Entropy Production Analysis in an Octagonal Cavity with an Inner Cold Cylinder: A Thermodynamic Aspect

Abstract

Understanding fluid dynamics and heat transfer is crucial for designing and improving various engineering systems. This study examines the heat transfer characteristics of a buoyancy-driven natural convection flow that is laminar and incompressible. The investigation also considers entropy generation (E_gen) within an octagonal cavity subject to a cold cylinder inside the cavity. The dimensionless version of the governing equations and their corresponding boundary conditions have been solved numerically using the finite element method, employing triangular mesh elements for discretization. The findings indicated that incorporating a cold cylinder inside the octagonal cavity resulted in a higher heat transfer (HT) rate than in the absence of a cold cylinder. Furthermore, using the heat flux condition led to a higher average Nusselt number (Nu_avg) and a lower Bejan number (Be) than the isothermal boundary condition. The results also showed that HT and E_gen were more significant in the Al₂O₃-H₂O nanofluid than the basic fluids such as air and water, and HT increased as χ increased. The current research demonstrates that employing the heat flux condition and incorporating nanoparticles can enhance the rate of HT and Egen. Furthermore, the thermo-fluid system should be operated at low $Ra$ to achieve greater HT effectiveness for nanofluid concerns.

1. Introduction

Scientists have been studying natural convection in cavities more frequently in the last few decades, as there are many real-world scenarios of it. Some of these are the cooling of electronic devices, heat exchangers [1,2], room ventilation [1], building insulation [1,2], and solar collectors [1,3], to name a few. To optimize the performance of these applications, it is crucial to understand the heat transfer mechanisms inside the cavity, which are influenced by several factors, such as the cavity’s geometry, fluid properties, and boundary conditions. Understanding these parameters is essential for designing and carrying out efficient heat transfer systems in natural convection.

Alsabery et al. [1] performed a numerical investigation of entropy generation (E_gen) in a square cavity containing a nanofluid and a solid object. They studied how E_gen and heat transfer (HT) are affected by different temperature distributions. They observed that optimizing temperature distributions in natural convection systems can significantly improve their performance and efficiency. Sheremet et al. [3] employed a computational approach to examine the impacts of incorporating an isothermal partition made of solid material into a cavity filled with nanofluid, which was cooled by an isothermal cooler positioned at one of the corners. Their study unveiled that E_gen decreased as the parameter χ increased under increasing $Ra$ conditions. Moreover, they discovered that utilizing nanofluids enhanced HT and lowered the $Be$ value. Selimefendigil et al. [4] studied E_gen in a nanofluid cavity with obstacles of different shapes. The researchers investigated the system while subject to the impact of a magnetic field and internal heat generation. Their investigation showcased that these techniques enhanced HT and resulted in a reduction of E_gen within the cavity. Their findings revealed that the shape of the obstacles significantly impacted HT and E_gen, and the use of nanofluids resulted in enhanced HT and reduced E_gen. Subsequently, Selimefendigil et al. [5] conducted a numerical examination of mixed convection and E_gen within a cavity filled with a power law fluid. The cavity featured a partial heater and a rotating cylinder, while a magnetic field also influenced the system. The research demonstrated that including a rotating cylinder led to augmented HT and diminished E_gen in the cavity. Furthermore, the study unveiled that implementing a magnetic field enhanced HT and reduced E_gen compared to scenarios where a magnetic field was not employed. Siavashi et al. [6] explored the realm of non-Darcy double-diffusive natural convection within inclined porous cavities, considering diverse source configurations. Their investigation unveiled that using inclined porous cavities with varying source arrangements enhanced HT and mitigated E_gen, in contrast to the absence of a source. Furthermore, the study emphasized the notable influence of the inclination angle on both HT and E_gen. Specifically, higher inclination angles yielded escalated heat transfer rates and reduced E_gen. Wang et al. [7] performed a computational study to analyze entropy generation in mixed convection within a square enclosure containing a rotating circular cylinder. Their study found results similar to Selimefendigil et al. [5], where a rotating cylinder increased HT and decreased E_gen in the cavity.

Rahimi et al. [8] analyzed E_gen and heatline visualization in a nanofluid-filled cavity with internal heaters. According to the research findings, the average Nusselt number (Nu_avg) increased when $Ra$ and χ both increased. Furthermore, they observed a positive correlation between the total E_gen and $Ra$, indicating an increase in E_gen with higher $Ra$ values. Conversely, the total E_gen n exhibited a negative relationship with χ, indicating a decrease in E_gen as χ values increased. In a subsequent study, Rahimi et al. [9] found results similar to that of Rahimi et al. [8]. The results of their study demonstrated that modifying the arrangement of rigid bodies within the refrigerant system is an effective strategy for managing the flow structure and temperature distribution. Kefayati et al. [10] performed a computational study to examine MHD natural convection and E_gen phenomena in a heated enclosure featuring two internally positioned cold cylinders filled with Carreau fluid. According to their findings, raising the buoyancy ratio increased E_gen and decreased Be_avg. They also discovered that an increase in $Ha$ decreased the total E_gen and caused a considerable change in Be_avg. Arun et al. [11] investigated E_gen in a square cavity containing a rectangular block placed at the centerline position of the cavity. They observed that$Ha$, buoyancy ratio, and $Ra$ influenced the flow patterns and rate of HT. Furthermore, their findings indicated that MHD and double-diffusive effects improved HT and decreased E_gen in the cavity. Bozorg et al. [12] numerically investigated the phenomenon of multi-phase mixed convection HT and E_gen within a cavity filled with a non-Newtonian fluid. The cavity had a rotating heater and cooler to facilitate the study. Their investigations indicated that the increase in Nu_avg that occurs with an increase in $Ra$ becomes irrelevant when $Ri$ is low. Moreover, they found that the heater and cooler cylinders’ rotation significantly impacted the flow patterns and rate of HT.

Alsabery [13] studied mixed convection multi-phase flow in a cavity filled with nanofluid. The research findings showed that both Nu_avg and the convective HT increase when both $Ra$ and the nanoparticle loading are increased. In a subsequent study, Alsabery [14] investigated the unsteady entropy generation in a flexible wall cavity containing a rotating cylinder. The findings indicated that the flexible wall impacted the flow, resulting in a rise in Nu_avg and a decrease in E_gen. These findings were consistent with those of Alsabery [13], which showed that using a rotating cylinder and nanofluid improved HT and reduced E_gen in the cavity. Ahrar [15] conducted a research study involving a numerical simulation of HT and E_gen in a cavity containing Al₂O₃-H₂O nanofluid. The impact of an external magnetic field source on the system was also investigated. According to the study results, Nu_avg and the total E_gen of the system decreased as $Ha$ increased. Tayebi and Chamkha [16] investigated E_gen analysis of MHD natural convection hybrid nanofluid flow in a cavity with a hollow cylinder. The study found that increasing buoyancy forces caused the rate of HT and E_gen due to HT, fluid flow, and magnetic effects, as well as the total E_gen to increase and Be_avg to decrease.

Tasmin et al. [17] studied the impact of non-Newtonian effects on HT and E_gen in natural convection flow inside a wavy nanofluid cavity. Their findings revealed that an increase in χ and $Da$ leads to higher HT and lower E_gen. The undulating cavity was also discovered to impact the heat transfer phenomenon substantially. Li et al. [18] investigated MHD natural convection and E_gen characteristics of a nanofluid near a circular baffle positioned within an inclined square cavity, considering the influences of thermal radiation (Rd). Their findings indicated that $Be$ decreased at low Ra values and increased at high Ra values when the baffle size was raised. They also found that increasing χ and $Rd$ led to an increase in E_gen. Kashyap et al. [19] investigated E_gen in a square cavity for different Pr values containing hot blocks. Their study revealed that the rate of HT and E_gen increased with higher $Ra$ and longer hot blocks. The research also showed that the flow behavior and E_gen are significantly impacted by $Pr$ with a rise in the rate of HT and a decrease in E_gen observed with higher$Pr$. Hamzah et al. [20] examined E_gen within a vented cavity containing a heated cylinder situated between two counter-rotating cylinders. According to the study, increasing $Ra$ and cylinder rotation increased the rate of HT and decreased E_gen.

Ahlawat et al. [21] explored the impacts of a heated block composed of a porous layer and nanofluid on convective HT and E_gen within a square cavity. The study findings indicated that the convective heat conduction from the heat source was enhanced as χ, $Ra$, and $Da$ values increased. Majeed et al. [22] examined the thermal flows and E_gen within a hexagonal cavity through numerical simulation. Their research indicated that applying a magnetic field can boost the rate of HT and decrease the system’s E_gen. Furthermore, their study suggests that magnetized hybrid nanomaterial could be served as effective heat transfer fluids. Acharya [23] conducted a study investigating the hydrothermal characteristics and entropy analysis of a buoyancy-driven MHD nanofluid flow within an octagonal cavity featuring fins. The study reveals that using fins enhanced HT and lower E_gen compared to a standard cavity case. Additionally, the research showed that raising Ra and χ led to an increase in the rate of HT and E_gen. Using a dynamic modulator, Ikram et al. [24] investigated the transient flow and HT analysis of forced convection in a partitioned, hexagonal air-filled cavity. Their findings showed that utilizing a dynamic modulator greatly enhanced heat transfer efficiency compared to a simple cavity configuration. Moreover, the study revealed that the rate of HT increased as $Re$ increased. Considering a multi-blade modulator, Ikram et al. [25] further explored the unsteady characteristics of conjugate heat transfer in a hexagonal cavity. Their findings confirmed that using a multi-blade dynamic modulator enhanced the HT efficiency of the cavity. In 2023, Saha et al. [26] presented a configurative systematic review of the research on heat transfer in cavities. The systematic review covered a wide range of topics related to cavity heat transfer, including natural or mixed or forced convection, radiation, and magnetic field effects. This review indicated that using different boundary conditions, heat sources, and fluid types significantly affects HT performance in cavities.

In their study, Sharma et al. [27] demonstrated that an increase in the radiation parameter and volume fraction of nanoparticles contributes to an enhancement in the temperature profile. The study by Almuhtady et al. [28] encompassed an entropy analysis of a non-inclined cavity with a triangular fin filled with a variable porous media. They observed that the HT rate exhibited a decreasing trend with an increase in the inclination angle of the cavity. Ahmed and Rashad [29] investigated the impact of an irregular left boundary of a container filled with porous elements in the presence of a constant magnetic source. The study findings indicated that the undulation number and the wavy contraction ratio contribute to increased HT. Ahmed and Raizah [30] examined the non-Newtonian power law nanofluid flow between inclined polygonal/cylinder polygonal/polygonal forms, employing the second law of thermodynamics. The key findings of the study revealed that the flow area and power-law index played significant roles in governing the convective and HT processes. Using a nanofluid, Raizah et al. [31] examined the flow and HT fields within a horizontal annulus partially saturated with a porous region. Their findings revealed that an increase in the solid volume fraction resulted in improved values of the Nu_avg while simultaneously reducing the average temperature and $Be.$

The author’s knowledge and the available literature demonstrate the importance of investigating HT and E_gen in enclosed systems for multiple real-life scenarios, such as cooling systems and heat exchangers. The novelty of our study lies in the specific focus on HT and E_gen analysis in an octagonal cavity with an inner cold cylinder. While previous works have explored heat transfer phenomena in various cavity geometries, including rectangular and cylindrical configurations, the investigation of HT and E_gen in complex geometry, like an octagonal cavity with an inner cold cylinder, is relatively limited. This unique geometry introduces novel HT characteristics and potential applications in thermal management systems. This research aims to explore the impact of three distinct fluids—namely, air, water, and Al₂O₃-H₂O nanofluid—on HT features and the E_gen of an octagonal cavity housing a cold cylinder. This study aims to analyze the impact of various factors, including $Ra$, $Pr$, isothermal boundary condition (IBC), heat flux boundary condition (HFBC), and χ, on HT and E_gen. The outcomes will comprise assessments of Nu_avg and $Be$, as well as graphical representations of streamlines, isotherms, E_gen due to temperature gradients and fluid friction, local E_gen, and Bejan number ($Be$) contours to help comprehend HT traits and E_gen phenomena within the cavity.

2. Physical and Mathematical Model

Octagonal cavities possess substantial utility in diverse domains. The intricate geometry of an octagonal cavity renders it an apt object for investigating fluid flow and heat transfer. For instance, replacing the conventional rectangular or triangular shapes with octagonal cavities in heat exchangers can significantly elevate the rate and efficiency of heat transfer. Moreover, in electronic cooling systems, octagonal cavities promote heat transfer and minimize the likelihood of thermal damage to electronic components. Overall, due to its diverse engineering applications, the octagonal cavity represents a crucial research topic for analysis. This study can reveal valuable observations regarding the heat exchange characteristics of octagonal voids, thereby influencing the configuration and enhancement of enclosed thermal management systems in diverse engineering domains. This knowledge may lead to improved cooling efficiency, reduced energy consumption, and increased reliability in electronic devices. Details of the physical model are presented in Figure 1, and the properties of fluids and particles are shown in

Table 1. Thermophysical properties at T = 293 K [32].

Physical Properties	H2O	Al2O3
Cp [J/kgK]	4179	773
$\rho$ [kg/m3]	997.1	3880
k [W/mK]	0.613	36
$\beta$ × 10−5 [1/K]	21	0.85

.

Figure 1 illustrates the octagonal cavity in which the octagon has a side length of L while the inner cylinder has a radius of 0.2 L. In the IBC case or HFBC case, the cylinder, along with the upper and lower horizontal walls, are regarded as cold (T_c), while the heated walls, located on the left and right sides, are maintained at a constant T_h (where T_c < T_h) in the IBC case or at a constant heat flux in the HFBC case. The remaining walls are adiabatic. The fluid flow being analyzed is assumed to be 2D, steady, incompressible, laminar, and Newtonian. The working fluids being studied are air, water, and Al₂O₃-H₂O nanofluid. The volume concentrations of nanoparticles (χ) being considered range from 1% to 5%. Consequently, the Navier–Stokes equations governing fluid flow and energy are presented in a dimensionless format, as depicted below:

Continuity Equation:

$$\frac{𝜕U}{𝜕X}+\frac{𝜕V}{𝜕Y}=0$$

(1)

U-momentum equation:

$$\frac{{\rho }_{nf}}{{\rho }_f}\left[U\frac{𝜕U}{𝜕X}+V\frac{𝜕U}{𝜕Y}\right]=-\frac{𝜕P}{𝜕X}+\frac{{\mu }_{nf}}{{\mu }_f}Pr\left(\frac{{𝜕}^{2}U}{𝜕X^2}+\frac{{𝜕}^{2}U}{𝜕Y^2}\right)$$

(2)

V-momentum equation:

$$\frac{{\rho }_{nf}}{{\rho }_f}\left[U\frac{𝜕V}{𝜕X}+V\frac{𝜕V}{𝜕Y}\right]=-\frac{𝜕P}{𝜕Y}+\frac{{\mu }_{nf}}{{\mu }_f}Pr\left(\frac{{𝜕}^{2}V}{𝜕X^2}+\frac{{𝜕}^{2}V}{𝜕Y^2}\right)+Ra Pr\frac{{\left(\rho {\beta }_T\right)}_{nf}}{{\left(\rho {\beta }_T\right)}_f} \Theta$$

(3)

Energy Equation:

$$\frac{{\left(\rho C_p\right)}_{nf}}{{\left(\rho C_p\right)}_f}\left[U\frac{𝜕\Theta }{𝜕X}+V\frac{𝜕\Theta }{𝜕Y}\right]=\frac{𝜕}{𝜕x}\left(\frac{k_{nf}}{k_f}\frac{𝜕\Theta }{𝜕x}\right)+\frac{𝜕}{𝜕y}\left(\frac{k_{nf}}{k_f}\frac{𝜕\Theta }{𝜕y}\right)$$

(4)

where X and Y represent the non-dimensional Cartesian coordinates system, U and V represent the non-dimensional velocity components in the X and Y directions, Θ represents the non-dimensional temperature, and P represents the non-dimensional pressure. Subscripts $f$ and $nf$ indicate pure fluid and nanofluid, respectively. Here, ${\rho }_{nf}={\rho }_f, {\mu }_{nf}={\mu }_f, k_{nf}=k_f, C{p}_{nf}=C{p}_f and {\left(\rho {\beta }_T\right)}_{nf}={\left(\rho {\beta }_T\right)}_f$ for pure fluids such as air or water in the present study. The non-dimensional numbers that showed up in the governing equations are $Ra$ and $Pr$.

$$\begin{gathered} \mathrm{Isothermal}\mathrm{boundary}\mathrm{condition}(\mathrm{IBC}),Ra=\frac{g\Delta T{L}^3{\beta }_f{\rho }_f}{{\alpha }_f{\mu }_f} \\ \mathrm{Heat}\mathrm{flux}\mathrm{boundary}\mathrm{condition}(\mathrm{HFBC}),Ra=\frac{g{L}^4{\beta }_f{\rho }_{f}q”}{{\alpha }_f{\mu }_f{k}_f} \\ Pr=\frac{{\nu }_f}{{\alpha }_f} \end{gathered}$$

(5)

To derive the non-dimensional Equations (1)–(4) from the associated dimensional equation, the following dimensionless parameters are utilized by Nag et al. [33]:

$$\begin{gathered} X=\frac{x}{L}, Y=\frac{y}{L}, U=\frac{uL}{{\alpha }_f}, V=\frac{vL}{{\alpha }_f}, P=\frac{p{L}^2}{{\rho }_f{\alpha }_f{}^2} \\ \Theta =\left\{\begin{array}{c}\frac{T-T_c}{\Delta T}, \mathrm{for} \mathrm{IBC}\\ \frac{\left(T-T_{ref}\right)k_f}{q”L}, \mathrm{for} \mathrm{HFBC}\end{array}\right. \end{gathered}$$

(6)

where, $\Delta T=T_h-T_c$ and $T_{ref}$ is a reference temperature can be considered the cylindrical surface temperature. Furthermore, ${\beta }_f$, ${\alpha }_f,$ and g represent the coefficient of volumetric expansion, thermal diffusivity, and gravitational acceleration, respectively.

Boundary conditions

• For horizontal walls,

$$U=V=\Theta =0$$

(7)

• For vertical walls,

$$U=V=0$$

(8)

$$\begin{gathered} \mathrm{For}\mathrm{IBC}:\Theta =1 \\ \mathrm{or}\mathrm{for}\mathrm{HFBC}:q”=-1 \end{gathered}$$

(9)

• For cylindrical surfaces,

$$U=V=\Theta =0$$

(10)

• For inclined walls,

  $$U=V=\frac{𝜕\Theta }{𝜕\widehat{n}}=0$$

  (11)

  where $\widehat{n}$ denotes the outward normal vector.

Thermal properties of nanofluid

Determining the thermophysical properties of nanofluids is a complex task as there is uncertainty about which models can produce reliable results, and these models significantly impact the solutions. Researchers have proposed various thermophysical property models for nanofluids. However, due to their diversity and complexity, the thermophysical characteristics of nanofluids are still subject to debate, and no definitive conclusion has been reached for their use in flow and heat transfer applications. To calculate the thermophysical parameters of the nanofluid, the current investigation employs classical relationships between the base fluid and nanoparticles, as represented by the following equations.

The density of nanofluid is defined as Corcione [34]:

$${\rho }_{nf}=\left(1-\chi \right){\rho }_f+\chi {\rho }_p$$

(12)

The heat capacitance of the nanofluid is defined as Corcione [34]:

$${\left(\rho C_p\right)}_{nf}=\left(1-\chi \right){\left(\rho C_p\right)}_f+\chi {\left(\rho C_p\right)}_p$$

(13)

The thermal conductivity of the nanofluid is defined as Corcione [34]:

$$\frac{k_{nf}}{k_f}=1+4.4R{e}_p{}^{0.4}P{r}_f{}^{0.66}{\left(\frac{T}{T_{fr}}\right)}^{10}{\left(\frac{k_p}{k_f}\right)}^{0.03}{\chi }^{0.66}$$

(14)

where $R{e}_p$ is the nanoparticles Reynolds number, defined as

$$R{e}_p=\frac{{\rho }_f{u}_B{d}_p}{{\mu }_f}=\frac{2{\rho }_f{k}_{b}T}{\pi {\mu }_f{}^2{d}_p}$$

(15)

The dynamic viscosity of the nanofluid is defined as Corcione [34]:

$$\frac{{\mu }_f}{{\mu }_{nf}}=1-34.87{\left(\frac{d_p}{d_f}\right)}^{-0.3}{\chi }^{1.03}$$

(16)

$$d_f=0.1{\left(\frac{6M}{N\pi {\rho }_f}\right)}^{\frac{1}{3}}$$

(17)

The thermal expansion coefficient of the nanofluid is defined as Corcione [34]

$$\frac{{\left(\rho {\beta }_T\right)}_{nf}}{{\left(\rho {\beta }_T\right)}_f}=\left(1-\chi \right)+\chi \frac{{\left(\rho {\beta }_T\right)}_p}{{\left(\rho {\beta }_T\right)}_f}$$

(18)

Nusselt number

The average Nusselt number (Nu_avg) as Ahlawat et al. [21] and Saha et al. [35]:

• For isothermal conditions:

$$N{u}_{avg}=\frac{k_{nf}}{k_f}\left[-{{\displaystyle \int }}_0^1{\left.\frac{𝜕\theta }{𝜕x}\right|}_{\mathrm{left}\mathrm{wall}}dy+{{\displaystyle \int }}_0^1{\left.\frac{𝜕\theta }{𝜕x}\right|}_{\mathrm{right}\mathrm{wall}}dy\right]$$

(19)

• For heat flux conditions:

$$N{u}_{avg}=\frac{k_{nf}}{k_f}\left[-{{\displaystyle \int }}_0^1{\left.\frac{1}{\theta }\right|}_{\mathrm{left}\mathrm{wall}}dy+{{\displaystyle \int }}_0^1{\left.\frac{1}{\theta }\right|}_{\mathrm{right}\mathrm{wall}}dy\right]$$

(20)

Entropy generation

In order to evaluate the thermal efficacy of the system, the generation of entropy is analyzed. Therefore, this research was utilized as a criterion for evaluating the thermal performance within the cavity. In our investigation, thermal impacts and viscous impacts were responsible for E_gen. The local E_gen, E_l,t is represented as follows in the non-dimensional form [36]:

$$\begin{gathered} E_{l,t} = S_{l,h}+S_{l,f} \\ {S}_{l,h} =\frac{k_{nf}}{k_f}\left[{\left(\frac{𝜕\theta }{𝜕x}\right)}^2+{\left(\frac{𝜕\theta }{𝜕y}\right)}^2\right] \\ {S}_{l,f} = \lambda \frac{{\mu }_{nf}}{{\mu }_f}\left[2{\left(\frac{𝜕u}{𝜕x}\right)}^2+2{\left(\frac{𝜕v}{𝜕y}\right)}^2+{\left(\frac{𝜕u}{𝜕y}+\frac{𝜕v}{𝜕x}\right)}^2\right] \end{gathered}$$

(21)

where

$$\lambda =\frac{{\mu }_f{T}_0{\alpha }_f{}^2}{k_f{L}^2\Delta T^2}$$

(22)

Here, $\lambda$ represents the irreversibility factor and is equal to${10}^{-4}$. It should be mentioned that when no particles are added, $k_{nf}=k_f$ and ${\mu }_{nf}={\mu }_f.$

The local Bejan number (B$e_l$) [36]:

$$B{e}_l=\frac{S_{l,h}}{S_{l,h}+S_{l,f}}$$

(23)

The average entropy generation (E_avg):

$$E_{avg}=\frac{{{\displaystyle \int }}_0^1{{\displaystyle \int }}_0^1{E}_{l,t} dx dy}{{{\displaystyle \int }}_0^1{{\displaystyle \int }}_0^{1}dx dy}$$

(24)

The average Bejan number ($Be$) [36]:

$$Be=\frac{{{\displaystyle \int }}_0^1{{\displaystyle \int }}_0^{1}B{e}_l dx dy}{{{\displaystyle \int }}_0^1{{\displaystyle \int }}_0^{1}dx dy}$$

(25)

3. Numerical Modelling

The governing Equations (1)–(4) with corresponding boundary conditions, Equations (7)–(11), have been performed using the finite element method (FEM). The governing PDEs are transformed into integral equations in this simulation methodology by applying the Galerkin weighted residual technique. The graphical results in our study are computed using the finite element method with a second-order approximation. It provides smooth solutions throughout the interior domain, even in the corner regions. This strategy will significantly improve fast convergence assurance. In this study, a non-uniform triangular mesh arrangement is implemented. The error criteria are evaluated with a relative tolerance of 10⁻⁶.

4. Meshing and Validation

Grid Independent Test:

A grid-independent test was conducted on the simulation to ensure accuracy and reliability by running it with different grid sizes while keeping other parameters constant. The test aimed to determine the optimal grid size. In this study, the test was performed using a triangular mesh and eight grid sizes ranging from 11,960 to 57,116 elements, and the results are presented in

Table 2. Comparison of Nu_avg for various grid dimensions for $Ra={10}^6$, at $Pr$ = 0.71 for air, $Pr$ = 7.00 for water, and $Pr$ = 7.30 for Al₂O₃-H₂O nanofluid.

	IBC		HFBC
Elements	$\mathit{P}\mathit{r}$ = 0.71 (Air)	$\mathit{P}\mathit{r}$ = 7.00 (Water)	$\mathit{P}\mathit{r}$ = 0.71 (Air)	$\mathit{P}\mathit{r}$ = 7.00 (Water)	$\mathit{P}\mathit{r}$ = 7.30 (χ = 0.05) Al2O3-H2O Nanofluid
11,960	19.69	21.18	25.01	26.11	39.32
15,132	19.78	21.23	25.05	26.14	39.39
16,990	19.80	21.18	25.14	26.21	39.34
20,452	19.92	21.25	25.17	26.30	39.33
20,462	20.03	21.32	25.18	26.34	39.31
24,474	20.08	21.36	25.20	26.39	39.34
29,704	20.22	21.39	25.22	26.46	39.57
57,116	20.23	21.40	25.23	26.48	39.58

. Based on the comparison, a grid size of 29,704 was selected for the simulations in this study.

Validation:

Figure 2 displays the streamlines and isotherms for incompressible flow at $Ra$ = 10⁵ and $Ra$ = 10⁶ in an octagonal cavity. The results obtained are qualitatively in good agreement with previous findings reported by Saha et al. [35]. Moreover, in Figure 3, a comparison is made between the current isotherm results and the numerical isotherm obtained through the CIP thermal LBM, as presented in the study conducted by Mussa et al. [37], as well as experimental results reported by Corvaro et al. [38]. The comparison was made for a rectangular cavity filled with air and heated by a small, localized heat source on the bottom wall. The figures demonstrate a clear agreement between the outcomes. To further validate the accuracy of the numerical simulation, the results for entropy generation are compared with those published in Ilis et al. [36] for square geometries. The excellent agreement between the present results for S_l,h, S_l,f, E_l,t, and Be_l and the previous studies is shown in Figure 4.

Streamline and isotherm profiles validation:

Figure 2. Comparison of the present streamlines and isotherms profiles with Saha et al. [35] for (a) $Ra$ = 10⁵ and (b) $Ra$ = 10⁶ with $Pr$ = 0.71.

Figure 2. Comparison of the present streamlines and isotherms profiles with Saha et al. [35] for (a) $Ra$ = 10⁵ and (b) $Ra$ = 10⁶ with $Pr$ = 0.71.

Isotherm profile validation:

Figure 3. Comparison of the Isotherm between the present results, Mussa et al. [37] and the experimental results of Corvaro et al. [38] for $Ra$ = 2.02 × 10⁵.

Figure 3. Comparison of the Isotherm between the present results, Mussa et al. [37] and the experimental results of Corvaro et al. [38] for $Ra$ = 2.02 × 10⁵.

Entropy profiles validation:

Figure 4. Comparison of the S_l,h, S_l,f, E_l,t, and Be_l between the (a) Ilis et al. [36] and the (b) present results.

Figure 4. Comparison of the S_l,h, S_l,f, E_l,t, and Be_l between the (a) Ilis et al. [36] and the (b) present results.

5. Use of Corcione [34] Empirical Correlation Model: A Justification

In 2011, Corcione [34] proposed a new empirical correlation model described in Equations (14)–(17) to determine the effective dynamic viscosity and thermal conductivity of nanofluids containing oxide and metal nanoparticles suspended in water or ethylene glycol-based solutions. The originality of this correlation is to demonstrate nanofluids’ thermal conductivity depending on the nanoparticle concentration, temperature, and size. The model proposed for nanoparticles ranging in size from 10–150 nm and for nanoparticle concentrations ranging from 0.02–9%, with temperatures ranging from 294–324 K. It is worth mentioning that a phenomenon like dispersion/agglomeration of a certain degree caused by the suspended nanoparticles of different shapes and sizes happens to be observed in a discrete experimental setup. While deriving the empirical correlation, Corcione [34] did not consider this type of data set because of too sharp of a contrast in the data sets with an absence of enough physical evidence or observed spurious experimental results, or the presence of using chemical dispersants/surfactants during experiments, which causes agglomeration or significant changes in the thermo-mechanical behaviour of the suspension.

In 2013, Corcione et al. [39] expanded their work by investigating the HT behaviour within a square cavity using a two-phase mixture model with alumina-oxide nanofluids. In this work, they utilize the same empirical correlation model to estimate the thermal properties and successfully validate their empirical model with other experimental data for nanoparticle concentrations of 1–4%. Their research indicated that the increase in HT rate depended on the optimal particle loading, which was influenced by factors such as particle size, concentration, and micro-mixing convection resulting from the Brownian dynamics of the loaded nanoparticles.

In 2015, a study published by Cianfrini et al. [40] considered single-phase nanofluid modelling with temperature-dependent physical properties for nanoparticle loading up to 6%, in which nanofluids are treated as pure fluids such that the solid–liquid interface is in a thermal equilibrium state and the no-slip condition is preserved in the interface. Subsequently, Corcione et al. [41] conducted a similar study using two-phase modelling considering different boundary conditions and examined alumina-oxide, copper-oxide, and titanium dioxide nanofluids employing the same empirical correlation. They observed that the HT rate peaked at the optimal particle loading and subsequently decreased associated with the nanofluids’ temperature and the difference between active walls.

Later, in 2018, Corcione et al. [41] also explored the two-phase modelling approach by investigating a slender vertical cavity filled with alumina-oxide nanofluids. They validated their findings for higher $Ra$ ranges of 10⁷ to 10⁸ and concentrations up to 4%. Their results aligned with the earlier findings of Corcione et al. [39,41] in 2013 and 2015. The above discussion asserts that the Corcione proposed empirical correlation can be effectively utilized in nanofluid modelling both in single-phase and multi-phase approaches for volume fractions up to 6%. To validate the nanofluid simulation conducted in the present study, there is a comparison with the experimental study reported by Ho et al. [42] for nanoparticle volume fractions of 3% and 4%, as shown in Figure 5. The present numerical results closely resemble the experimental data, indicating that our model accurately captures the HT characteristics of the nanofluid. This agreement validates the reliability of our numerical approach and enhances our understanding of nanofluid HT phenomena.

6. Findings and Analysis

This numerical investigation examines the heat transfer and flow phenomena occurring within an enclosed space with an octagonal shape containing a cold cylinder. The study aims to explore the heat transfer characteristics of air, water, and Al₂O₃-H₂O nanofluid under different parameters, such as $Ra$ ranging from 10³ to 10⁶, $Pr$ = 0.71 for air and 7.0 for water, $Pr=6.77–7.30$ for Al₂O₃-H₂O nanofluid, and χ varying between 1% to 5%. The nanofluid has a nanoparticles diameter (d_p) of 10 nm. This study will also analyze and compare the effects of IBC and HFBC on entropy generation. The subsequent sections will demonstrate the impact of the parameters mentioned above on the flow and temperature fields through the utilization of streamlines and isotherms. Furthermore, this study will investigate E_gen due to temperature gradient and viscous dissipation, as well as local E_gen and Be_l.

Table 3. Variation of Nu_avg, E_avg, and $Be$ using the isothermal boundary condition without cylinder and with cylinder for air and water.

	Without Cylinder			With Cylinder
	Air						Pr = 0.71
$\mathit{R}{\mathit{a}}_{\mathit{f}}$	Nuavg	Eavg	Be	Nuavg	Eavg	Be	% Increase of Nuavg
103	2.42	2.62	0.95	3.43	3.73	0.94	41.88
104	4.20	7.38	0.58	6.70	11.18	0.61	59.70
105	6.81	58.75	0.12	11.42	88.32	0.13	67.61
106	11.22	1085.94	0.01	20.22	1373.95	0.015	80.16
	Water						Pr = 7.0
$\mathit{R}{\mathit{a}}_{\mathit{f}}$	Nuavg	Eavg	Be	Nuavg	Eavg	Be	% Increase of Nuavg
103	2.44	2.65	0.94	3.44	3.74	0.94	40.95
104	4.52	8.78	0.53	6.87	12.26	0.57	51.91
105	7.43	84.26	0.09	12.21	103.29	0.12	64.32
106	12.18	1370.57	0.01	21.39	1570.48	0.01	75.58

displays the changes in Nu_avg, E_avg, and $Be$under isothermal boundary conditions, with and without a cylinder, for air and water. For air without a cylinder, the Nu_avg value increases from 2.42 to 11.22, the E_avg value increases from 2.62 to 1085.94, while the Be value decreases from 0.95 to 0.01, as the $Ra$value increases from 10³ to 10⁶. In contrast, for water without a cylinder, the Nu_avg value increases from 2.44 to 12.18, the E_avg value increases from 2.65 to 1370.57, while the $Be$ value decreases from 0.94 to 0.01, as the $Ra$ value increases from 10³ to 10⁶. Moreover, Table 3 shows that the Nu_avg and E_avg increase while $Be$ decreases with an increase in $Ra$ and the presence of a cylinder in the case of air. For all cases with $Ra$ = 10³ and 10⁴, the $Be$ values are greater than 0.5, indicating that E_gen due to temperature gradient is more significant than E_gen due to fluid friction. However, for $Ra$ = 10⁵ and 10⁶, Be is less than 0.5, indicating that E_gen due to fluid friction is more dominant than E_gen due to temperature gradient. Furthermore, in the case of air subjected to a cylinder under the same isothermal boundary condition, the Nu_avg value rises from 3.43 to 20.22, the E_avg value increases from 3.73 to 1373.95, while the $Be$ value drops from 0.94 to 0.015 with increasing $Ra$values from 10³ to 10⁶. For water subjected to a cylinder under the same isothermal boundary condition, the Nu_avg value rises from 3.44 to 21.39, the E_avg value increases from 3.74 to 1570.48, while the $Be$ value drops from 0.94 to 0.01 with increasing $Ra$values from 10³ to 10⁶.

This suggests that the presence of the cylinder has a substantial effect on improving convective HT in air and water. The presence of the cold cylinder disrupts the flow patterns and creates flow recirculation zones. This leads to increased fluid mixing and enhanced HT through convective processes. In addition, the presence of the cylinder leads to an increase in Nu_avg and E_avg as the $Ra$ value increases. As the $Ra$ value increases from 10³ to 10⁶, the percentage increase in Nu_avg ranges from 41.88% to 80.16% in air and 40.95% to 75.58% in water, which further verifies the cylinder’s capacity to improve convective HT and E_gen in air and water. When comparing the results between air and water, we can observe that Nu_avg values are generally larger for water, indicating that convective HT is more efficient in water than in air. Nevertheless, the impact of the cylinder on Nu_avg is similar for both air and water, with a significant enhancement observed in both cases. According to Table 3, the cylinder’s presence substantially improves convective HT in air and water. To better understand this impact, we will concentrate on the results acquired with the cylinder case rather than those without it.

6.1. Variation of Streamline and Isotherm Profiles for Air/Water Case

Figure 6 displays the streamlines and isotherms of air and water for different $Ra$ using IBC. In the case of $Ra$ = 10³, the streamline profiles present two identical rolls within the cavity of air and water. However, the isotherms are only symmetrical concerning the central vertical axis, while asymmetry is observed in the horizontal axis. This asymmetry is attributed to distinct convection effects encountered by the cavity’s top and cold bottom walls and the cold cylinder. The temperature distribution inside the cavity is non-uniform and demonstrates significant variations across different regions. However, the temperature difference between neighboring regional points is small. It is worth noting that the isotherms located near the top cold wall and cylinder exhibit closely spaced contours, whereas those near the cold bottom wall possess flatter contours. Due to its higher density and thermal conductivity, water can transfer heat effectively more effectively than air. Therefore, at the same $Ra$, water generates larger and more stable convection cells than air, and its streamlines tend to be smoother and more regular than air flow patterns. As depicted in Figure 6, an increase in $Ra$ significantly alters the isotherms and streamlines for both air and water cases. When $Ra$ increases, the buoyancy forces that drive convection circulation become stronger, leading to a substantial deformation of the isotherms within the cavity. Specifically, at $Ra$ = 10⁴, there is an increase in the flow intensity, which is further amplified at $Ra$ = 10⁵ and $Ra$ = 10⁶. As for the air case, at $Ra$ = 10⁵, streamlines begin to form vertices, while in water, streamlines form two vertices near the cold bottom wall. Moreover, at $Ra$ = 10⁶, air streamlines form two vertices, and water streamlines form four vertices near the cold bottom wall. At higher$Ra$, streamlines exhibit more complexity and chaos than lower values, although they tend to be smoother and more regular than those observed in the air. With increasing $Ra$, convection circulation becomes stronger, causing the isotherms to become more tightly spaced near the two heat walls, the top cold wall, and the cylinder, as is the case for $Ra$ = 10⁵ and $Ra$ = 10⁶. The isotherms become even more closely spaced than at lower values, denser than those observed in the air.

6.2. Variation of Different Entropy Production for Air/Water Cases

In Figure 7, E_gen due to temperature gradient and viscous dissipation, as well as the local E_gen and local Be for $Ra$ = 10⁶ for air and water using IBC, are shown. At this high $Ra$, the convective effects become dominant in determining the fluid flow and the amount of entropy generated, causing noticeable differences in E_gen between water and air. The higher heat conductivity and more significant convective effects of water result in higher magnitude changes in E_gen compared to air. The local E_gen and Be_l contours also display greater values in water than in air at $Ra$ = 10⁶, again attributed to the stronger convective effects in water leading to more significant inefficiency in fluid flow and greater entropy production.

Table 4. Variation of Nu_avg, E_avg, and Be using IBC and HFBC for air and water.

	IBC			HFBC
	Air						Pr = 0.71
$\mathit{R}{\mathit{a}}_{\mathit{f}}$	Nuavg	Eavg	Be	Nuavg	Eavg	Be	% Increase of Nuavg
103	3.43	3.73	0.94	5.90	0.37	0.92	72.11
104	6.70	11.25	0.61	9.78	0.84	0.25	45.97
105	11.42	88.32	0.13	16.00	6.18	0.021	40.13
106	20.22	1373.95	0.015	25.22	58.67	0.001	24.75
	Water						Pr = 7.0
$\mathit{R}{\mathit{a}}_{\mathit{f}}$	Nuavg	Eavg	Be	Nuavg	Eavg	Be	% Increase of Nuavg
103	3.44	3.74	0.94	5.91	0.37	0.92	71.75
104	6.87	12.26	0.57	9.92	0.85	0.25	44.32
105	12.21	103.29	0.12	16.63	6.81	0.018	36.26
106	21.39	1570.48	0.014	26.46	65.22	0.0012	23.70

shows that for both air and water under IBC and HFBC, Nu_avg increases and $Be$ decreases as $Ra$ increases. This is expected, as higher $Ra$ values indicate greater temperature differences between the wall and fluid, resulting in increased heat transfer rates. Table 4 also shows that as the $Ra$ value increases from 10³ to 10⁶, the average E_gen increases for air and water. The table also indicates that the average E_gen is lower when using HFBC than IBC for air and water. Furthermore, the average E_gen is higher for water than for air under both heating conditions. Table 4 demonstrates that the average E_gen is lower at lower values of Ra and higher at higher values of $Ra$. Regarding $Be$ values, it can be observed that for both air and water at $Ra$ = 10³ under HFBC, Be is greater than 0.5. This implies that E_gen, due to temperature gradient, is more dominant than E_gen due to fluid friction. However, at Ra values of 10⁴ to 10⁶, $Be$ values are less than 0.5, indicating that E_gen due to fluid friction is more dominant than E_gen due to the temperature gradient.

Under IBC, in the case of air subjected to a cylinder, the Nu_avg value increases from 3.43 to 20.22, the E_avg value increases from 3.73 to 1373.95, while the $Be$ value decreases from 0.94 to 0.015 with increasing Ra values from 10³ to 10⁶. Similarly, for water subjected to a cylinder, when the Ra value increases from 10³ to 10⁶, the Nu_avg value increases from 3.44 to 21.39, the E_avg value increases from 3.74 to 1570.48, and $Be$ value decreases from 0.94 to 0.014. Under HFBC, in the case of air subjected to a cylinder, the Nu_avg value increases from 5.90 to 25.22, and the E_avg value increases from 0.37 to 58.67. In contrast, the $Be$ value decreases from 0.92 to 0.001 values when $Ra$ increases from 10³ to 10⁶. Similarly, for water subjected to a cylinder, when the $Ra$ value increases from 10³ to 10⁶, the Nu_avg value increases from 5.91 to 26.46, the E_avg value increases from 0.37 to 65.22, and the $Be$ value decreases from 0.92 to 0.0012.

Table 4 also reveals that the highest percent increase in Nu_avg for air and water occurs at the lowest $Ra$ value of 10³. This indicates that at lower $Ra$ values, using HFBC has a more substantial effect on HT. When the $Ra$ value increases, the percent increase in Nu_avg decreases for both air and water, indicating that the impact of HFBC is less pronounced at higher $Ra$ values. HFBC involves maintaining a constant heat flux on the wall, causing the wall temperature to adjust accordingly. This generates a greater temperature difference between the wall and the fluid, resulting in a higher convective HT coefficient and, ultimately, a higher Nu_avg. By comparing the results for both air and water in Table 4, it is apparent that Nu_avg is generally greater for HFBC than for IBC. This suggests HT is more effective using HFBC.

Table 4 provides significant findings on the effect of boundary conditions on convective heat transfer and suggests that HFBC is more effective in enhancing convective heat transfer than IBC. For a better understanding of the effect of HFBC on convective heat transfer, we will focus on the results obtained with HFBC rather than the results obtained from IBC.

Figure 8 demonstrates that no vortices are formed near the bottom wall when the heat flux boundary condition is applied to air with $Ra$ = 10⁶. This is because HFBC controls the heat transfer rate at the walls, allowing for variation in temperature distribution. As a result, the heat transfer rate is more uniform near the bottom wall, leading to less temperature variation in the flow. This results in a smoother flow without any vortices near the bottom wall. Figure 8 shows that under HFBC, two vortices are formed near the bottom wall of the water. This is due to the higher thermal conductivity of water, which results in a greater heat transfer rate near the bottom wall. The streamlines and isotherms contours for air and water under HFBC indicate a more uniform heat distribution throughout the cavity. In addition, Figure 7 shows that using IBC leads to less efficient and uniform heat transfer throughout the cavity, while using HFBC (shown in Figure 8) results in higher local E_gen contours near the walls. This is because the smoother flow near the walls generates higher levels of viscous dissipation, which is a significant source of E_gen. Likewise, the local $Be$ contours indicate higher values near the walls when implementing HFBC than IBC. When using HFBC, the elevated Be_l near the walls suggests more efficient HT in those areas.

We will explore the potential benefits of using nanofluids in natural convection to enhance HT and reduce E_gen. We will investigate the impact of different concentrations of nanoparticles on system performance, as well as the influence of a cold cylinder on the natural convection system, which is highly effective in improving HT and reducing E_gen. Moreover, we will examine how the application of HFBC can further enhance HT and reduce E_gen in natural convection. Finally, we will assess the system’s environmental impact by evaluating its Ecological Coefficient Performance (ECOP). All of these details will be presented in the following analysis.

6.3. Variation of Streamline, Isotherm, Local E_gen, and Be for Al₂O₃-H₂O Nanofluid Subject to Different $Ra$

Figure 9 plots the streamlines, isotherms, E_l,t, and Be_l for various$Ra$while maintaining a constant $\chi$ of 5%. The streamline profiles indicate the presence of two core vortices inside the cavity. As $Ra$ increases, the buoyancy force increases, causing the core vortices to rise. This leads to a faster rate of convective HT, resulting in a more efficient HT process within the cavity. The isotherm contours reveal that as $Ra$ increases, the contours become denser in proximity to the heated and cold walls and in the vicinity of the cold cylinder. This suggests that the denser contours near the non-insulated walls play a crucial role in influencing the HT efficiency of the cavity. In addition to the streamline and isotherm contours, Figure 9 demonstrates that altering $Ra$ impacts the local entropy and Be_l contours for $\chi$ of 5%. Examining the local entropy contours, we observe that as $Ra$ increases, the graphs become denser in the vicinity of the heated and cold cylinders while decreasing in the lower half. Moreover, the local entropy contours show that the graphs become denser in the upper half of the cavity, resulting in an increase in entropy in that region. As for local $Be,$ for the lower $Ra$ values ($Ra$= 10³ and $Ra$ = 10⁴), the graphs are uniformly distributed throughout the cavity, indicating a uniform HT distribution. However, as $Ra$ increases, the graphs become denser near the lower cold wall, suggesting a decrease in the rate of HT in that particular region.

6.4. Variation of Entropy for Different$\chi$

The behavior of E_gen due to the temperature gradient, viscous dissipation, local E_gen, and local $Be$ contours can vary significantly when using nanofluids, depending on $\chi$ as shown in Figure 10. Including nanoparticles in the fluid increases its thermal conductivity, resulting in a reduced temperature gradient and E_gen caused by the temperature gradient. As $\chi$ in the nanofluid increases from 1% to 5%, the effective thermal conductivity of the fluid also increases, leading to further reductions in temperature gradient and E_gen due to the temperature gradient. Figure 10 exhibits that at a constant $Ra$ = 10⁶, increasing $\chi$ has a minimal impact on E_gen due to temperature. However, as $\chi$ increases from 1% to 5%, the entropy generated due to viscous dissipation and local E_gen decreases. Additionally, Be_l for nanofluids increases with increasing$\chi$, indicating a less effective HT process. The rise in Be_l with an increase in $\chi$ suggests a decrease in the efficiency of HT in the cavity.

Table 5. Variation of Nu_avg and heat transfer enhancement (E%) with different $\chi$ for HFBC.

	Ra = 103		Ra = 104		Ra = 105		Ra = 106
	Nuavg	E (%)	Nuavg	E (%)	Nuavg	E (%)	Nuavg	E (%)
$\chi$ = 1%	6.34	7.38	10.64	7.25	18.08	8.69	28.69	8.45
$\chi$ = 2%	6.85	16.01	11.03	11.17	18.79	13.00	29.80	12.64
$\chi$ = 3%	7.20	21.85	11.13	12.18	19.22	15.54	30.57	15.54
$\chi$ = 4%	7.37	24.72	11.25	13.40	19.64	18.11	31.29	18.28
$\chi$ = 5%	7.77	31.57	11.29	13.78	19.86	19.42	31.89	20.56

displays the relationship between Nu_avg and varying $\chi$ under different $Ra$ for HFBC. Additionally, the percentage of Nu_avg enhancement ($\mathsf{{\rm E}}\%=\frac{N{u}_{avg} \left(\chi \ne 0\right)-N{u}_{avg} \left(\chi =0\right)}{N{u}_{avg} \left(\chi =0\right)}\times 100)$in comparison to the base fluid is provided for each scenario. When $Ra$ is 10³, increasing $\chi$ from 1% to 5% results in a substantial increase in both Nu_avg and enhancement percentage. Specifically, Nu_avg increases from 6.34 to 7.77, and the enhancement percentage increases from 7.38% to 31.57%. Similarly, for $Ra$ of 10⁴, Nu_avg increases from 10.64 to 11.29, and the enhancement percentage increases from 7.25% to 13.78%. The pattern observed in Table 5 is consistent for $Ra$ values of 10⁵ and 10⁶. Specifically, for $Ra$ = 10⁵, Nu_avg increases from 18.08 to 19.86, and the enhancement percentage increases from 8.69% to 19.42% when $\chi$ increases from 1% to 5%. For $Ra$ = 10⁶, Nu_avg increases from 28.69 to 31.89, and the enhancement percentage increases from 8.45% to 20.56% with the same increase in$\chi$. This increase in $\chi$ results in a corresponding increase in thermal conductivity and convective HT, ultimately leading to higher Nu_avg values. Additionally, Table 5 illustrates that the enhancement percentage in Nu_avg increases with increasing$\chi$, indicating that adding nanoparticles to the base fluid enhances the rate of HT.

As the nanoparticle’s$\chi$ increases, the overall thermal conductivity of the nanofluid also increases. This enhancement in thermal conductivity facilitates more efficient HT within the cavity, leading to higher HT rates. This also increases Brownian motion, which aids in dispersing heat more uniformly throughout the cavity, thereby enhancing HT.

Table 6. Variation of $Be$ with different $\chi$ for HFBC.

	$\mathit{R}\mathit{a} = {10}^3$	$\mathit{R}\mathit{a} = {10}^4$	$\mathit{R}\mathit{a} = {10}^5$	$\mathit{R}\mathit{a} = {10}^6$
$\chi$ = 1%	0.93396	0.24515	0.01722	0.00114
$\chi$ = 2%	0.94602	0.25493	0.01724	0.00114
$\chi$ = 3%	0.95657	0.27131	0.01738	0.00115
$\chi$ = 4%	0.96666	0.29741	0.01778	0.00116
$\chi$ = 5%	0.97569	0.35121	0.01890	0.00131

presents the relationship between $Be$ and varying $\chi$ and $Ra$ under HFBC. When $Ra$ is 10³, increasing $\chi$ from 1% to 5% increases $Be$ from 0.93 to 0.98. Similarly, for Ra = 10⁴, $Be$increases from 0.25 to 0.35. The trend continues for $Ra$ values of 10⁵ and 10⁶, where $Be$ increases with increasing$\chi$. For Ra = 10⁵, $Be$ increases slightly from 0.017 to 0.019, whereas for $Ra$ = 10⁶, $Be$ appears to remain constant, increasing from 0.0011 to 0.0013. This indicates that when $Ra$ is held constant, increasing $\chi$ results in higher $Be$ values, signifying a greater rate of convective HT than the rate of E_gen. Table 6 further reveals that for $Ra$ = 10³, all $\chi$ (1–5%) yield Be values greater than 0.5, indicating that E_gen due to temperature gradient is more significant than E_gen due to fluid friction. However, for $Ra$ = 10⁴, $Ra$ = 10⁵, and $Ra$ = 10⁶, all $\chi$ result in $Be$values less than 0.5, suggesting that E_gen due to fluid friction is more dominant than E_gen due to temperature gradient.

Table 7. Variation of average entropy with different $\chi$ for HFBC.

	$\mathit{R}\mathit{a} = {10}^3$	$\mathit{R}\mathit{a} = {10}^4$	$\mathit{R}\mathit{a} = {10}^5$	$\mathit{R}\mathit{a} = {10}^6$
$\chi$ = 1%	0.3811	0.8437	6.6832	64.7459
$\chi$ = 2%	0.3503	0.8274	6.6609	64.3995
$\chi$ = 3%	0.3370	0.8236	6.5913	64.1616
$\chi$ = 4%	0.3308	0.8073	6.4776	63.1396
$\chi$ = 5%	0.3208	0.7976	6.3161	61.7198

depicts the variation of average entropy with varying $\chi$ for HFBC at different $Ra$. The E_avg decreases for all $Ra$ as $\chi$ increases from 1% to 5%. For instance, at $Ra$ = 10³, the E_avg decreases from 0.38 to 0.32 as $\chi$ increases from 1% to 5%. Similarly, for $Ra$ = 10⁴, the E_avg decreases from 0.84 to 0.79, for $Ra$ = 10⁵, the E_avg decreases from 6.68 to 6.32, and for $Ra$ = 10⁶, the E_avg decreases from 64.75 to 61.72. This indicates that increasing $\chi$ decreases the cavity’s overall entropy. The reason behind this reduction in entropy is attributed to the enhanced thermal conductivity and convective heat transfer due to the presence of nanoparticles. These properties facilitate more efficient heat transfer within the cavity, leading to a reduction in entropy. Therefore, adding nanoparticles can improve the system’s thermal performance but at the expense of a decrease in the overall entropy.

Table 8. Ecological Coefficient Performance (ECOP) with different $\chi$ for HFBC.

	Ra = 103	Ra = 104	Ra = 105	Ra = 106
$\chi$ = 1%	16.6405	12.7796	2.6997	0.4431
$\chi$ = 2%	19.5577	13.6847	2.8410	0.4627
$\chi$ = 3%	21.3561	13.6978	2.8787	0.4764
$\chi$ = 4%	21.6150	14.0579	3.0077	0.4956
$\chi$ = 5%	24.2235	14.2051	3.0283	0.5168

depicts how the Ecological Coefficient Performance $\left(ECOP=\frac{N{u}_{avg}}{E_{avg}}\right)$ is affected by varying$Ra$, given a heat flux boundary condition. Results show that, at $Ra$ = 10³, increasing from 1% to 5% causes ECOP to increase from 16.64 to 24.22. Similarly, at $Ra$ = 10⁴, ECOP rises from 12.78 to 14.21. The pattern of ECOP values for $Ra$ = 10⁵ and 10⁶ is also similar, with an increase in ECOP from 2.70 to 3.03 at $Ra$= 10⁵, and from 0.44 to 0.52 at $Ra$ = 10⁶. In general, an increase from 1% to 5% leads to a higher ECOP, indicating that adding nanoparticles to the fluid enhances energy efficiency and reduces environmental impact. By increasing ECOP through volume concentration, nanoparticles can improve energy and environmental efficiency. Additionally, Table 8 reveals that as $Ra$ increases from 10³ to 10⁶, there is a decrease in ECOP values, implying reduced energy efficiency and a higher environmental impact. This trend suggests that higher $Ra$ values may result in lower ECOP values, potentially indicating a reduction in energy efficiency and an increase in the system’s environmental impact.

6.5. Proposed Correlation for Nu_avg

In this section, a formula is introduced that establishes a correlation between Nu_avg and three factors: $Ra$, P$r$, and χ. The construction of the correlation model for Nu_avg is based on the values for ${10}^3\le Ra\le {10}^6,1\%\le \chi \le 5\%,Pr=6.77\mathrm{to}7.30$. Based on the mentioned assumptions, we propose the following general form for the correlation:

$$N{u}_{avg}=f+a\left(R{a}^b+P{r}^c+100 \chi d \right)$$

(26)

where $a$ = 1.918599, b = 0.199469, c = 0.179773, $d$ = 0.29138858, and $f$ = −4.78256978.

Furthermore, the accuracy of the predicted data acquired from the proposed model in relation to the actual data for Nu_avg is visually depicted in Figure 11. The correlation between the predicted and actual data, as indicated by an R² value of 0.99, demonstrates a strong agreement. Notably, the comparison between the actual values and Equation (26) reveals a satisfactory level of concordance. Critically, the analysis of residual plots confirms that the non-linear multiple regression model proposed in Equation (26) exhibits a bias concerning the actual values.

7. Conclusions

The current investigation analyzed ways to improve HT and E_gen in an octagonal cavity with a cylinder by varying$\chi$,$Pr$,$Ra$, and the boundary conditions (IBC and HFBC). The presence of the cold cylinder within the cavity has been found to have a significant impact on the system’s heat transfer characteristics, resulting in enhanced heat transfer performance. This observation emphasizes the importance of considering the geometrical arrangement of the cavity and its influence on heat transfer processes. Implementation of HFBC for both air and water leads to higher Nu_avg and lower E_avg in comparison to IBC. This highlights the effectiveness of HFBC in improving heat transfer and reducing entropy generation, thereby enhancing the overall thermal performance of the system. The study’s key findings are the following:

• The presence of a cold cylinder within the cavity significantly impacts the system’s heat transfer characteristics, increasing heat transfer;
• Using HFBC for air and water results in a higher Nu_avg and lower E_avg than IBC;
• At lower $Ra$ ($Ra$ = 10³), Nu_avg increases the most for air and water when HFBC is applied;
• The addition of nanoparticles enhances HT, and as $\chi$ increases, the rate of HT also increases;
• The average E_gen rises with an increase in Ra, and at higher values of $Ra$, such as $Ra$ = 10⁶, the average E_gen is higher;
• Increasing $\chi$ in a fluid leads to improved energy efficiency and reduced environmental impact;
• The rise in $Ra$ from 10³ to 10⁶ results in a reduction in energy efficiency and an increased environmental impact.

In the present study, the observation of particle clustering and its impact on the effective viscosity of the nanofluids has not been considered. This phenomenon may have a substantial impact on the convective heat transfer phenomenon. Our study serves as a preliminary investigation, providing insights into natural convection behavior under simplified conditions. Further studies considering the clustering effect would be necessary for a more comprehensive understanding of nanofluid behavior in realistic scenarios. The future perspective of the present study can be considered to be the following:

• The study is focused on the natural convection flow and heat transfer phenomenon and does not account for forced convection;
• $Ra$ is limited to the range of 10³ to 10⁶ for laminar flow in this study, but higher values of $Ra$ could be considered in future research;
• The investigation is limited to specific parameters for fluid flow and heat transfer characteristics. Other factors, such as cylinder shape, size, and cavity orientation, could impact the system’s behavior and should be studied in future research;
• The study only considers the Al₂O₃-H₂O nanofluid, and other types of nanofluids with different properties and characteristics may exhibit different behavior under similar conditions;
• In future research, the possibility of investigating the effects of moving the fixed cold cylinder towards the walls of the cavity could provide valuable insights into the system’s thermal behavior;
• Future research can explore the clustering effects and mass transport of particles to gain a deeper understanding of their impact on the system.