Thermal Analysis of Magneto-Natural Convection Flows within a Partially Thermally Active Rectangular Enclosure

Abstract

In this study, we numerically investigate heat transfer enhancement in a partially thermally active rectangular enclosure. The enclosure is filled with a ternary hybrid nanofluid (water, Carbon Nanotube, ${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$, and Graphene). It is subjected to a magnetic field and uniform internal heat generation. The study also investigates the effect of magnetic field strength and direction on the natural convection flow, which arises from density fluctuations caused by partial heating of the left vertical wall. To solve the dimensionless governing equations, the finite element approach is employed. The parameters studied in detail include Rayleigh number $\left(Ra\right),$ Hartmann number $\left(Ha\right),$ nanoparticle volume fraction $\left(\varphi \right)$, and heat generation coefficient $\left(\lambda \right).$ The findings are presented graphically for the range of the parameters as follows: ${10}^3\le Ra {\le 10}^6,$ $0\le Ha\le 20$, $0.01\le \varphi \le 0.05$, and $0\le \lambda \le 15$. It is noted that these parameters have an impact on heat transfer enhancement, flow patterns, and temperature fields. The results show that the average Nusselt number ($\overline{Nu}$) increases with an increasing value of $\varphi$. Moreover, it has been noted that $\overline{Nu}$ decreases as the value of $Ha$ increases, and the impact becomes more obvious at higher $Ra$ values. Finally, the influence of the heat generation coefficient on the heat transfer rate inside an enclosure is examined.

1. Introduction

Due to its diverse range of applications, including electronic appliances, microelectromechanical system appliances (MEMS), nuclear power plants, reactor cooling, buildings and thermally insulated systems, solar cells, the food storage industry, and geophysical fluid mechanics, the research of magnetohydrodynamic natural convection flows in an enclosure has lately acquired great attention [1]. Natural convection is influenced by magnetic fields in various practical instances, such as the formation of crystals in fluids, the casting of metals, the extraction of geothermal energy, and fusion reactors. Understanding the flow behavior and heat transport mechanism under the influence of magnetic field in an enclosure filled with electrically conducting fluid has become increasingly important [2,3,4,5]. When working with such fluids, two opposing forces, Lorentz and buoyancy forces (body forces), begin to interact with one another, thus altering the transport dynamics because of their mutual impacts.

External magnetic fields are increasingly used in the material manufacturing industry to control convection currents and improve product quality. Therefore, analysis of momentum and heat transport in such processes is essential. In the past, various researchers have investigated natural convection for different geometries and fluids subjected to magnetic fields [6]. For instance, Aydin and Yesiloz [7] investigated natural convection in a quadrantal enclosure filled with nanofluid with thermally active adjacent walls. Sathiyamoorthy and Chamkha [8] studied natural convection in a square cavity filled with liquid gallium and subjected to a magnetic field at an inclination angle $\varphi$ to the horizontal plane. A numerical study of magneto-natural convection flow was conducted by Rudraiah et al. [9] in a rectangular enclosure with vertically isothermal walls and insulated horizontal walls.

Nanofluids have gained significance in heat transfer as the demand for heat transfer enhancement has grown. Choi and Eastman [10] proposed nanofluid, which several researchers are now considering. The use of nanofluids has been investigated in various engineering applications to enhance heat transfer performance. Researchers have extensively studied nanofluids with enhanced thermal properties to increase thermal efficiency in numerous engineering applications. Most researchers have found that incorporating high-thermal-conductivity nanoparticles into the base fluid enhances the thermal efficiency of the resulting nanofluid [11,12,13]. Some researchers believe that nanoparticles added to the base fluid can reduce heat transmission significantly. When it comes to enhanced enclosure design that is subjected to these factors, researchers should consider these factors. Thus, it is crucial to gain a better knowledge of the phenomenon.

A relatively new class of nanofluids (NFs), known as “hybrid” nanofluids, has recently appeared in parallel to the ongoing developments of the common nanofluids. This new class is produced by suspending many types of nanoparticles in a base fluid. A key objective of hybridization is compromised qualities between the advantages and disadvantages of the properties of specific nanoparticles. Additionally, the pricing of various nanoparticle varieties is substantially varied from one to another among providers of nanoparticles. For instance, copper nanoparticles cost around ten times more than alumina nanoparticles. Therefore, it is economical if one obtains the qualities of costly nanoparticles using a minimal amount. In light of this, scientists have developed modern-designed equipment by combining a mixture of three nanoparticles in the natural convection techniques. This has led to a significant improvement in thermal efficiency (heat transfer augmentation). Ternary hybrid nanofluid (THNF) is the term given to this new fluid. The creation of such a composite nanofluid is innovative since it significantly enhances thermophysical and heat transfer properties. Experimental studies show that these fluids’ thermal conductivities are higher than those of hybrid and mono nanofluids [14,15].

A numerical analysis of magnetohydrodynamic natural convection was performed by Raisi et al. [16] in a square enclosure containing nanofluid (water-${Al}_2{O}_3$). The study examined multiple parameters and demonstrated that heat transfer improved with increasing Rayleigh number $\left(Ra\right)$ but decreased with increasing Hartmann number $\left(Ha\right)$. Kahveci and Oztuna [17] found that the heat transfer performance of an enclosure can be affected by the magnetic field’s direction. Their investigation indicated that the orientation of the magnetic field had a substantial impact on the heat transfer rate within the enclosure. Specifically, their findings showed that a magnetic field aligned in the x-direction was more effective at inhibiting heat transfer than a vertically oriented magnetic field. The numerical analysis carried out by P. Kandaswamy et al. [18] focused on magnetoconvection in a partially thermally active square cavity. The study found that low $Ha$ did not affect heat transfer rates at low $Ra.$ A numerical investigation by A. Mahmoudi et al. [19] examined the impact of a magnetic field and heat generation/absorption on heat transfer rates and entropy generation in a water-${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$ nanofluid-filled square cavity using numerical methods. The study concluded that heat transfer rate was influenced by the heat production/absorption coefficient in the range of 10 < q < 5, but the entropy generation rate remained unaffected.

The present study aims to investigate flow behavior and heat transfer enhancement in a partially thermally active rectangular enclosure filled with a ternary hybrid nanofluid consisting of water (base fluid) and Carbon Nanotubes, ${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$, and Graphene nanoparticles in natural convection flows. The study aims to analyze the influence of Rayleigh number $\left(Ra\right),$ heat generation coefficient $\left(\lambda \right),$ and Hartmann number $\left(Ha\right)$ on flow patterns and temperature fields, as well as on the heat transfer rate within an enclosure. The magnetic field can alter the natural convection phenomena, which can be used in microgravity conditions as well as for suppressing the natural convection effect under gravity if it is needed.

2. Mathematical Formulation

2.1. Problem Description

Consider a steady, laminar flow of an incompressible ternary hybrid nanofluid in a two-dimensional rectangular enclosure exposed to a uniform magnetic field of strength $B_0$ at an inclination $\gamma$ in the presence of uniform internal heat generation. The left vertical wall is partly subjected to constant high temperature ${\theta }_h$ while the right vertical wall is partly subjected to cold temperature ${\theta }_c$ as shown in Figure 1. The enclosure’s remaining boundaries are thermally insulated. Inside the enclosure, the radiation heat transfer effect is insignificant. Constant fluid properties are considered, and the variation in density that causes buoyant forces is approximated using the Boussinesq approximation.

2.2. Governing Equations

The continuity, change of linear momentum, and energy equations for steady, two-dimensional, laminar, incompressible flow in Cartesian coordinates in dimensional form are given below:

$$\frac{\partial \mathrm{u}}{\partial \mathrm{x}}+\frac{\partial \mathrm{v}}{\partial \mathrm{y}}=0$$

(1)

where u and v denote the x and y components of velocity, respectively.

$$u\frac{\partial \mathrm{u}}{\partial \mathrm{x}}+v\frac{\partial \mathrm{u}}{\partial \mathrm{y}}=-\frac{1}{{\rho }_{thnf}}\frac{\partial p}{\partial x}+\frac{{\mu }_{thnf}}{{\rho }_{thnf}}{\nabla }^{2}u+\frac{1}{{\rho }_{thnf}}{\sigma }_{thnf}{B}_0^2(v\mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\gamma }\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\gamma }-u{\mathrm{s}\mathrm{i}\mathrm{n}}^2\mathsf{\gamma })$$

(2)

$$u\frac{\partial v}{\partial \mathrm{x}}+v\frac{\partial v}{\partial \mathrm{y}}=-\frac{1}{{\rho }_{thnf}}\frac{\partial p}{\partial x}+\frac{{\mu }_{thnf}}{{\rho }_{thnf}}{\nabla }^{2}v+\frac{({\rho \beta )}_{thnf}}{{\rho }_{thnf}}g\left(\theta -{\theta }_c\right)+\frac{1}{{\rho }_{thnf}}{\sigma }_{thnf}{B}_0^2(u\mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\gamma }\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\gamma }-v{\mathrm{c}\mathrm{o}\mathrm{s}}^2\mathsf{\gamma })$$

(3)

$$u\frac{\partial \mathsf{\theta }}{\partial \mathrm{x}}+v\frac{\partial \mathsf{\theta }}{\partial \mathrm{y}}={\mathsf{\alpha }}_{\mathrm{t}\mathrm{h}\mathrm{n}\mathrm{f}}{\nabla }^2\mathsf{\theta }+\frac{\mathrm{Q}}{{(\mathsf{\rho }{\mathrm{C}}_{\mathrm{p}})}_{\mathrm{t}\mathrm{h}\mathrm{n}\mathrm{f}}}(\mathsf{\theta }-{\mathsf{\theta }}_{\mathrm{c}})$$

(4)

where the fluid velocity and temperature are represented by $v,T,$ and $p$ being the pressure of fluid, respectively, where $x$ and $y$-components of velocity are indicated by 𝑢 and 𝑣, respectively. ${\rho }_{thnf}$ is the effective density and ${\sigma }_{thnf}$ represents electrical conductivity, ${\alpha }_{thnf}$ being the thermal diffusivity of the ternary hybrid nanofluid and ${\mu }_{thnf}$ indicating the dynamic viscosity of the ternary hybrid nanofluid. Thermal expansion coefficient is represented by 𝛽 and magnetic field strength density as $B_0$, respectively.

The dimensional boundary conditions for the governing equations are

$$u=0,v=0,\hspace{1em}\hspace{1em}\theta ={\theta }_h at x=0, a\le y\le b$$

(5a)

$$u=0,v=0,\hspace{1em}\hspace{1em}\theta ={\theta }_c at x=1.5, c\le y\le d$$

(5b)

$$u = 0, v = 0,\hspace{1em}\hspace{1em}\frac{\partial \theta }{\partial x}= 0 at y = 0 , 0 \le y \le a, b \le y\le 1$$

(5c)

$$u = 0, v= 0,\hspace{1em}\hspace{1em}\frac{\partial \theta }{\partial x} = 0 at y = 1.5 , 0 \le y \le c, d \le y \le 1$$

(5d)

$$u=0,v=0,\hspace{1em}\hspace{1em}\frac{\partial \theta }{\partial y}=0 at y=0 and y=1$$

(5e)

2.3. Thermophysical Properties

Nanofluids demonstrate altered viscosity, density, specific heat capacity, and surface tension, among other properties. These modifications are highly dependent on factors like nanoparticle concentration, size, shape, and material. As a result, nanofluids offer immense potential in fields like electronics cooling, solar energy systems, automotive industries, and thermal management of electronic devices.

In this study, the thermophysical properties of nanofluid are calculated based on ref. [20]

$${\rho }_{thnf}={\varphi }_1{\rho }_{ps1}+{\varphi }_2{\rho }_{ps2}+{\varphi }_3{\rho }_{ps3}+\left(1-{\varphi }_1-{\varphi }_2-{\varphi }_3\right){\rho }_{bf}$$

(6)

$${\beta }_{thnf}={\varphi }_1{\beta }_{s1}+{\varphi }_2{\beta }_{s2}+{\varphi }_3{\beta }_{s3}+\left(1-{\varphi }_1-{\varphi }_2-{\varphi }_3\right){\beta }_{bf}$$

(7)

$$({\rho C_p)}_{thnf}={\varphi }_1{\left(\rho C_p\right)}_{s1}+{\varphi }_2{\left(\rho C_p\right)}_{s2}+{\varphi }_3{\left(\rho C_p\right)}_{s3}+\left(1-{\varphi }_1-{\varphi }_2-{\varphi }_3\right){\left(\rho C_p\right)}_{bf}$$

(8)

$$\begin{gathered} {\mu }_{thnf}=\frac{{\varphi }_1{\mu }_{nf1}+{\varphi }_2{\mu }_{nf2}+{\varphi }_3{\mu }_{nf3}}{\varphi }, k_{thnf}=\frac{{\varphi }_1{k}_{nf1}+{\varphi }_2{k}_{nf2}+{\varphi }_3{k}_{nf3}}{\varphi } \\ {\alpha }_{thnf}=\frac{k_{thnf}}{({\rho C_p)}_{thnf}} \end{gathered}$$

(9)

${\mu }_{thnf}, k_{thnf}$, ${\beta }_{thnf, }{\alpha }_{thnf}$, and ${\rho }_{thnf}$ represent the dynamic viscosity, thermal conductivity, thermal expansion coefficient, thermal diffusivity, and density of the ternary hybrid nanofluid, respectively.

$${\mu }_{nf1}=(1+{2.5\varphi }_1+{{\varphi }_1}^2){\mu }_{bf}$$

(10)

$${\mu }_{nf2}=(1+{13.5\varphi }_2+{{904.4\varphi }_2}^2){\mu }_{bf}$$

(11)

$${\mu }_{nf3}=(1+{37.1\varphi }_3+{{612.6\varphi }_3}^2){\mu }_{bf}$$

(12)

$$k_{nf1}=k_{bf}\left[\frac{\left(k_1+2k_{bf}\right)+2{\varphi }_1\left(k_1-k_{bf}\right)}{\left(k_1+2k_{bf}\right)-{\varphi }_1\left(k_1-k_{bf}\right)}\right]$$

(13)

$$k_{nf2}=k_{bf}\left[\frac{\left(k_2+3.9k_{bf}\right)+3.9\left(k_2-k_{bf}\right)}{\left(k_2+3.9k_{bf}\right)-{\varphi }_2\left(k_2-k_{bf}\right)}\right]$$

(14)

$$k_{nf3}=k_{bf}\left[\frac{\left(k_3+4.7k_{bf}\right)+4.7\left(k_3-k_{bf}\right)}{\left(k_3+4.7k_{bf}\right)-{\varphi }_3\left(k_3-k_{bf}\right)}\right]$$

(15)

where 1, 2, and 3 represent ${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3,$ CNT, and Graphene, respectively. ${{\mu }_{nf}}_i$, ${k_{nf}}_i$ where i = 1, 2, and 3 represent the viscosity and thermal conductivity of nanoparticles. ${\varphi }_i$ and ${C_p}_i$ represent the volume fraction and specific heat of the nanoparticles, respectively.

For n and $\psi$, representing the shape factor and sphericity of nanoparticles, respectively, the relationship is as follows.

$$n=\frac{3}{\psi }$$

(16)

$\psi =1,n=3$ (Spherical-shaped nanoparticle)

$\psi =0.61,n=4.9$ (Cylindrical-shaped nanoparticle)

$\psi =0.52,n=5.7$ (Platelet-shaped nanoparticle)

The thermophysical properties of thermal conductivity and dynamic fluid viscosity of the ternary hybrid nanofluid also vary with particle shape. This means that the shape of particles (cylindrical, spherical, and platelet-shaped particles) can have a significant effect on these properties.

2.4. Non-Dimensional Form

The following are the dimensionless groups used in the governing equations, which are presented in their dimensionless form as:

$$\begin{gathered} \left(X,Y\right)=\frac{\left(x,y\right)}{L},U=\frac{uL}{{\alpha }_f},V=\frac{vL}{{\alpha }_f},P=\frac{p{L}^2}{{\rho }_{nf}{{\alpha }_f}^2},T=\frac{\theta -{\theta }_C}{{\theta }_h-{\theta }_c}, \\ Ra=\frac{{\beta }_{f}g{L}^3\left({\theta }_h-{\theta }_c\right)}{{\nu }_f{\alpha }_f},Pr=\frac{{\nu }_f}{{\alpha }_f},Ha=B_{0}L\sqrt{\frac{{\sigma }_{nf}}{{\rho }_{nf}{\nu }_f}} \end{gathered}$$

(17)

$$\frac{\partial \mathrm{U}}{\partial \mathrm{X}}+\frac{\partial \mathrm{V}}{\partial \mathrm{Y}}=0$$

(18)

$$U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=-\frac{{\rho }_f}{{\rho }_{thnf}}\frac{\partial P}{\partial X}+Pr\left[\frac{\frac{{\mu }_{thnf}}{{\mu }_f}}{\frac{{\rho }_{thnf}}{{\rho }_f}}\right]{\nabla }^{2}U+{Ha}^{2}Pr(Vsin\gamma cos\gamma -U{sin}^2\gamma )U$$

(19)

$$U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{{\rho }_f}{{\rho }_{thnf}}\frac{\partial P}{\partial Y}+Pr\left[\frac{\frac{{\mu }_{thnf}}{{\mu }_f}}{\frac{{\rho }_{thnf}}{{\rho }_f}}\right]{\nabla }^{2}V+RaPr\left[\frac{{\beta }_{thnf}}{{\beta }_f}\right]T+{Ha}^2\mathrm{Pr}(Usin\gamma cos\gamma -V{cos}^2\gamma )V$$

(20)

$$U\frac{\partial T}{\partial X}+V\frac{\partial T}{\partial Y}=\frac{{\alpha }_{thnf}}{{\alpha }_f}{\nabla }^{2}T+\lambda T$$

(21)

2.5. Boundary Conditions

The dimensionless boundary conditions used to solve the governing equations are

$$\begin{array}{cc}\hspace{0.17em} U=0,V=0,\hfill & T=T_h at X=0, a\le Y\le b\hfill \end{array}$$

(22a)

$$\begin{array}{cc} \hspace{0.17em} \hspace{0.17em}U=0,V=0,\hfill & T=T_c at X=1.5, c\le Y\le d\hfill \end{array}$$

(22b)

$$\begin{array}{cc} U = 0, V = 0, \hfill & \frac{\partial T}{\partial X}= 0 at X = 0 , 0 \le Y \le a, b \le Y \le 1\hfill \end{array}$$

(22c)

$$\begin{array}{cc} \hspace{0.17em}\hspace{0.17em}U = 0, V = 0, \hfill & \frac{\partial T}{\partial X} = 0 at X = 1.5 , 0 \le Y \le c, d \le Y \le 1\hfill \end{array}$$

(22d)

$$\begin{array}{cc}U=0,V=0,\hfill & \frac{\partial T}{\partial Y}=0 at Y=0 and Y=1\hfill \end{array}$$

(22e)

2.6. Nusselt Number

Heat transfer enhancement within an enclosure is indicated by the dimensionless Nusselt number, with an increase in Nusselt number corresponding to thermal enhancement. The local and mean Nusselt number are given by the following expressions, respectively:

$$Nu\left(Y\right)=-\left(\frac{k_{thnf}}{k_{bf}}\right){\left.\frac{\partial T}{\partial X}\right|}_{X=0}$$

(23)

$$\overline{Nu}=\frac{1}{l}\underset{0}{\overset{l}{\int }}Nu\left(Y\right)dY$$

(24)

where $l$ represents the length of thermally active vertical walls.

The thermophysical properties for ternary hybrid nanofluid are influenced by the properties of water and nanoparticles, as outlined in

Table 1. Numerical values of thermo-physical properties of base fluid (Water) and nanoparticles (CNT, ${Al}_2{O}_3$, and Graphene) [21].

Property	Water	Al2O3	Graphene	CNT
k (W/(m K))	0.613	40	5000	3007.4
$\rho (\mathrm{k}\mathrm{g}/{\mathrm{m}}^3)$	997.1	3970	2200	2100
$\beta \left({\mathrm{K}}^{-1}\right)$	2.1 × 10−4	5.80 × 10−6	−0.000008	0.00002
Cp (J/(kg K))	4179	765	790	410
μ (kg/(ms))	8.91 × 10−5	-
Shape	-	spherical	platelet	cylindrical

.

2.7. Numerical Scheme and Code Validation

The Navier–Stokes and heat energy equations are mathematically modeled as partial differential equations using COMSOL Multiphysics v6.0, a commercial software that employs the finite element method. The computational domain is partitioned into several elements and a convergence criterion of $\overline{\left|\frac{{\xi }^{n+1}-{\xi }^n}{{\xi }^n}\right|\le {10}^{-6}}$ is set for all cases where $\xi$ depicts the pressure, velocity, and temperature components. Mesh independence test is performed for the rectangular cavity with partially thermally sensitive vertical walls for $Ra={10}^5$ as shown in

Table 2. Mesh dependency for rectangular enclosure, $Ra={10}^5$.

RL	#Elements	#DOF	Avg Nu	% Error
Normal	2088	5044	2.3759	0.88
Fine	3592	8268	2.3972	2.20
Finer	8972	20,288	2.4510	1.45
Extra Fine	22,644	49,792	2.4872	0.004
Extremely Fine	37,430	79,364	2.4871	-

. A grid sensitivity test is performed by taking into consideration five different elements, 2088, 3592, 8972, 22,644, and 37,430, and the grid density containing 37,430 elements is selected for all computations. The present scheme is first validated by verifying the published work of S. Dutta et al. [22]. The $Nu$ variation on the bottom heated wall for $Ra={10}^6$ and $\varphi =0.05$ (a) for numerical work [22] and (b) for the present work is shown in Figure 2.

Table 3. $\overline{Nu}$ for several $Ha$ where $\varphi =0.05$ and $Ra={10}^4$.

Ha	$\overline{\mathit{N}\mathit{u}}$
	S. Dutta et al. [22]	Present Work	Relative Difference %
0	4.945	4.9033	0.84
20	4.273	4.1749	2.30
40	4.029	3.9726	1.40
60	4.000	3.9544	1.14
80	3.995	3.9510	1.10
100	3.993	3.9501	1.07

presents the results of the validation of the method, where $\overline{Nu}$ values for various $Ha$ values at a fixed $\varphi =0.05$ and $Ra={10}^6$ were compared to the findings of S. Dutta et al. [22]. The obtained values are in good agreement with that of S. Dutta et al. [22].

3. Results and Discussion

A numerical analysis was performed in this work and the results are displayed in terms of isotherms and streamlines for $Ha=0,5,10,15 \mathrm{a}\mathrm{n}\mathrm{d} 20$, $A=1.5,$ $\varphi =0.03$, and ${10}^3\le Ra\le {10}^6$. The flow of fluid and heat transfer enhancement within an enclosure depend on parameters such as Rayleigh number ($Ra$), the strength of the magnetic field $\left(Ha\right)$, and nanoparticle volume fraction $\varphi .$

3.1. Analysis of Flow Fields

The influence of $Ra$ and $Ha$ on flow behavior was investigated in terms of streamlines and is shown in Figure 3, Figure 4, Figure 5 and Figure 6. The Rayleigh number plays a crucial role in determining the flow pattern during natural convection. When there are temperature differences within a fluid, free convection can occur due to changes in density caused by these temperature variations. Specifically, when the temperature increases, the density of fluid decreases, leading to an upward motion of the fluid driven by buoyancy forces. For $Ra\le {10}^4$, flow is not induced as the viscous forces predominate the buoyant and the Lorentz force, and the heat transfer is primarily through the conduction mechanism. It can be seen that the fluid near to the left vertical wall becomes heated and moves to the right vertical cold wall. The fluid descends after coming in contact with the cold vertical wall and then rises again along the left vertical heated wall, thus developing recirculation zones. Convection is considerably reduced near insulated boundaries. The recirculation cell becomes smaller with an increasing value of $Ha$ for $Ra\le {10}^4$, as can be seen in Figure 3 and Figure 4. On the other hand, with increasing $Ha$, the cell formed in the enclosure moves to the side walls and becomes bigger for higher values of $Ra\ge {10}^5$. Thus, the dominance of convective effects can be seen at higher values of $Ra$ which eventually increases the strength of vortices. It is worth noting that at lower $Ha$, convective effects cannot be reduced by the magnetic field.

3.2. Analysis of Temperature Fields

A detailed investigation of the isotherms can help to understand the physics of heat transfer and thus is presented in Figure 3, Figure 4, Figure 5 and Figure 6. Isotherms are evenly distributed and nearly parallel to one another with a slight twisting, particularly for $Ra\le {10}^4$, which shows the dominance of the conduction heat transfer mechanism in an enclosure at low $Ra\le {10}^4.$ An intriguing observation is made that for larger values of $Ra$, the isotherms approach the cold vertical side wall with an increasing Hartmann number, which can be ascribed to stronger convective effects within an enclosure with the warm fluid moving to the heated wall and the corresponding low temperature fluid to the right colder wall. For $Ra={10}^6$, the isothermal lines are twisted for each $Ha$, indicating the predominant convective effects within an enclosure.

3.3. Combined Effect of $Ra,Ha$, and $\varphi$ on Heat Transfer Rate

Table 4. $\overline{Nu}$ along heated vertical wall for varying Ra, Ha and $\varphi$.

$\overline{\mathit{N}\mathit{u}}$
$\mathit{\varphi }$	$\mathit{H}\mathit{a}$	$\mathit{R}\mathit{a}={10}^3$	$\mathit{R}\mathit{a}={10}^4$	$\mathit{R}\mathit{a}={10}^5$	$\mathit{R}\mathit{a}={10}^6$
0.01	0	0.64429	1.3910	2.9575	5.5226
	5	0.63117	1.3379	2.9104	5.4951
	10	0.60855	1.2098	2.7823	5.4154
	15	0.59313	1.0620	2.6022	5.2904
	20	0.58521	0.92965	2.3989	5.1294
0.03	0	0.65537	1.2182	2.7273	5.1881
	5	0.65117	1.1823	2.6887	5.1642
	10	0.64316	1.0930	2.5823	5.0947
	15	0.63686	0.98583	2.4303	4.9853
	20	0.63312	0.88758	2.2557	4.8439
0.05	0	0.69215	1.0678	2.4871	4.8571
	5	0.69097	1.0456	2.4562	4.8364
	10	0.68846	0.98914	2.3702	4.7760
	15	0.68615	0.91960	2.2450	4.6807
	20	0.68456	0.85474	2.0986	4.5570

presents the impact of nanoparticle volume fraction $\left(\varphi \right),$ Hartmann number $\left(Ha\right)$, and Rayleigh number $\left(Ra\right)$ on the average Nusselt number $(\overline{Nu})$ for the heated left vertical wall. The relationship between $\overline{Nu}$ and $Ha$ is heavily influenced by $Ra$. This can be observed from the variations in $\overline{Nu}$ as $Ra$ changes. At low $Ra$, heat transfer via conduction dominates, and the magnetic field has a minor impact. For $Ra\le {10}^3$, the magnetic field has no significant effect on $\overline{Nu}.$ For $Ra={10}^4$, the effect of increasing $Ha$ on $\overline{Nu}$ is smaller, and this is true for all values of $\varphi$. However, at higher values of $Ra={10}^6$, there is a significant impact of magnetic field in reducing $\overline{Nu}$. When the Rayleigh number $\left(Ra\right)$ is sufficiently high, convection heat transfer becomes more dominant than the conduction mode of heat transfer, especially in the absence of a magnetic field. This is because the buoyancy force, which increases with $Ra,$ becomes stronger than viscous forces, leading to the transition from a conduction-dominated mechanism to a convection-dominated mechanism. Nonetheless, in the presence of a magnetic field within the enclosure, the drop in $\overline{Nu}$ values is more significant for larger $Ha$ values at $Ra$ values of ${10}^5$ and ${10}^6$ compared to $Ra$ values of ${10}^3$ and ${10}^4.$ Therefore, the magnetic field plays a critical role in reducing $\overline{Nu}$ values, and this reduction is clearly evident at larger $Ra$ values. Moreover, nanofluid presence causes a rise in $\overline{Nu}$ with increasing $\varphi$, resulting in an increased heat transfer rate.

3.4. Influence of Magnetic Field Inclination Angle on Heat Transfer Rate

The effect of magnetic field inclination angle on average Nusselt for the given $Ra$ values, $Ha$ and ${\varphi }_{}$, is shown in Figure 7. Fluid flow in a natural convection-dominated regime is caused by the difference in temperature between the thermally active right and left enclosure walls. When the applied magnetic field is 90° (or 270°) inclined, the magnetic field direction coincides with that of the high-temperature fluid (or low-temperature fluid).

This results in improved heat energy transportation, which also improves the efficiency of heat transfer. However, the magnetic effect prevents the high/low-temperature fluid from flowing when the applied magnetic field has an inclination angle of 0° (or 180°). Consequently, $\overline{Nu}$ drops. The magnetic field’s vertical component facilitates fluid flow and heat energy transportation effects inside an enclosure for all other magnetic field orientation angles. Mean value of Nusselt number therefore exhibits a slight improvement. Hence, a magnetic field directed in a horizontal direction result in significant decrease in $\overline{Nu}$.

3.5. Effect of Hartmann Number on Heat Transfer Rate

The impact of $Ha$ on heat transfer efficiency within an enclosure filled with a ternary hybrid nanofluid, with a volume fraction of nanoparticles $\varphi =0.05$, is depicted in Figure 8. As $Ra$ rises, the heat transfer rate also increases. Conversely, an increase in $Ha$ from 0 to 20 leads to a decrease in heat transfer rate. The Hartmann number has a notable impact on heat transfer within an enclosure due to its influence on the strength and orientation of the magnetic field, which in turn affects fluid flow and heat transfer. At low $Ha$, natural convection flow predominates, and the magnetic field is weak, while at high Hartmann numbers, the magnetic field dominates over viscous, resulting in significant suppression of fluid motion and decrease in convective heat transfer. In fact, the $\overline{Nu}$ remains constant, as $Ha$ is increased for $Ra={10}^3$ due to the predominance of the conduction heat transfer mechanism, and the magnetic field has no noticeable impact on heat transfer enhancement. When $Ra={10}^6$, convection heat transfer predominates and the $\overline{Nu}$ decreases as $Ha$ increases.

3.6. Influence of Rayleigh Number on Heat Transfer Performance

The impact of $Ra$ on $\overline{Nu}$ for distinct values of $Ha$ is shown in Figure 9. Clearly, a rise in $Ra$ corresponds to a rise in $\overline{Nu}$. $Ra$ is the crucial variable that governs how the flow is configured in natural convection. Free convection occurs when a fluid’s density changes owing to a temperature difference in the fluid. Buoyancy forces are what cause the fluid to rise when the temperature rises since the density of the fluid changes. Thus, it can be inferred that when $Ra$ rises, $\overline{Nu}$ along the vertically heated section rises as a result of improved free convection heat transfer. The effect of $Ha$ on $\overline{Nu}$ at $Ra={10}^3$, where heat transfer by conduction is dominant, is negligible. Contrarily, it may be observed that $Ha$ only significantly affects the $\overline{Nu}$ at higher $Ra\ge {10}^6$ values when convective flow is prevented by the magnetic field.

3.7. Influence of Hartmann Number on Velocity Profile

Figure 10 shows the impact of $Ha$ on vertical velocity component for different values of Rayleigh number. With an increase in $Ha$, the velocity component’s amplitude initially increases and then decreases. This is because magnetic effects have a resistive nature, which reduces the velocity profile and, as a result, reduces the amount of heat transported by convection in the regime. In other words, buoyancy forces are negatively impacted by magnetic forces, which causes conduction to predominate the convection mechanism at higher values of $Ha$. The magnetic field exerts Lorentz forces that resist the flow distribution, leading to a decrease in velocities and a suppression of convection as the $Ha$ increases. Increasing $Ha$ results in a decrease in velocity due to the impact of a strong magnetic field on fluid flow.

3.8. Influence of Rayleigh Number on the Velocity Profile

The effect of $Ra$ on vertical velocity component for different values of $Ha$ is shown in Figure 11. The velocity component is nearly linear for $Ra\le {10}^4$. For $Ra={10}^5$ there is a slight change in amplitude of the velocity curve. The amplitude of the velocity curve increases drastically for $Ra={10}^6$. In natural convection within an enclosed system, the Rayleigh number has a significant impact on the fluid’s velocity. With an increase in Rayleigh number, the buoyancy forces become more dominant than the viscous forces, resulting in faster fluid movement. For instance, when the left wall is heated and the right wall is cooled, the fluid near the hot surface expands and becomes less dense, causing it to rise due to buoyancy force. As it rises, it cools and becomes denser, resulting in it descending toward the cooler surface. This creates a convection current that propels the fluid motion within an enclosure. With an increase in the $Ra$, the buoyancy forces become even stronger relative to the viscous forces, causing a more vigorous convection current, and therefore an increase in the fluid’s velocity.

3.9. Impact of Internal Heat Generation

The variation of $\overline{Nu}$ along the heated wall is depicted in Figure 12, as a function of the heat generation coefficient $\left(\lambda \right)$ and Rayleigh number $\left(Ra\right)$ for $Ha=20$ and $\varphi =0.05.$ According to the results, the rate of heat transfer along the heated vertical wall reduces as the heat generation coefficient ($\lambda$) increases, regardless of the given $Ra$ values. When $\lambda \ge 0$, the temperature of nanofluids is more elevated than that of hot and cold vertical walls for $Ra={10}^3$ and ${10}^4$, respectively. The nanofluids heat up both side walls as a result. This results in a negative Nusselt number at the heated wall. Figure 12b illustrates the influence of the heat production coefficient and the solid volume fraction on the heat transfer rate at the heated wall. With an increase in the heat generation parameter’s value, the $\overline{Nu}$ decreases. For the high value of the heat-generation condition i.e., $\lambda \ge 0$, the addition of the nanoparticle has a significant effect. Figure 12c depicts the impact of heat generation or absorption coefficient on the $\overline{Nu}$ of the hot wall for several $Ha.$ The figure shows that for any $Ha$, the heat transfer rate at the heated wall decreases as the heat generation coefficient increases. The temperature in the cavity is higher under the heat-generating condition than it is under the non-heat generation condition $(\lambda =0).$ As a result, when the internal heat generation is present, the rate of heat transfer close to the heated walls of the cavity is lower than it would be otherwise. Thus, a decrease in $Ha$ is observed for increasing value of $\lambda$.

The impact of heat generation on temperature and flow fields, respectively, is presented in Figure 13. Initially, a single clockwise circulation cell that fills the whole cavity initially defines the flow field, and the isotherms exhibit a minor twisting toward the cold vertical wall, showing a dominance of conduction. This fact can be attributed to the external heating (temperature gradient). Internal heat generation eventually has an effect that is comparable to external heating. The formation of small secondary cells on the opposing sides of an enclosure, to the right and left of the heated and cold walls, is generated by increasing the heat generation parameter. This causes intense flow circulation inside an enclosure. An internal heating speeds up the enclosure’s rate of heat transmission in the temperature field. The temperature of the enclosure’s upper liquid layers tends to rise as a result of additional internal heat in the fluid. This alters the isothermal lines’ shape, and the thermal fields are concentrated towards an enclosure’s right half. The temperature of the nanofluid rises as the coefficient of heat generation increases, resulting in decreased heat transfer near the heated wall and promoting it towards the cold wall. The concentration of the isotherms near the cold wall for $\lambda =5$ suggests a significant temperature gradient.

3.10. Comparison among Different Nanofluids

Figure 14 illustrates the variation of $\overline{Nu}$ with solid volume fraction $\varphi =$ 0.01, 0.03, and 0.05 at $Ha=20$ and $Ra={10}^3$ for various nanoparticles (${Al}_2{O}_3$, CNT, Graphene)$.$ Table 1 lists the thermophysical characteristics of the mentioned nanoparticles. Figure 14 demonstrates that nano thermal conductivity is a key factor in heat transfer, and that water–Graphene nanofluid enhances the heat transport performance more than water–${Al}_2{O}_3$ and water–CNT nanofluid. Increasing the volume fraction of nanoparticles has a noticeable impact when natural convection is very low, or when the $Ha$ is higher, and the $Ra$ is lower.

4. Conclusions

This study uses numerical simulations to investigate natural convection in a rectangular enclosure filled with ternary hybrid nanofluid that is partially thermally active and subject to an inclined magnetic field with internal heat generation. The study demonstrates that the Rayleigh number $\left(Ra\right),$ Hartmann number $\left(Ha\right)$, and nanoparticle volume fraction $\left(\varphi \right)$ affect the flow characteristics and heat transport rate in the enclosure. The following is a brief overview of the key findings.

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The streamlines are concentrated on active sections of the vertical walls, and isotherms are distorted for higher $Ra$ values due to the dominant convection effects in the enclosure.

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The rate of heat transport inside the enclosure is affected by flow parameters including $Ra,Ha,$ and $\varphi$.

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The $\overline{Nu}$ decreases with increasing $Ha$ and at larger $Ra$ values. Rayleigh numbers play a significant role in controlling flow characteristics in natural convection due to stronger buoyancy effects.

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At higher $Ra$ values, $\overline{Nu}$ shows a decreasing trend with high $\varphi$ due to the dominant Lorentz force.

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The rate of heat transfer was found to be most significantly reduced when the magnetic field was directed horizontally.

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Increasing the value of $\lambda$, the rate of heat transmission at the hot wall decreases for all $Ra$ values, while it increases close to the cold wall.