

_{th}(See Equation (22)) and S

_{viscous}(See Equation (23)). Finally, Figure 6c represents the final result that the volume fraction of the hybrid nanoparticles, as well as the Ra number, have a direct influence on the total generated entropy in the cavity; however, the impact of the Rayleigh number (buoyancy force) is more substantial than the ϕ

_{hnf}.

## 6. Conclusions

_{2}O

_{3}in an enclosed cavity was investigated. The enclosed cavity concluded two vertical and horizontal parts, and the heat was transferred from the left vertical border and bottom horizontal border toward other directions. The Cu and Al

_{2}O

_{3}nanoparticles were chosen as composite nanoparticles. Then, the different volume fractions of hybrid nanoparticles and their proportion were investigated on the energy transference. Furthermore, the effect of Ra and wall width and their relationship with nanoparticle volume fractions were studied as two other important factors, which affect the energy transference and generated entropy in the chamber. The conclusions are:

- -
- Hybrid nanoparticles enhanced energy transport when the conduction mechanism was dominant. Conversely, they had no significant influence on convective transport;
- -
- The wall-width ratio (WR) is a parameter that can have a different influence on the energy transference rate in different conditions. Increasing the wall width led to a reduction of the energy transference rate at low Ra (10
^{3}) owing to the dominant conduction heat transfer mechanism; - -
- WR had a positive influence on energy transference at high Ra (10
^{5}) when WR was increased from 0.2 to 0.3. However, a further increase of wall width reduced the heat transfer in the cavity when WR > 0.4, and therefore, an optimum wall width can enhance the heat transfer at high Ra; - -
- At Ra = 10
^{3}, the nanoparticles of Cu and Al_{2}O_{3}had a similar effect on nanofluid in the range of ϕ_{hnf}= 0–0.05, and they enhanced the strength of energy transference to be the same as each other. Conversely, Cu nanoparticles had a stronger impact on heat transfer compared to Al_{2}O_{3}in convection heat transfer at Ra = 10^{5}; - -
- Ra and ϕ
_{hnf}both could enhance generated thermal and viscous entropy, however, Ra had a more intensive influence on generated entropy in the cavity.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Numerical Approach and Grid Check

_{avg}and the total generated entropy has been obtained as:

**Table A1.**Grid check for Nu

_{avg}and S

_{gen}(Ra = 10

^{5}, Pr = 6.2, ϕ

_{hnf}= 0.05, WR = 0.3, ϕ

_{r}= 0.5).

Case Study (n) | No. of Elements | Nu_{avg} | Err_{Nu} (%) | S_{gen} | Err_{Sgen} (%) |
---|---|---|---|---|---|

1 | 3264 | 11.025 | – | 593.70 | – |

2 | 5100 | 11.022 | 0.027 | 596.68 | 0.499 |

3 | 7344 | 11.020 | 0.018 | 598.33 | 0.276 |

4 | 9996 | 11.019 | 0.009 | 599.33 | 0.167 |

5 | 13,056 | 11.019 | 0.001 | 599.98 | 0.108 |

## Appendix B. Validation of the Numerical Code

**Figure A2.**Comparison between the isolines of air (Ra = 10

^{6}, Pr = 0.71, WR = 0.3) for the present work and another one presented by Nithiarasu et al. [35].

_{2}O

_{3}and TiO

_{2}) on the circulation pattern and the rate of energy transference has been evaluated. Figure A3 depicts the comparison between the outcomes of Kahveci [36] and the present research for the dependency of the Nu

_{avg}on the concentration of Al

_{2}O

_{3}nano-sized particles. It is apparent that the obtained values of the Nu

_{avg}coincide with the data of Kahveci [36], representing a finite element code that can appropriately simulate the thermal convective energy transference of nanoliquids in a confined chamber.

**Figure A3.**Influence of Al

_{2}O

_{3}nanoparticles volume fraction on the Nu

_{avg}(Pr = 6.2, β = 0.02, θ = 0°): Comparison between the present work and outcomes of Kahveci [36].

_{gen}and Be are correctly calculated, and the values of similar isolines are quite similar. Finally, the survey of Deng and Tang [38], shown in Figure A5, has been utilized to check the validity of the solved heat function equation. The employed boundary conditions in [38] are exactly the same as [36,37], except that a solid rectangular solid is placed in the center of the cavity, and an extra energy equation for the solid block needs to be solved. As seen, the heat lines are in acceptable agreement with those presented by Deng and Tang [38].

**Figure A4.**Comparison between (left) the total entropy production and (right) the local Bejan number (Ra = 10

^{6}, Pr = 0.71, WR = 0.3): Ilis [37] and the present research.

**Figure A5.**Comparison between the patterns of heat lines obtained by Deng and Tang [38] and the present work (Ra = 10

^{5}, Pr = 0.71).

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**Figure 2.**The impact of WR and Ra on (

**a**) isotherms, (

**b**) heat lines, (

**c**) streamlines and (

**d**) total isentropic lines when Pr = 6.2, ϕ

_{hnf}= 0.05 and ϕ

_{r}= 0.5.

**Figure 3.**The variation of average (

**a**) Nu and (

**b**) Be numbers against nanofluid volume fractions in various Ra for Pr = 6.2, WR = 0.3 and ϕ

_{r}= 0.5.

**Figure 4.**The variation of (a) Nu

_{avg}and (b) Be numbers against ϕ

_{r}with different values of wall width ratio (ϕ

_{hnf}= 0.05, Ra = 10

^{3}and Pr = 6.2).

**Figure 5.**The variation of (a) Nu

_{avg}and (b) Be numbers against ϕ

_{r}with different values of wall width ratio (ϕ

_{hnf}= 0.05, Ra = 10

^{5}and Pr = 6.2).

**Figure 6.**The impact of Ra and hybrid nanoparticles concentration on (

**a**) entropy production due to energy transference, (

**b**) the portion of generated entropy caused by friction of the hybrid nanofluid, and (

**c**) total entropy production for WR = 0.3, Pr = 6.2 and ϕ

_{r}= 0.5.

ρ (kg·m^{−3}) | c (J·kg^{−1}·K^{−1}) | k (W·m^{−1}·K^{−1}) | α (m^{2}·s^{−1}) × 10^{7} | β (K^{−1}) × 10^{6} | |
---|---|---|---|---|---|

Al_{2}O_{3} | 3970 | 765 | 40 | 131.7 | 25.5 |

Cu | 8933 | 385 | 400 | 1163.1 | 50.1 |

Host fluid (water) | 997.1 | 4179.0 | 0.613 | 1.47 | 210 |

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**MDPI and ACS Style**

Ghalambaz, M.; Hashem Zadeh, S.M.; Veismoradi, A.; Sheremet, M.A.; Pop, I.
Free Convection Heat Transfer and Entropy Generation in an Odd-Shaped Cavity Filled with a Cu-Al_{2}O_{3} Hybrid Nanofluid. *Symmetry* **2021**, *13*, 122.
https://doi.org/10.3390/sym13010122

**AMA Style**

Ghalambaz M, Hashem Zadeh SM, Veismoradi A, Sheremet MA, Pop I.
Free Convection Heat Transfer and Entropy Generation in an Odd-Shaped Cavity Filled with a Cu-Al_{2}O_{3} Hybrid Nanofluid. *Symmetry*. 2021; 13(1):122.
https://doi.org/10.3390/sym13010122

**Chicago/Turabian Style**

Ghalambaz, Mohammad, Seyed Mohsen Hashem Zadeh, Ali Veismoradi, Mikhail A. Sheremet, and Ioan Pop.
2021. "Free Convection Heat Transfer and Entropy Generation in an Odd-Shaped Cavity Filled with a Cu-Al_{2}O_{3} Hybrid Nanofluid" *Symmetry* 13, no. 1: 122.
https://doi.org/10.3390/sym13010122