Numerical Analysis of Natural Convection in an Annular Cavity Filled with Hybrid Nanofluids under Magnetic Field

Abstract

This paper presents a numerical study of natural convection in an annular cavity filled with a hybrid nanofluid under the influence of a magnetic field. This study is significant for applications requiring enhanced thermal management, such as in heat exchangers, electronics cooling, and energy systems. The inner cylinder, equipped with fins and subjected to uniform volumetric heat generation, contrasts with the adiabatic outer cylinder. This study aims to investigate how different nanoparticle combinations (Fe₃O₄ with Cu, Ag, and Al₂O₃) and varying Hartmann and Rayleigh numbers impact heat transfer efficiency. The finite volume method is employed to solve the governing equations, with simulations conducted using Fluent 6.3.26. Parameters such as volume fraction (ϕ₂ = 0.001, 0.004, 0.006), Hartmann number (0 ≤ Ha ≤ 100), Rayleigh number (3 × 10³ ≤ Ra ≤ 2.4 × 10⁴), and fin number (N = 0, 2, 4, 6, 8) are analyzed. Streamlines, isotherms, and induced magnetic field contours are utilized to assess flow structure and heat transfer. The results reveal that increasing the Rayleigh number and magnetic field enhances heat transfer, while the presence of fins, especially at N = 2, may inhibit convection currents and reduce heat transfer efficiency. These findings provide valuable insights into optimizing nanofluid-based cooling systems and highlight the trade-offs in incorporating fins in thermal management designs.

1. Introduction

Hybrid nanofluid has emerged as a promising heat transfer medium due to its enhanced heat transfer capabilities compared to conventional fluids.

These nanofluids are carefully engineered mixtures in which nanoscale solid particles are meticulously dispersed in a base fluid, typically water, oil, or ethylene glycol. These particles can be composed of a wide range of materials, including metals such as copper, silver, or gold; non-metals like carbon nanotubes or graphene oxide; and metal oxides such as titanium dioxide or alumina. The primary goal of nanofluid formulation is to improve the thermal and physical properties of the base fluid. As a result, nanofluids are widely used in heat transfer applications, including the cooling of electronics, automotive engines, and industrial processes, as well as in solar thermal systems and various manufacturing processes where precise temperature control and efficient heat transfer are crucial. They also find applications in the automotive and aerospace industries.

Since the introduction of the concept by Choi and Eastman [1], who suggested enhancing a base fluid’s thermal conductivity by dispersing nanoparticles, the field has seen significant advancements. Research has focused on understanding the behavior of nanofluids in various configurations and their role in enhancing heat transfer [2,3,4,5].

Table 1. Literature review of improvement of thermal conductivity.

Nanoparticle	Base Fluid	Enhancement	Ref.
MWCNT	water	58%	[6]
Al2O3	water	15%	[7]
MWCNT	water	48%	[8]
SiO2	methanol	10.3%	[9]
TiO2	water	8.2%	[10]
Fe3O4	water	48%	[11]
TiO2	Oil	7.08	[12]
Al2O3	water	20%	[13]
Diamond	EG	17.23%	[14]

[6,7,8,9,10,11,12,13,14] summarizes nanoparticles, base fluid, and improvement of thermal conductivity reported by researchers and scientists in recent studies.

Recent research has increasingly focused on hybrid nanofluids, which are fluids where nanoparticles of varying compositions are combined to create a hybrid system. These hybrid nanofluids are particularly useful in heat transfer applications requiring high thermal conductivity. Over the past few decades, extensive numerical and experimental research has been conducted on hybrid nanofluids, encompassing their preparation, characterization, modeling, convective heat transfer, and various applications. For instance, Esfe et al. [15] experimentally investigated the thermal conductivity of CNTs-Al₂O₃/water nanofluid using a statistical approach, revealing that the thermal conductivity of the nanofluid increases with both temperature and solid volume fraction.

Numerous experimental and numerical studies have been conducted to investigate the natural convection of hybrid nanofluids in enclosures, focusing on parameters such as nanoparticle concentration, fluid properties, and enclosure geometry, all of which significantly impact heat transfer performance. For instance, Esfe et al. [16] experimentally examined the thermal conductivity and dynamic viscosity of Ag-MgO/water hybrid nanofluid. Scott et al. [17] performed detailed measurements and analyses of temperature distributions and flow patterns within a square cavity containing Al₂O₃-MWCNT/water hybrid nanofluids. Similarly, Al-Maliki et al. [18] investigated heat transfer phenomena in an inclined rectangular cavity filled with (CuO-Al₂O₃/water) hybrid nanofluids, focusing on the combined effects of inclined orientation, hybrid nanofluid composition, and PCM integration on heat transfer performance. Through experimental work, Zadkhast et al. [19] developed a novel correlation to estimate the thermal conductivity of MWCNT-CuO/water hybrid nanofluid. In another study, Afrand et al. [20] investigated the effect of temperature and nanoparticle concentration on the rheological behavior of Fe₃O₄-Ag/EG hybrid nanofluids. Giwa et al. [21] explored the thermoconvection performance of Al₂O₃-MWCNT hybrid nanofluids at various nanoparticle concentrations within a differentially heated square cavity, aiming to understand how hybrid nanoparticle mixtures influence convective heat transfer under different thermal boundary conditions. Additionally, an extensive literature review by Scott et al. [22] summarized recent research and discussed strategies for improving heat transfer with hybrid nanofluids in cavities. Many studies have also been conducted on natural convection of hybrid nanofluids in complex geometries [23,24,25].

In parallel, other researchers have focused on Magnetohydrodynamics (MHD). For instance, Nepal Chandra Roy [26] investigated (MHD) natural convection of hybrid nanofluids in a multi-heat source enclosure. This study utilized mathematical modeling and numerical simulations with the finite difference method to analyze the behavior of hybrid nanofluids under the influence of magnetic fields, fluid flow, and heat transfer. The results offered valuable insights into optimizing multi-source thermal management systems using hybrid nanofluids and (MHD) effects. Another study, led by Abdel-Nour and colleagues [27], examined (MHD) natural convection of an Al₂O₃-Cu/water hybrid nanofluid in a porous enclosure with sinusoidally varying boundary temperatures. The Navier–Stokes equations were discretized using the Galerkin finite element method, with a particular focus on analyzing entropy generation as a measure of irreversibility within the system. Mansour et al. [28] conducted a numerical investigation on the effects of heat sources and sinks on entropy generation and (MHD) natural convection in a square porous cavity filled with Al₂O₃-Cu/water hybrid nanofluid. The study employed the finite difference method to solve the dimensionless partial differential equations governing the problem. The results were analyzed in terms of entropy production, considering the Hartmann number, nanoparticle volume fraction, and the geometrical parameters of the cavity. Similarly, Rashad et al. [29] numerically studied (MHD) natural convection in a triangular cavity filled with Al₂O₃-Cu/water hybrid nanofluid and subjected to constant heat flux. Their research focused on the impact of localized heating from below and internal heat generation on fluid flow and heat transfer phenomena. The governing equations, formulated in terms of the stream function and vorticity, were solved using the finite difference method. The results indicated that under conditions of low natural convection, increasing the nanoparticle volume fraction significantly enhances heat transfer. Furthermore, a wealth of valuable information on natural convection can be found in the works of Roy et al. [30] and Majeed et al. [31,32].

Furthermore, the effects of Fe₃O₄ nanoparticles on hydrothermal processes under a magnetic field have been investigated in several studies [33,34,35,36,37].

The primary objective of this study is to simulate natural convection in an annular cavity filled with hybrid nanofluids under the influence of a magnetic field. This configuration is relevant to applications such as heat exchangers, electronic cooling, and solar energy systems. The aim of the study is to investigate how the interaction between hybrid nanoparticles, fins, and magnetic fields affects the natural convection and heat transfer performance within the cavity. By analyzing parameters such as the Hartmann number, the number of fins, and the volume fraction of nanoparticles, this research will provide insights into the optimization of the use of hybrid nanofluids in practical applications where magnetic fields and complex geometries are crucial.

2. Physical Model

Figure 1 shows a schematic view of a 2D annular cavity equipped with fins and filled with hybrid nanofluid. It is a cross-section of a heat exchanger with longitudinal fins. As can be seen in this figure, the inner cylinder Di is submitted to volumetric heat generation. The outer cylinder Do is maintained adiabatic with a magnetic field applied in a horizontal direction. The aspect ratio AR is defined as (AR = D₀/D_i = 3), and the fin dimensions are (0.0139, 0.02).

The heat transfer takes place from the heated wall of the inner diameter to the fluid filling the cavity initially at ambient temperature T₀ = 298 K.

The fluid enclosed between the inner and outer cylinders is water-based and contains various nanoparticle combinations of Fe₃O₄, Cu, Ag, and Al₂O₃ that are in thermal equilibrium. Nanoparticles have spherical and uniform shapes. Thermophysical properties of the fluid and nanoparticles are defined in the following

Table 2. Water and nanoparticles thermophysical properties [30,34].

	ρ (kg/m3)	K (w/mK)	Cp (J/kg·K)	σ (s/m)	β (1/K)
Water (bf)	997.1	0.608	4182	5.5 × 10−6	21 × 10−5
Fe3O4 (ϕ1)	5810	6	670	2.5 × 10−5	1.3 × 10−5
Al2O3 (ϕ2)	3970	40	765	35 × 106	0.85 × 10−5
Cu (ϕ2)	8933	401	385	59.6 × 106	1.67 × 10−5
Ag (ϕ2)	10,500	429	235	6.30 × 107	1.89 × 10−5

.

3. Governing Equations

The flow under consideration is Newtonian, incompressible, steady, and laminar. The dimensional governing equations under Boussinesq’s approximation are continuity, momentum, and the energy equations in the two-dimensional Cartesian coordinate system for the case of a single-phase fluid, are as follows [38]:

• Continuity equation

$${{\rho }_h}_{nf}\left({\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{\partial v}{\partial y}}\right)=0$$

(1)

• Momentum equations

x-Component

$$\begin{array}{l}{\rho }_{hnf}\left(u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}\right)=-{\displaystyle \frac{\partial p}{\partial x}}+{\mu }_{hnf}\left({\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}\right)\end{array}$$

(2)

y-Component

$${\rho }_{hnf}\left(u{\displaystyle \frac{\partial v}{\partial x}}+v{\displaystyle \frac{\partial v}{\partial y}}\right)=-{\displaystyle \frac{\partial p}{\partial y}}+{\mu }_{hnf}\left({\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}v}{\partial y^2}}\right)-{\rho }_{hnf}g{\beta }_{hnf}(T-T_0)-{\sigma }_{hnf}{B}_0^{2}v$$

(3)

• Energy equation

$${\rho }_{hnf}\left(u{\displaystyle \frac{\partial T}{\partial x}}+v{\displaystyle \frac{\partial T}{\partial y}}\right)={\displaystyle \frac{{k_h}_{nf}}{C{p}_{hnf}}}\left({\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}\right)$$

(4)

• Energy equation for solid wall

$$k_s\left({\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}\right)+Q_v=0$$

(5)

Q_v is the uniform volume heat generation imposed on the inner cylinder.

The adopted boundary conditions are as follows:

The fluid is initially at ambient temperature T_o = 298 K;

The no-slip boundary condition is applied at walls: (u = 0, v = 0);

An adiabatic condition for the outer cylinder D₀.

$${\displaystyle \frac{\partial T}{\partial x}}={\displaystyle \frac{\partial T}{\partial y}}=0$$

(6)

The conduction heat-flux continuity is considered in the solid–fluid interface.

The thermophysical properties of mono- and hybrid nanofluids are assumed in accordance with the Tiwari–Das model and are presented as follows [30] (

Table 3. Thermophysical characteristics of mono and hybrid nanofluid [30].

	Mono Nanofluid	Hybrid Nanofluid
Density	${\rho }_{nf}=(1-\varphi ){\rho }_f+\varphi {\rho }_s$	${\rho }_{hnf}=(1-\varphi ){\rho }_f+{\varphi }_1{\rho }_1+{\varphi }_2{\rho }_2$
Heat capacitance	$(\rho Cp{)}_{nf}=(1-\varphi \left)\right(\rho Cp{)}_f+\varphi (\rho Cp{)}_s$	$(\rho Cp{)}_{hnf}=(1-\varphi \left)\right(\rho Cp{)}_f+{\varphi }_1\left(\rho C{p}_1\right)+{\varphi }_2\left(\rho C{p}_2\right)$
Thermal expansion	$(\rho \beta {)}_{nf}=(1-\varphi \left)\right(\rho \beta {)}_f+\varphi (\rho \beta {)}_s$	$(\rho \beta {)}_{hnf}=(1-\varphi \left)\right(\rho \beta {)}_f+{\varphi }_1\left(\rho {\beta }_1\right)+{\varphi }_2\left(\rho {\beta }_2\right)$
Electrical conductivity	${\displaystyle \frac{{\varphi }_{nf}}{{\sigma }_f}}=1+{\displaystyle \frac{3\left[\frac{{\sigma }_s}{{\sigma }_f}-1\right]\varphi }{\left(\frac{{\sigma }_s}{{\sigma }_f}+2\right)-\left(\frac{{\sigma }_s}{{\sigma }_f}-1\right)\varphi }}$	${\displaystyle \frac{{\varphi }_{hnf}}{{\sigma }_f}}=1+{\displaystyle \frac{3\left[\frac{({\varphi }_1{\sigma }_1+{\varphi }_2{\sigma }_2)}{{\sigma }_f}-({\varphi }_1+{\varphi }_2)\right]}{\left(\frac{({\varphi }_1{\sigma }_1+{\varphi }_2{\sigma }_2)}{\varphi {\sigma }_f}+2\right)-\left(\frac{({\varphi }_1{\sigma }_1+{\varphi }_2{\sigma }_2)}{{\sigma }_f}-({\varphi }_1+{\varphi }_2)\right)}}$
Thermal conductivity	${\displaystyle \frac{k_{eff}}{k_f}}={\displaystyle \frac{(k_s+2k_f)-2\varphi (k_f-k_s)}{(k_s+2k_f)+\varphi (k_f-k_s)}}$	${\displaystyle \frac{k_{hnf}}{k_f}}=1+{\displaystyle \frac{\left[\frac{({\varphi }_1{k}_1+{\varphi }_2{k}_2)}{\varphi }+2k_f+2({\varphi }_1{k}_1+{\varphi }_2{k}_2)-2\varphi k_f\right]}{\left(\frac{({\varphi }_1{k}_1+{\varphi }_2{k}_2)}{\varphi }+2k_f\right)-\left(({\varphi }_1{k}_1+{\varphi }_2{k}_2)+\varphi k_f\right)}}$
Thermal diffusivity	${\alpha }_{nf}={\displaystyle \frac{k_{nf}}{{\left(\rho C_p\right)}_{nf}}}$	${\alpha }_{hnf}={\displaystyle \frac{k_{hnf}}{{\left(\rho C_p\right)}_{hnf}}}$
Dynamic viscosity	${\mu }_{nf}={\displaystyle \frac{{\mu }_f}{{\left(1-\varphi \right)}^{2.5}}}$	${\mu }_{hnf}={\displaystyle \frac{{\mu }_f}{{\left(1-({\varphi }_1+{\varphi }_2)\right)}^{2.5}}}$

):

Data acquisitions

The key parameters controlling the phenomenology of natural convection appear to be as follows:

The Rayleigh number: $Ra={\displaystyle \frac{g\beta Q_v{\delta }^5}{{\nu }_f{\alpha }_f{k}_f}}$;

The Prandtl number: $Pr={\displaystyle \frac{{\nu }_f}{{\alpha }_f}}$;

The Hartmann number: $Ha=B_0\delta \sqrt{{\displaystyle \frac{{\sigma }_f}{{\mu }_f}}}$;

The Nusselt number: $Nu={\displaystyle \frac{h{D}_h}{k_{hnf}}}$.

h is the heat transfer coefficient defined as $h={\displaystyle \frac{\ddot{q}}{(T_w-T_b)}}$.

$\ddot{q}$: is the average surface heat flux.

T_w and T_b are, respectively, the mean temperature of the wall and the mean bulk temperature of the fluid calculated as follows: $T_w={\displaystyle \frac{1}{A}}{\displaystyle \iint Tdxdy}$, $T_b={\displaystyle \frac{{\displaystyle \iint uTdxdy}}{{\displaystyle \iint u}dxdy}}$.

The Nusselt number becomes: $Nu={\displaystyle \frac{\ddot{q}{D}_h}{k_{hnf}(T_w-T_b)}}$.

4. Numerical Procedure

The finite volume method is used to solve the governing equations using the commercially available software Fluent 6.3.26. The Simple algorithm proposed by Patankar [39] is used for coupling between pressure and velocity with the third-order MUSCL scheme, which improves on the second-order scheme in terms of more effective gradient resolution and less numerical diffusion. The convergence requirement for the iteration of a non-linear system is given by: ${\left|{\displaystyle \frac{({\mathrm{\Phi }}^{k+1}-{\mathrm{\Phi }}^k)}{{\mathrm{\Phi }}^{k+1}}}\right|}_{\max }\le {10}^{-6}$.

Φk are u, v, *T, and p, k is the number of iterations. The steady-state computations are performed on a 64 GB RAM8.00 Go, 2.5 GHz, Intel^®, Core(TM) i5-3210 M CPU desktop computer. Six meshes were considered in order to study the effect of the mesh on the numerical solution.

Table 4. Nu_avg, T_max, deviation Nu_avg% and T_max% for various grid resolution, N = 2, Ra = 1.2 × 10⁴.

Grid Resolutions	8100	15,000	16,500	18,000	22,500	27,000
Nuavg	18,222	19,014	19,044	19,278	19,782	19,913
Deviation Nuavg %	5.795	1.38	1.22	-	2.54	3.18
Tmax (K)	547,797	395,922	383,500	335,522	316,593	315,406
Deviation Tmax %	38.75	15.255	12.51	-	5.97	6.377

presents the average Nusselt number Nu_avg and le maximum temperature of the fluid in the annulus for pure water and Ra = 1.2 × 10⁴. The grid size (22, 500) is found to meet the requirement of both the grid independence study and the computational time limits. Figure 2 illustrates the mesh representation (a) for the annulus without fins, (b) for the annulus with two fins, and (c) for the annulus with four fins.

The quantitative results are compared with those of Vahl-Davis et al. [40] and Kumar [41] for an annular vertical cylinder with the geometric parameters (A: aspect ratio and λ = re/ri: radius ratio). The results of

Table 5. Comparison of $\overline{\mathrm{N}\mathrm{u}}$ for annular cylinder (A = 1, λ = 1).

Rayleigh Number Ra	103	104
De Vahl Davis [40]	2.242	4.523
Kumar, Kalam [41]	2.242	4.523
Present study	2.471	4.968

show that with a percentage between 8% and 9%, our results are in agreement with those of the literature. Also, the present study is compared with other works reported in [42,43], as seen in Figure 3 and Figure 4.

5. Results and Discussion

The focus of this section is on natural convection in a finned annular cavity filled with hybrid nanofluid. The base fluid is water with a Prandtl number Pr = 5.63, containing a combination of nanoparticles; the Fe₃O₄ with volume fraction ϕ₁ = 0.2% mixed with Al₂O₃; and Cu or Ag with volume fraction ϕ₂ = 0.1%, 0.4%, and 0.6%. The numerical results are presented in the form of isotherms, stream function, velocity contours, and average Nusselt number distribution for different Hartmann numbers (0 ≤ Ha ≤ 100), Rayleigh numbers (3 × 10³ ≤ Ra ≤ 2.4 × 10⁴), and the number of fins (N = 2, 4, 6, 8).

5.1. Hartmann Number

The effect of the Hartmann number (Ha = 0, 25, 55, 100) on the streamlines, velocity contours, and isotherms is shown in Figure 5, Figure 6 and Figure 7 for the three mixtures of hybrid nanofluid. Firstly, it is shown that Ha has no effect on the velocity and on the temperature when Ha is equal to zero. The streamlines are smooth, indicating stable natural convection. The temperature distribution is uniform with smooth gradients. The isotherms are relatively dense, indicating a high rate of heat transfer and moderate velocities driven purely by temperature differences. As the Hartmann number increases, the effect of the Lorentz force in the annulus becomes more pronounced for the three mixtures, indicating a suppression of fluid motion and the formation of recirculation zones by a slight decrease in temperature and increase in velocity fields. The isotherms in Figure 5, Figure 6 and Figure 7 show that with increasing Ha, the temperature contours become more spaced out, indicating a thicker thermal boundary layer. Each nanoparticle type (Cu, Ag, Al₂O₃) responds differently to the increasing Hartmann number. The differences between the nanofluids Ag, Cu, and Al₂O₃ are primarily due to their varying thermal conductivities. It is shown that the isotherms become denser near heated surfaces, showing effective heat transfer maintained or even enhanced. The velocity contours may show localized increases in velocity gradients in specific regions due to the complex interaction between the magnetic field and fluid flow, aiding in maintaining adequate heat transfer rates.

Copper, with the highest thermal conductivity, shows the most significant reduction in temperature gradients as Ha increases, making it the most effective at achieving thermal uniformity in the presence of a magnetic field. The extension of these figures allows us to see the full effect of the magnetic field strength on the temperature profile along a vertical line in the annular cavity in Figure 8. As it is seen in this figure, in the absence of a magnetic field (Ha = 0), natural convection dominates, leading to a more significant temperature gradient. For the Fe₃O₄-Cu and Fe₃O₄-Ag mixtures, Figure 8a,b show that the temperature at the center line of the cavity decreases significantly with increasing Ha. This trend suggests that the magnetic field has a significant effect on the heat distribution, probably due to the high thermal conductivity of Cu and Ag, which enhances the effect of suppressed convection. On the other hand, the temperature profile of the Fe₃O₄-Al₂O₃ mixture shows the least variation with Ha, reflecting the lowest thermal conductivity among the three nanofluids.

The induced magnetic field is a response to the movement of the fluid within the applied magnetic field; their interaction results in the Lorentz force, which influences flow patterns and heat transfer. Figure 9, Figure 10 and Figure 11 illustrate the contours of the induced magnetic field B for different Hartmann numbers (Ha = 25, 55, 70, and 100) for the three mixtures cited above. From these figures, it is seen that, as the Hartmann number increases, the influence of the induced magnetic field on the fluid flow becomes more pronounced. This results in more complex and intense convection patterns, with stronger gradients and localized high-intensity areas in the contours of the induced magnetic field B. The variations in the contours reflect the distinct thermal and magnetic properties of the nanoparticles used, affecting the natural convection and magnetic field distribution uniquely for each mixture.

For a better understanding of the phenomenon, Figure 12 depicts the variations in the induced magnetic field B along a vertical line in the cavity for the three mixtures of hybrid nanofluids at various Hartmann numbers (Ha). The curves indicate that adding different nanoparticles significantly alters the distribution of the induced magnetic field in the cavity. Increasing the Hartmann number amplifies these effects, making the variations more pronounced. The differences among the mixtures highlight the importance of the specific properties of the nanoparticles in natural convection and the distribution of the magnetic field. The percentage differences between the induced magnetic field B values at Ha = 100 and Ha = 70 for Fe₃O₄-Ag, Fe₃O₄-Cu, and Fe₃O₄-Al₂O₃ mixtures is around 18.18% for the Fe₃O₄-Cu hybrid nanofluid and around 15.38% for both Fe₃O₄-Ag and Fe₃O₄-Al₂O₃ hybrid nanofluid.

Figure 13 depicts the variation in the average Nusselt number with the magnetic parameter Ha for the three different mixtures at different nanoparticle volume fractions (ϕ₂ = 0.001, 0.004, and 0.006). It is shown that the Nusselt number increases not only with Ha but also with the nanoparticle volume fraction ϕ₂. A higher concentration of nanoparticles leads to a greater increase in thermal conductivity, thereby improving the fluid’s ability to conduct heat. This effect is particularly important in regions where the magnetic field suppresses fluid motion. The enhanced conduction compensates for the reduced convection.

The magnetic properties of nanoparticles such as Fe₃O₄ can interact with the applied magnetic field and affect the overall flow and heat transfer. This interaction can lead to changes in the velocity field, as seen in Figure 5, Figure 6 and Figure 8, and may contribute to localized improvements in heat transfer.

From these results, the observation that the Nusselt number Nu increases with the Hartmann number Ha despite the suppression of fluid motion and a shift toward conduction-dominated heat transfer might seem contradictory at first glance. However, this phenomenon can be understood by considering several key factors:

• The thermal conductivity of the nanoparticles is higher than the base fluid, allowing the fluid to transfer heat more efficiently. This increase in thermal conductivity reduces the thermal resistance of the fluid, improving heat transfer even when convective currents are suppressed by the magnetic field—a higher Hartmann number.
• The inclusion of nanoparticles can thin the thermal boundary layer or maintain a steep temperature gradient across it, increasing the overall heat transfer rate.

Silver (Ag) and copper (Cu) typically possess higher thermal conductivity than aluminum oxide (Al₂O₃). Consequently, copper nanoparticles in the Fe₃O₄-Cu/water hybrid nanofluid can transfer heat to the fluid more efficiently. Compared to silver or alumina nanoparticles, copper nanoparticles may interact differently with the fluid.

The relationship between the thermal conductivity ratio k_hnf/k_f and the Hartmann number Ha is an essential aspect of understanding how magnetic fields influence the thermal performance of nanofluids. Figure 14 illustrates the variation in the thermal conductivity ratio (k_hnf/k_f) with Ha for different concentrations ϕ₂. The three specific concentrations of ϕ₂ are 0.001, 0.004, and 0.006. In all figures, an increase in Ha generally leads to an increase in the thermal conductivity ratio k_hnf/k_f. This indicates that applying a magnetic field enhances the thermal conductivity of the hybrid nanofluid relative to the base fluid. Contrary to what might be expected, the thermal conductivity ratio (k_hnf/k_f) decreases as the concentration of the second nanoparticles ϕ₂ increases from 0.001 to 0.006 for all types of mixtures.

For ϕ₂ = 0.001, 0.004 and 0.006, the thermal conductivity ratio starts at a higher value and increases with Ha. Fe₃O₄-Cu and Fe₃O₄-Ag nanofluids show a higher increase compared to Fe₃O₄-Al₂O₃.

5.2. Nanoparticles Concentration

As the concentration of nanoparticles in the hybrid nanofluid increases, this can lead to an increase in the average Nusselt number Nu_avg. Figure 15 shows how the Nu_avg varies with the concentration ϕ₂. The analysis is conducted for a Rayleigh number Ra of 1.2 × 10⁴ and a Hartmann number Ha of 35. For the three mixtures, Nu_avg increases with an increase in nanoparticle concentration ϕ₂. Increasing nanoparticle concentration in a nanofluid generally leads to enhancements in thermal conductivity and convective heat transfer, which contribute to higher Nusselt numbers [29]. The increase in Nusselt number with higher nanoparticle concentration is primarily due to the improved thermal conductivity provided by the nanoparticles. The specific type of nanoparticle used (Fe₃O₄ combined with Cu, Ag, or Al₂O₃) influences the extent of this improvement, with Fe₃O₄-Cu providing the most significant improvement.

Among the three hybrid nanofluids, Fe₃O₄-Cu shows the highest enhancement in Nusselt number with increasing ϕ₂, followed by Fe₃O₄-Ag and then Fe₃O₄-Al₂O₃. This reflects the inherent differences in the thermal conductivities and heat transfer enhancement capabilities of the Cu, Ag, and Al₂O₃ nanoparticles.

5.3. Rayleigh Number

The parameter of natural convection is the Rayleigh number, which is defined as a function of volumetric heat generation and the thickness of the inner diameter.

Figure 16 represents the variation in the rate of heat transfer with the Rayleigh number Ra for the three hybrid nanofluids at a fixed nanoparticles concentration ϕ₂ = 0.004, N = 2, and a Hartmann number Ha = 35. The Rayleigh number represents the influence of buoyancy forces on natural convection. As expected, Nu_avg increases with Ra for all nanofluids, indicating enhanced convective heat transfer with stronger buoyancy effects. The difference in Nuavg among the three nanofluids becomes more pronounced as the Rayleigh number increases, suggesting that the effectiveness of these nanofluids in enhancing natural convection is more significant at higher Ra values. Similar to Figure 15, Fe₃O₄-Cu shows the highest Nu_avg across the range of Rayleigh numbers, followed by Fe₃O₄-Ag, and then Fe₃O₄-Al₂O₃. This consistent trend further reinforces the superior heat transfer capability of the Cu-based nanofluid.

5.4. Fin Number

Figure 17 depicts the effect of varying the number of fins (N = 0, 2, 4, 6) on the temperature distribution on the left, the velocity field in the middle, and the induced magnetic field on the right inside the cavity filled with Fe₃O₄-Cu hybrid nanofluid under a magnetic field characterized by Ha = 55 at Ra = 1.2 × 10⁴ and ϕ₂ = 0.004. This figure shows a noticeable reduction in the temperature within the cavity as fin numbers increase. The fins improve heat transfer by increasing the surface area available for conduction, allowing for more efficient cooling. The increased number of fins disrupts the thermal boundary layer, resulting in a more uniform temperature distribution and preventing the formation of hot spots within the fluid. The velocity of the fluid is also affected by the number of fins. With more fins, the flow becomes more constrained, resulting in a reduction in velocity. This is because the fins act as barriers, restricting the free movement of the fluid, especially near the walls where the boundary layers form. The magnetic field further suppresses fluid motion due to the Lorentz force, which becomes more pronounced as the number of fins increases. On the other hand, the induced magnetic field tends to become stronger as the number of fins increases. This can be attributed to the interaction between the magnetic field and the fluid flow, where the suppression of the velocity by both the fins and the magnetic field increases the concentration of the magnetic field distribution.

Table 6. Variation in the Nusselt number with the nanoparticles concentration and the fin numbers for Ha = 55, Ra = 1.2 × 10⁴.

Nanoparticles		N	0	2	4	6	8
	ϕ2
Fe3O4-Cu	0.001		16,510	14,558	12,635	7919	6452	Nuavg
	0.004		25,441	19,227	17,612	11,838	7676
	0.006		38,620	26,374	23,865	17,860	9104
Fe3O4-Ag	0.001		16,562	10,080	9243	8307	6623	Nuavg
	0.004		26,906	18,687	13,848	12,301	6920
	0.006		32,254	22,805	20,784	15,043	8083
Fe3O4-Al2O3	0.001		16,199	10,130	9040	8286	5657	Nuavg
	0.004		19,687	15,005	14,714	10,729	7020
	0.006		21,270	17,114	14,922	12,531	10,314

presents the variation in the average Nusselt number with different nanoparticle concentrations and fin numbers under a magnetic field. As shown in this table, for each nanoparticle type Fe₃O₄-Cu, Fe₃O4-Ag, and Fe₃O₄-Al₂O₃, Nu_avg tends to decrease as the number of fins N increases. This indicates that the addition of more fins generally results in a reduction in heat transfer efficiency. The reduction could be due to the increased obstruction to fluid flow within the cavity, which reduces the convective currents responsible for heat transfer. Higher concentrations of nanoparticles (ϕ₂ = 0.006) typically result in higher Nusselt numbers compared to lower concentrations (ϕ₂ = 0.001). This effect is more pronounced when the number of fins is lower, indicating that the improvement in thermal conductivity due to the nanoparticles is more effective when the number of fins is lower. Among the mixtures, Fe₃O₄-Cu has the highest Nusselt numbers over all fin numbers. This suggests that this combination is the most effective in improving heat transfer within the cavity. This aligns with the decrease in Nusselt number observed in Table 6.

Table 7. Percentage decrease in the Nusselt PD% for each fin number for Ha = 55, Ra = 1.2 × 10⁴.

Nanoparticles		N	0	2	4	6	8
	ϕ2
Fe3O4-Cu	0.001		-	11.84	23.47	52.05		60.93	PD%
	0.004		-	24.42	30.77	53.46		69.82
	0.006		-	27.97	38.20	53.75		76.46
Fe3O4-Ag	0.001		-	39.13	44.19	49.84		60.01	PD%
	0.004		-	31.53	48.53	54.28		74.28
	0.006		-	29.29	35.65	53.36		74.93
Fe3O4-Al2O3	0.001		-	37.48	44.19	48.84		65.078	PD%
	0.004		-	23.78	25.26	92.474		64.34
	0.006		-	19.53	29.84	41.08		51.50

presents the percentage decrease (PD%) across various fin numbers N and the concentrations ϕ₂. The percentage decrease in the average heat transfer rate (PD%) for different nanoparticle types compared to the baseline case of N = 0 (no fins) is calculated from the formula cited below.

$$PD\%=\left|{\displaystyle \frac{N{u}_{N=0}-N{u}_N}{N{u}_{N=0}}}\right|\times 100$$

It is clear from these values that introducing fins and nanoparticles actually leads to a decrease in Nu_avg compared to the baseline case without fins (N = 0). The presence of nanoparticles may alter the thermal boundary layer in such a way as to reduce the effectiveness of the fins. As can be seen in this table, in the case of the mixture Fe₃O₄-Cu, for ϕ₂ = 0.001 and N = 8, PD% is around 60.93%, 60.01% for Fe₃O₄-Ag, and 65.078% for Fe₃O₄- Al₂O₃. Regarding N = 6 and the mixture Fe₃O₄-Ag, the PD% is around 49.84% and 53.36%, respectively, for ϕ₂ = 0.001 and ϕ₂ = 0.006. We can say that at higher concentrations, nanoparticles may agglomerate. This reduces their surface area and thermal conductivity. This may counteract the benefits of the additional fin, resulting in a net reduction in heat transfer. In conclusion, in engineering applications, simply adding fins or nanoparticles does not always guarantee enhanced performance.

6. Conclusions

This article presents a 2D numerical study on natural convection in an annular cavity filled with a hybrid nanofluid under the influence of a magnetic field. The geometry analyzed for the cavity represents a cross-section of a heat exchanger equipped with longitudinal fins. The inner cylinder is subjected to uniform volumetric heat generation, while the outer cylinder is kept adiabatic, subjected to uniform volumetric heat generation, while the outer cylinder is kept adiabatic. The objective of this study was to analyze the flow structure and quantify the heat transfer rate within the annular cavity, while also evaluating the effect of nanoparticles under the action of the magnetic field. To achieve this, a single-phase model was used to consider the following particle mixtures: Fe₃O₄-Cu, Fe₃O₄-Ag, and Fe₃O₄-Al₂O₃, taking into account key parameters such as nanoparticle concentration, ϕ₂; Hartmann number, Ha; Rayleigh number, Ra; and the number of fins, N. The results of this study revealed the following:

➢

As the Hartmann number increases, temperature decreases while velocity increases, indicating a stronger magnetic influence that intensifies convective currents. Each type of nanoparticle (Cu, Ag, Al₂O₃) reacts differently to this increase, with Cu nanoparticles appearing to be the most sensitive to magnetic fields. The Nusselt number and the thermal conductivity ratio also increase with the rise in Hartmann number, nanoparticle concentration, and Rayleigh number, leading to an improvement in heat transfer rate.

➢

The addition of fins up to N = 2 can slow down natural convection currents, thereby reducing heat transfer efficiency. A configuration with two fins (N = 2) might therefore be the most suitable for this study. Thus, the magnetic field not only influences the contribution of nanoparticles to heat transfer but can also enhance it.

Finally, as a perspective and continuation of this study, experimental work could be carried out to validate the numerical results. This would involve constructing a physical model of the annular cavity with hybrid nanofluids on the one hand and, on the other, measuring heat transfer rates under different magnetic field intensities. Furthermore, examining the influence of different orientations and intensities of the magnetic field on heat transfer and fluid flow could provide deeper insights for optimizing magnetic control in practical applications. We also recommend that future studies incorporate multiphase models to better understand the effects of nanoparticles under more realistic conditions.