Numerical Simulation of the Natural Convection with Presence of the Nanofluids in Cubical Cavity

)is article consists of a numerical study of natural convection heat transfer in three-dimensional cavity filled with nanofluids. )is configuration is heated by a partition maintained at a hot constant and uniform temperature TH. )e right and left vertical walls are kept at a cold temperature TC while the rest is adiabatic. )e fluid flow and heat transfer in the cavity are studied for different sets of the governing parameters, namely, the nanofluid type, the Rayleigh number Ra� 10, 10, 10, and 10, and the volume fraction V varying between V � 0 and 0.1. )e obtained results show a positive effect of the volume fraction and the Rayleigh number on the heat transfer improvement. )e analysis of the results related to the heat transfer shows that the copperbased nanofluid guarantees the best thermal transfer. In addition, the increase of the heating section size and Ra leads to an increased amount of heat. Similarly, increasing the volume fraction improves the intensification of the flow and increases the heat exchange.


Introduction
e flow and heat transfers induced by natural convection have been considerably studied during these last decades because of their direct application in various fields of engineering such as air conditioning, energy efficiency, cooling of electronic components, etc. e previous works had mostly focused on the improvement of heat transfer induced by natural convection, which led to the development of a new research area based on the molecular structure of the fluid: modern nanotechnology, which can produce metallic or nonmetallic particles of nanometer dimensions. ese nanomaterials, with average sizes below 100 nm, have unique mechanical, optical, electrical, magnetic, and thermal properties and nanofluids are mixtures of these particles and traditional fluids such as water, oil, and ethylene glycol. Hence, a minimal amount of guest nanoparticles, when dispersed uniformly and suspended stably in host fluids, can provide improvements in the thermal properties of the host fluids. e primary early interest in nanofluids from a technological viewpoint was the possibility of using these fluids for cooling purposes. Although the higher conductivity is an encouraging phenomenon of the cooling capabilities of such fluids, it is also important not only to reveal the convective behavior of nanofluids but also to understand the dynamics of fluids and the theories of heat transfer of nanofluids. Till now, convective studies of nanofluids have been very limited compared to experimental and theoretical studies on conduction.
Numerical work on the natural convection of nanofluids was carried out by Khanafer et al. [1]. ey studied a differentially heated cavity with hot and cold vertical walls and adiabatic horizontal walls. ey showed that the rate of heat transfer increases with an increase in the volume fraction of the nanoparticles for all the considered values of the Grashof number. In addition, they have shown that the heat transfer rate is favored when the volume fraction of the nanoparticles increases. Among the works dealing with heat transfer in the presence of nanoparticles, Murshed et al. [2] showed that the thermal conductivity increases with the increase of the volume concentration, and they also found that, for a volume concentration of 5%, the improvement of the effective thermal conductivity is about 33% for the nanotubes and of 30% for spherical nanoparticles. ey found that the size and shape of the particles have a significant influence on the thermal conductivity of nanofluids.
Similar work was carried out by Jou and Tzeng [3] inside a differentially heated cavity. ey also used the stream function vorticity formulation in a way identical to that used in a previous study by Khanafer et al. [1]. In addition to the Grashof number effects, they then described the effect of the cavity aspect ratio (width/height) on thermal behavior. However, they concluded that, at 20% volume fractions, Newtonian behavior of the fluid is doubtful and it is extremely difficult to make stable nanofluids to use. Also, at such volume fractions, Newtonian behavior of the fluid is doubtful. A recent numerical study by Ho et al. [4] in a vertical square enclosure filled with a nanofluid (water-Al 2 O 3 ) shows clearly the effects of different parameters on flow simulation. ey examined the effects of uncertainties due to the adoption of different models for thermal conductivity and dynamical viscosity. ese authors found that the heat transfer through the enclosure could be enhanced or attenuated depending on the models used for the dynamical viscosity of the nanofluid. Hwang et al. [5] have made several measurements on the thermal conductivity of nanofluids and have shown that this parameter depends strongly on the volume fraction of the suspended particles and the thermal conductivity of the base fluid. Recently, Oztop and Abu-Nada [6] have numerically studied heat transfer and fluid flow as a function of buoyancy forces in a partially heated enclosure filled with nanofluids containing different types of nanoparticles. ey showed that the improvement of the heat transfer was more pronounced for low values of the aspect ratio. ey found that, for all the considered Rayleigh numbers, the average Nusselt number increases as the volume fraction of the nanoparticles increases.
Anilkumar and Jilani [7] presented a numerical investigation on the natural convection of a mixture of ethylene glycol and aluminum oxide in an enclosure heated by an isothermal partition located on its lower wall. e study indicates the influence of the nanoparticle volume fraction, the Rayleigh number, and the shape ratio of the enclosure on the dynamical and thermal behavior of the fluid. e results obtained show that the heat transfer improves with the increase of the volume fraction of the nanoparticles, and the presence of the nanoparticles in the base fluid modifies the structure of the flow. Natural convection in an inclined square enclosure heated by a source placed in the center of the left wall was studied byÖgüt [8]. e cavity was filled with a mixture of water and nanoparticles (Cu, Ag, CuO, Al 2 O 3 , and TiO 2 ), and the results show that the type of nanoparticle is a major factor in improving heat transfer.
Hussein et al. [9] investigated numerically the magnetohydrodynamic (MHD) natural convection flow of Cu-water nanofluid in an open enclosure using lattice Boltzmann method (LBM) scheme. ey found that the absolute values of stream function decline significantly by increasing Hartmann numbers while these values rise by increasing Rayleigh numbers. In addition, they have shown that, depending on the value of the Hartmann and Rayleigh numbers, the solid volume fraction has a significant influence on the function of flux and heat transfer. e authors in [10] studied numerically the natural convection in three types of inclined wavy cavities filled with Al 2 O 3 -water and Ag-water nanofluids and subjected to a discrete isoflux heating from its left sidewall. ey found that the geometry of the wavy cavity has a cursive role on the flow and thermal fields pattern. Also, the local Nusselt number along the heat source increases with the solid volume fractions and wave amplitudes and the heat functions increase for both nanofluids and base fluids with these parameters. e same authors have other articles in this context (Hussein et al. [11][12][13]). e effects of the inclination angle of an external magnetic force on the natural convection inside a cubical cavity filled with a carbon nanotube-(CNT-) water nanofluid are investigated by Al-Rashed et al. [14]. ey found that, for larger Rayleigh number, the heat transfer is increased significantly with the percentage of CNT particles. Moreover, they observed that a repelling effect of the magnetic force inhibits the heat transfer by 50% when Ha is increased from 50 to 100. In another work [15], they investigated the effect of magnetic field on convective heat transfer of highly electrically conductive fluids taking into consideration the case of a low-conductive LiNbO 3 oxide melt. Chand et al. [16] presented a numerical investigation of the thermal instability in a low Prandtl number nanofluid in a porous medium. ey indicated that the Prandtl and Darcy numbers have a destabilizing effect while the Lewis number and modified diffusivity ratio have a stabilizing effect for the stationary convection.
Hussein et al. [17,18] have studied two numerical cases in a three-dimensional cavity filled with nanofluids. e first study concerns natural convection and the second the mixed convection. For the first one, they found that the generation of total entropy becomes insignificant under the effect of the tilt angle for the low Rayleigh number. In addition, the flow circulation and the average Nusselt number increase with Rayleigh number. In the second study, they indicated that the average Nusselt number increases with the increase in the Richardson number and the solid volume fraction of the nanoparticles.
Kolsi et al. [19] performed a numerical investigation on the effect of a periodic magnetic field inside a cubical enclosure filled with nanofluids (Multi-Walled Carbon Nanotubes). ey mentioned that the oscillation period, the magnitude of the magnetic field, and adding nanoparticles have an important effect on heat transfer, temperature field, and flow structure. Haq et al. [20][21][22] analyzed the heat transfer with a presence of the Magnetohydrodynamic (MHD) water-based Single-Wall Carbon Nanotubes (SWCNTs), the water-based copper oxide (CuO) nanofluid, and uniform magnetic field. ey, respectively, concluded that the increase of the Rayleigh number enhances the stream flow and isotherms behaviors. However, the heat transfer rate inside the cavity is decreasing with the increase in volume fraction, Hartmann number, and L T due to dominant convection. In addition, this transfer rate is increased due to increase in Rayleigh number and wavelength parameter. In addition, they observed that Darcy and Hartmann number do not have significant effects on the temperature distribution.
Muhammad et al. [23] investigated numerically the mixed convection flow of Ag-ethylene glycol nanofluid flow in a cavity having thin central heater. eir results, which are expressed by isotherms, streamlines, velocity field, and average Nusselt numbers, show that the heat transfer for Fourier's law is improved as compared to the results obtained for the case of Cattaneo-Christov heat flux. e augmentation of heat transfer in most of these studies was attributed to the virtue of the abnormal enhancement of the nanofluid thermal conductivity [24][25][26].
In a study by Guiet et al. [27], the laminar natural convection of a mixture of water and copper nanoparticles was numerically studied by the Boltzmann method in a square cavity heated from below and cooled laterally. e heating is provided by a block, which is either kept at a constant temperature or subjected to constant heat flow.
is authors used the Brinkman [28] and the Patel et al. [29] models to calculate ,respectively, the effective viscosity and thermal conductivity of the nanofluid. Almeshaal et al. [30] had studied numerically the effect of water-based hybrid nanofluid of CNT-aluminum oxide on natural convection inside the T-shaped cavity. ey found that the heat transfer performance manifests as a function of the size, volumetric percentage of nanoparticles, fraction of CNT composites, and Rayleigh number. Similarly, with the explosive advances in nanofluids in recent years, adding nano-sized metal particles in impinging cooling fluids has also received great interest from the community of process intensification for heat dissipation equipment [31][32][33][34].
e industrial applications of the heat transfer by natural convection are the electronic component cooling, the nuclear reactors, and the heat losses in solar collectors. Our study is part of this framework, particularly related to the intensification of heat change in the electronic component. e major studies cited considered two-dimensional cavity, but in our case, we are going to consider three-dimensional cavity. e fundamental parameters of the problem are the Rayleigh number, the volume fraction of the nanoparticles, and the dimensions and position of the heating block. e results of the study show that the heat transfer improves with the increase of the volume fraction of the nanoparticles independently of the boundary conditions applied on the heating block. Flow and heat transfer are significantly affected by the size of the heating source. Hence, the purpose of the present investigation is to study numerically the laminar natural convection in a cubical enclosure filled with nanofluid. e temperature distributions, the velocity patterns, and the heat transfer rates are analyzed and discussed in this paper. Figure 1. It consists of three-dimensional cavity of length (H), filled with nanofluids and heated by the partition. e mathematical model used in this paper is based on the Navier-Stokes and energy equations. ese equations are discretized by the finite volume method, taking into account the Boussinesq approximation and neglecting the viscous dissipation. e physical parameters are given by the following equations:

Problem Formulation e physical model considered is shown schematically in
(i) e density [35]: (1) (ii) e heat capacitance of the nanofluid [35]: (iii) ermal expansion coefficient [35]: (iv) e dynamical viscosity of the nanofluid: Brinkman [28] has extended the Einstein formula to cover a wide range of volumetric concentrations, which gives the following relation: (v) ermal diffusivity of nanofluids: (vi) e thermal conductivity of nanofluids: e formula of Maxwell [36] is given by k nf , k f , and k np are, respectively, the thermal conductivities of the nanofluid, the basic fluid, and the solid Mathematical Problems in Engineering 3 nanoparticles.
e thermophysical properties of the basic fluid, water, and the considered nanoparticles are given in Table 1. e governing equations can be written as follows: In order to limit the number of variables involved in the problem under study, the system of equations is dimensionalized by considering dimensionless parameters based on reference quantities as presented in Table 2. erefore, the governing equations of continuity, momentum, and energy conservation for stable laminar flow in Cartesian coordinates can be written in the following dimensionless form: e adopted thermal boundary conditions are as follows: θ � −0.5, on the cold wall, and θ � 0.5, on the heated partition.
(zθ/zn) � 0 on the adiabatic walls. e adopted hydrodynamic boundary conditions are as follows: e average Nusselt number, Nu a , is defined as the integral of the temperature flux through the vertical right cold wall and formulated as

Mesh.
It is the subdivision of the studied field into volume elements in the X direction, Y direction, and Z direction whose intersection represents a node, where the variables P and T are found. e U, V, and W components of the velocity vector are located in the middle of the faces connecting two adjacent nodes. e discretization of the domain is obtained by a mesh consisting of a network of points (nodes). us, a volume element (control volume) is defined around each node. In order to obtain a fine mesh at the level of the active walls (X � 0 and X � 1), the computational domain has been discretized by adopting a mesh that is non-uniform in the X direction and uniform in the Y and Z directions. To test the influence of the number of nodes on the obtained results, we solved the transport equations for mesh sizes of 41 × 41 × 41, 51 × 51 × 51, 61 × 61 × 61, 71 × 71 × 71, and 81 × 81 × 81 and Ra � 10 5 (Table 3). Following this mesh sensitivity test, we chose the 61 × 61 × 61 mesh.

Validation.
A numerical code was developed based on the finite volume method to discretize the governing equations.
e conservation equations of momentum coupled with the continuity equation are solved using the algorithm SIMPLE. e resolution of the obtained discrete algebraic system is based on the Alternating Direction Implicit (ADI) scheme. e code was then validated by comparing our results with those obtained in two different cases for a differentially heated three-dimensional cavity: the first case concerns a three-dimensional cavity filled with nanofluids (Ravnik et al. [37]) and the second is related to a three-dimensional cavity filled with air (Lo et al. [38] (Table 4)). It is clear that the results of our code are in good agreement with those proposed by Ravnik et al. [37] (Figure 2 and Table 5). Indeed, the maximum deviation is of the order of 3.59%.

Convergence Criterion.
e partial differential equations and the method applied to solve the resulting algebraic system require iterative calculation. In an iterative process, we must start with an evaluation of the solution, which does not automatically check the equation to be solved. When subsequent iterations produce no significant change in the values of the dependent variables ϕ, convergence is said to have been achieved. In practice, this convergence is expressed by a test used to stop the iterative process, also called the convergence criterion, which depends on the nature of the problem and the objectives of the calculation.
By adopting the procedure used by Penot [39], this test can be carried out, for a given variable, as follows: where ϕ represents the field variables (U, V, W, T, P). e subscripts i, j, and k are the points of the mesh. e Mathematical Problems in Engineering 5 exponents n and n + 1 represent the previous iteration and the current iteration, respectively. e parameter ε 1 � 10 − 5 has been chosen small enough to avoid truncation errors while remaining above rounding errors [40,41].

Results and Discussion
e results presented in this article are obtained for Rayleigh numbers Ra ranging between 10 3 and 10 6 and the     volume fraction equal to 0 and 0.1. e Prandtl number is fixed at Pr � 6.2 (water). Isotherms and streamlines are presented in order to illustrate the nanofluid motion and the heat transfer within the cavity. Interesting results have been found, in terms of fluid flow and heat transfer through the cavity, depending on the adopted values of the governing parameters and also the nature of the used nanofluid. Figure 3 shows the distribution of the temperature in the three-dimensional cavity for different lengths d, Ra � 10 5 , and ϕ � 0.04. e figure illustrates that fluid movement occurs from the partition to the cold walls so that the heat transfer rate is maintained permanently in the cavity and this phenomenon is done by convection due to the high thermal gradients in the active walls, for that value of Ra. e increase in the heating surface, resulting from the increase in the length d, naturally leads to an increase in the energy supplied to the heat transfer fluid. In order to have a better view of the temperature profile within the cavity (in 3D), different planes were considered. us, the plan chosen for the rest of our study will be that corresponding to Z � 1/2 and which passes perpendicularly through the middle of component 1. is plane suitably represents the flow and heat transfer within the cavity and being the most representative. e results presented are the results of simulations carried out by taking a Rayleigh number varying between 10 3 and 10 6 and a heating partition varying from 20 to 80% of the left vertical wall. Figure 4 shows the current lines (left) and isotherms (right) for the nanofluid (water-Al 2 O 3 ) in the plane Y � 0.5, for different dimensionless partition lengths, Ra � 10 5 , and volume fraction ϕ � 4%. It can be seen that the increase in the length of the heating partition leads to an increase in flow resistance and consequently the intensity of the longitudinal flow decreases. For a length d � 0.75, the distance between the partition and the vertical wall of the cavity (located at Z � 1) becomes small and consequently causes a decrease in the intensity of the longitudinal flow of the fluid passing through the passage located between the planes located between Z � 0.75 and Z � 1. e fluid that is heated by the block moves towards the adiabatic upper wall, where it splits into two flows heading towards the cold vertical walls. is configuration leads to the formation of almost identical counter-rotating cells and a symmetrical flow structure compatible with the symmetry of the boundary conditions. Note also that the increased length of the partition contributes to the generalization of the flow in almost the entire cavity through larger, resistant convective cells. e corresponding isotherms become more constricted near the cold vertical walls. Figure 5 illustrates the structure of the left streamlines and the right isotherms in plan Z for nanofluid (Al 2 O 3 ) and pure water with the length d � 0.5; the temperature distribution has different behavior near the active zones for different values of the Rayleigh number. In fact, increasing "Ra" improves the heat transfer from the warm walls of the cavity to the cold ones as shown in this figure.

Isotherms and Streamlines.
For the low value of Rayleigh number (Ra � 10 3 and 10 4 ), the viscous forces are greater than the buoyancy forces and diffusion is the main heat transfer mode. e figure shows that the flow consists of two cells occupying the entire cavity whose nuclei are located in the center. For low Ra, the viscosity determines the intensity of the convection motion, due to the decreasing of the driving force for both fluids, that is, the water, which has the greatest intensity for these moderate Rayleigh numbers. e fluid that is heated by the heating walls moves to the adiabatic upper wall, where the flow goes to the right and left vertical cold walls, because the density of the heated fluid decreases and so moves upwards, and that is the flow intensity of the pure fluid which is the strongest. e temperature field is stratified along the diagonal of the cavity and the thermal gradients are weak in the vicinity of the active walls. On the other hand, for high Ra, the nanofluid circulates faster than water due to increasing of the buoyancy force. e shape of the isotherms changes under the effect of the viscosity, which is due to the change in the heat transfer mode, which goes from conduction to convection when increasing the Ra. e fluid motion drives the heat of the active areas through the cavity. When Ra � 10 6 , the thermal boundary layers become thinner and the isotherms become laminated in the central region of the cavity. e increase of the stratification is a function of the position and the dimension of the heated source ( Figure 4). For Ra � 10 3 , the isotherms display a symmetrical structure with respect to the vertical plane passing through X � 1/2. For a fixed value of the Rayleigh number, the degree of increase of the stratification with the increase of the dimensionless length of the heating source becomes important. e profiles show that, for the low values of Ra (Ra ≤ 10 4 ), the temperature increases away from the left cold wall (θ � −0.5) with the increase of Ra until reaching the value θ � 0.175 for X � 1/2 above the heating partition. is temperature decreases gradually until reaching the value −0.5 on the opposite cold wall of the enclosure. For Ra ≥ 10 5 , the temperature increases from the value −0.5 at the level of the cold wall located at X � 0 until reaching its maximum value. is behavior is due to the buoyancy force that becomes important and subsequently the heat transfer mode goes from conduction to convection. is will lead to significant heat transfer within the cavity for the nanofluids and the base fluid (water) and much of this transfer is by convection. It is also noted that, for different values of the Rayleigh number, as the heat exchange surface increases, consequently the temperature increases significantly.
In this part, the influence of different types of nanoparticles (Al 2 O 3 , Cu, and TiO 2 ) dispersed in the base fluid on the heat transfer is studied. Numerical simulations are performed for different Ra and volume fraction ϕ � 0.04. e comparisons conducted between the three types of nanoparticles concern primarily the dynamic and thermal fields,  and then the heat transfer. Figure 6 illustrates the isotherms and the streamlines for different types of nanoparticles (Al 2 O 3 , Cu, and TiO 2 ) and a pure fluid in the Z � 1/2 plane, the volume fractions varying between 0 and 0.04, and the Rayleigh number (Ra � 10 4 , 10 5 , and 10 6 ). e comparison between the nanofluids and the pure fluid shows a significant deviation of the isotherms by increasing the Rayleigh number and the volume fraction. For Ra � 10 3 , these     isotherms are nearly parallel to the vertical walls, indicating that, for small values of the Rayleigh number, the heat transfer is practically conductive. For Ra � 10 5 , the isotherms are slightly deformed with a small variation following y. Beyond this value of Ra, the deformation of the isotherms becomes more evident and horizontal stratification begins to settle in the center of the cavity. For Ra � 10 6 , we notice that the vertical gradient of the temperature becomes almost constant at the heart of the cavity and subsequently a horizontal stratification settles there. e values of ψ (the function of the current) increase with the increasing of Rayleigh number. us, it is shown that the flow of the pure fluid is stronger than that of the nanofluid for Ra � 10 3 . e streamlines become tighter close to the sidewalls, and the shape of the cells changes completely for Ra ≥ 10 5 . e flow of the nanofluids becomes stronger. According to this figure, the water and the different mixtures have the same dynamic and thermal behavior. Furthermore, Figure 6 confirms that the copper nanoparticles allow a better heat exchange by comparing with Alumina and Titanium and the type of nanoparticles is a main factor for the improvement of the heat transfer.

Velocity.
In order to justify more the flow of the fluid in the cavity, Figure 7 shows the profiles of the vertical component of the velocity (V) along the horizontal median of the enclosure, for different lengths and Rayleigh numbers; it increases under the effect of the buoyancy force with the increase of Ra. We observe a decrease in the intensity of the fluid flow at the cold side walls and an increase in the heated walls. is rise increases with Rayleigh number and reaches its maximum at the center of the heated walls, where the temperature is maximum.
is is due to the increase in the intensity of the thermal thrust forces and therefore to the predominance of convective heat transfer. In addition, it is useful to mention that the flow becomes stagnant in the 0.35 ≤ X ≤ 0.65 zone where the heating partition is located, and this is for all lengths d ≥ 0.5. It is also observed that the behavior of the dynamic field is symmetrical with respect to the plane X � 0.5. erefore, the fluid is heated significantly and relaxes with a decrease in density. It follows a rise of the fluid upwards and the speed of the flow becomes important. Figure 8 shows the vertical component velocity profiles at the center line for different Rayleigh number values (Ra � 10 3 , 10 4 , 10 5 , and 10 6 ) and different nanofluids. From this figure, we notice that the vertical velocity (V) reaches a maximum value near the adiabatic sidewalls, for all types of nanoparticles, where the fluid changes direction and V decreases towards negative values. e difference between the three types of nanofluids is small and the vertical component of the velocity is not significantly affected by the type of nanoparticles. For Ra � 10 3 , we observe that the values of the speed are higher in the case of base fluid, and the addition of nanoparticles contributes to slowing down of the fluid in the vertical direction. Decreasing speed results in a decrease in convective heat transfer. However, since much of the heat exchange is conducted by conduction, the decrease in the speed of the nanofluids has no visible influence on the overall heat transfer, which remains better than that of the water due to the high value of the thermal conductivity of the nanofluid. In the case of dominant convection (Ra � 10 6 ), we observe that the velocities reached by nanofluids are greater than those corresponding to base fluid.
us, the use of nanofluids modifies the velocity profiles and, as a result, the temperatures and the heat transfer are increased. When comparing the velocity profiles between the nanofluid and the base fluid, only small differences are observed. It is concluded that the V (y) components of the velocity are not sensitive to the type of nanofluid.

Average Nusselt Number.
e results presented in this part are obtained for Rayleigh numbers Ra between 10 3 and 10 6 , different volume fractions of the nanoparticles, and length of the partition. Subsequently, we will focus on the effect of these parameters on the average Nusselt number. e variation of the average Nusselt number as a function of the length of the heated sources for different Rayleigh numbers is shown in Figure 9. It is noted that, regardless of length and form of the heated sources, the average Nusselt number increases with the increase of the volume fraction. is increase is due to the improvement of the thermal conductivity of the nanofluid. e effect of the nanoparticles is greater at low Ra values, so that the 6% increase in the volume fraction for a length of 0.5 leads to a relative increase in the average Nusselt number of about 9% for Ra � 10 6 and about 26% for Ra � 10 3 . e figure also shows that, for a given volume fraction, the average Nusselt increases with the increase in Ra and the length of the block. Figure 10 shows the variation of the average Nusselt number for different values of Ra, as a function of the volume fraction of the Al 2 O 3 , Cu, and TiO 2 . We note that Nu av increases with Ra and the volume fraction of the nanoparticles. is is due to the improvement of the k nf , when the volume fraction of the nanoparticles increases. We observe, in a general way, that the intensification of the thermal thrust forces, through the increase of the number of Rayleigh, favors the thermal exchanges within the cavity and this is because of the increase of the speed ( Figure 7) following the intensification of convection currents. is figure also shows the effect of the nanoparticles type on the variation of the average Nusselt number. is heat transfer rate is maximum in the case of water-Cu. A large number of Nusselt, that is to say, a stronger exchange, is favored by a higher conductivity to evacuate heat, a lower heat capacity to reduce storage, and a higher density to promote convection (force Archimedes). ese parameters play in favor of copper in comparison with Al 2 O 3 and TiO 2 nanoparticles. Indeed, the latter nanoparticles have very close heat capacities and densities (difference not exceeding 12%), but the thermal conductivity of Al 2 O 3 is greater than that of TiO 2 by about 34.6%, according to the results of Glades [42]. As a result, Al 2 O 3 nanoparticles provide better heat transfer than TiO 2 nanoparticles.

Conclusion
e study presented in this article deals with the natural laminar convection of nanofluids within a three-dimensional cavity containing two heating portions located on the left wall. In the first part of this paper, we analyzed the effect of various parameters on heat transfer, namely, the volume fraction, the Rayleigh number, and the height of the heat source. is allowed us to highlight the effect generated by each parameter and to identify the optimal situations favoring heat transfer. In the second part, we made a comparison between three