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The present investigation deals with the case of an Ag (CuO or Al2O3)water nanofluid’ convection phenomenon within a square enclosure, which including two heated sources, of a square shape. To do so, a computer code, based on the finite volume approach and the SIMPLER algorithm, is adopted to solve the general coupled equations. By studding the impact of some pertinent parameters such the Rayleigh number, the heaters’ vertical location within the enclosure, as well as the nanoparticles’ volume fraction and its type on the nanofluid flow and heat behaviors, the numerical predictions proved the nanoparticles type is the key factor for the heat transfer enhancement within such industrial geometry, side by side with the increasing function of the Rayleigh number and the nanoparticles’ volume fraction. Unlike the latter, the increase in the heaters’ vertical location within the enclosure affects the nanofluid flow, but it has no significant impact on the mean transfer rate.
natural convection, square enclosure, Agwater nanofluid, CuOwater nanofluid, AL_{2}O_{3}water nanofluid, square heaters, finite volume approach
Heat convection of nanoﬂuids, which are a mixture of nanoparticles and a base ﬂuid; such Water and Oil, has been recently an active ﬁeld of research since they are used to improve the transfer rate of industrial processes. Compared to other techniques for enhancing the heat transfer in practical applications, nanoﬂuids have the advantage of behaving like pure ﬂuids reason to the nanometric size of the introduced particles. Thus, it’s adopted in various applications such advanced nuclear systems or micro/mini channel heat sinks, electronic equipment as power transistors, printed wiring boards and chip packages mounted on computer motherboards.
Back in time, and up to 1995, some papers dealing with natural convection phenomenon of nanofluids within differentially heated enclosures were very helpful for our beginning [14]. A review of the literature indicates that numerous industrial geometries were investigated [59] but almost works were done using rectangular configurations [1012].
Khanafer et al. [13] studied the transfer performance of some nanofluids into a rectangular enclosure. Their results proved that the heat transfer rate, at any given Grashof number, is an increasing function of the nanoparticles’ volume fraction. Additionally, the variances within different models, based on the thermophysical properties of the used nanofluid, have substantial effects on the numerical predictions. The authors proposed then, their new correlations to predict the mean transfer rate within such enclosure for various values of the Grashof number and the nanoparticles’ volume fraction. Tiwari and Das [14] investigated numerically the behavior of nanofluids inside a twosided’ liddriven square enclosure, governed by a horizontal thermal gradient. Using the finite volume method, the researchers developed a new model to predict the thermal behavior of nanofluids within such industrial geometry; taking into account the solid volume fraction.
A year after, Oztop and AbuNada [15] undertook a numerical study of natural convection phenomenon within a rectangular enclosure, partially heated from it left side wall, by considering various nanofluids, such as Cuwater, TiO_{2}water and Al_{2}O_{3}water as well. The right wall was maintained at a constant temperature; lower than that of the left discrete source, when the other walls were supposed adiabatic. Their results displayed the increasing function of the heat transfer with the nanoparticles’ volume fraction, proving at the same time, that using Cu nanoparticles is more required for better heat transfer within such configuration. As far as the left discrete source is increased, heat transfer rate is found to be pronounced as the source surface is getting larger. Then, higher aspect ratio between the active walls is more desired.
Muthtamilselvan et al. [16] studied the mixed convection phenomenon within a liddriven enclosure; filled by a Cuwater nanofluid. The side walls were assumed to be insulated, when the bottom wall is kept at a constant temperature higher than that of the moving top one. Regarding these conditions, it was found that both; the aspect ratio and the solid volume fraction affect the flow pattern and the heat transfer of the convective fluid within the enclosure. Going far, Mahmoodi [17] investigated numerically the natural convection phenomenon within a partitioned square enclosure filled by nanofluids such as Cuwater, Agwater and TiO_{2}water as well. The cavity’ side walls were assumed to be cold when a thin heater, for which both location and length were varying, was positioned inside the latter. With such investigation, Mahmoodi proved that the rate of heat transfer increases with the increase Rayleigh number and the nanoparticles volume fraction. At low Rayleigh values, the type of nanofluid does not affect the heat transfer whereas for high values of the latter, the heat transfer reaches its maximum values with the Agwater nanofluid. The effect of the heater’s position on the mean transfer is found depends on the Rayleigh value and the increased length as well. A year after, Mahmoodi and Sebdani [18] inspected numerically the free convection of the Cuwater nanofluid inside a differentially heated square enclosure with an adiabatic square body; located in its centre. For various Rayleigh values, except the case of Ra = 10^{4}, the mean Nusselt number is found increases with the increase nanoparticles volume fraction.
Motivated by these works and many others, the purpose of our present work is to maintain the natural convection phenomenon within a square enclosure, filled by an Ag, CuO, or Al_{2}O_{3}water nanofluid, and including two square heaters which mounted nearer the cold side walls at affixed distance of about 15% the cavity’ height. The numerical results will be presented in terms of pertinent parameters such as the Rayleigh number, the nanoparticles volume fraction as well as the heaters’ vertical distance. Such studied configuration is an interesting work for many industrial applications, especially in the Refining and Petrochemical fuels’ exchangers.
The problem under investigation, as shown in Figure 1, is a laminar, two dimensional natural convection within a square enclosure; having two isothermal square heaters, (of a width w equals 10% the cavity’ length), mounted nearer the side cold walls at a distance X_{p} = 0.15. the enclosure is assumed is assumed to be filled with an Ag, CuO, or Al_{2}O_{3}water nanofluid, where the water and the chosen nanoparticles are in thermal equilibrium. The computational results are obtained for a wide range of the Rayleigh number and the nanoparticles volume fraction, when the heaters’ location Y_{p} is ranging between 0.15 and 0.75. The constant thermophysical properties of the base fluid and the nanoparticles are given in Table 1, whereas the nanofluid density variation in the buoyancy forces is determined using the Boussinesq approximation [19].
Figure 1. Simulation domain with its boundary conditions
Table 1. Thermophysical properties of the base fluid and the nanoparticles at T = 25°C
Thermophysical properties 
Fluid Phase (water) 
Ag 
CuO 
Al_{2}O_{3} 
C_{p} (J/kg K) 
4179 
235 
531.8 
761.55 
r (kg/m^{3}) 
997.1 
10500 
6320 
3962 
k (W/m K) 
0.613 
429 
20 
37.17 
b (1/K) 10^{6} 
210 
19.7 
18 
75 
The nanofluid density, noted as ρ_{nf}, the heat’ capacity, (ρ C_{p})_{nf}, the thermal expansion’ coefficient, (ρ β)_{nf}, as well as the thermal diffusivity, α_{nf}, are defined respectively, as follows [13]:
ρ_{nf} = (1  φ) ρ_{f} + φ ρ_{s } (1)
(ρC_{p})_{nf} = (1  φ) (ρ C_{p})_{f} + φ (ρ C_{p})_{s} (2)
(ρ β)_{nf} = (1  φ) (ρ β)_{f} + φ (ρ β)_{s } (3)
$\alpha_{\mathrm{nf}}=\frac{\mathrm{k}_{\mathrm{nf}}}{\left(\rho \mathrm{C}_{\mathrm{p}}\right)_{\mathrm{nf}}}$ (4)
For the dynamic viscosity, μ_{nf}, and the thermal conductivity, k_{nf}, Brinkman [20] and MaxwellGarnett formulations [21] are adopted, respectively, as:
$\mu_{\mathrm{nf}}=\frac{\mu_{\mathrm{f}}}{(1\varphi)^{2,5}}$ (5)
$\frac{\mathrm{k}_{\mathrm{nf}}}{\mathrm{k}_{\mathrm{f}}}=\frac{\left(\mathrm{k}_{\mathrm{s}}+2 \mathrm{k}_{\mathrm{f}}\right)2 \varphi\left(\mathrm{k}_{\mathrm{f}}\mathrm{k}_{\mathrm{s}}\right)}{\left(\mathrm{k}_{\mathrm{s}}+2 \mathrm{k}_{\mathrm{f}}\right)+\varphi\left(\mathrm{k}_{\mathrm{f}}\mathrm{k}_{\mathrm{s}}\right)}$ (6)
The dimensionless conservation equations, describing the transport phenomenon inside the square, can be written as:
$\frac{\partial \mathrm{U}}{\partial \mathrm{X}}+\frac{\partial \mathrm{V}}{\partial \mathrm{Y}}=0$ (7)
$\mathrm{U} \frac{\partial \mathrm{U}}{\partial \mathrm{X}}+\mathrm{V} \frac{\partial \mathrm{U}}{\partial \mathrm{Y}}=\frac{\partial \mathrm{P}}{\partial \mathrm{X}}+\frac{\mu_{\mathrm{nf}}}{\rho_{\mathrm{nf}} \alpha_{\mathrm{f}}}\left[\frac{\partial^{2} \mathrm{U}}{\partial \mathrm{X}^{2}}+\frac{\partial^{2} \mathrm{U}}{\partial \mathrm{Y}^{2}}\right]$ (8)
$\mathrm{U} \frac{\partial \mathrm{V}}{\partial \mathrm{X}}+\mathrm{V} \frac{\partial \mathrm{V}}{\partial \mathrm{Y}}=\frac{\partial \mathrm{P}}{\partial \mathrm{Y}}+\frac{\mu_{\mathrm{nf}}}{\rho_{\mathrm{nf}} \alpha_{\mathrm{f}}}\left[\frac{\partial^{2} \mathrm{V}}{\partial \mathrm{X}^{2}}+\frac{\partial^{2} \mathrm{V}}{\partial \mathrm{Y}^{2}}\right]+\frac{(\rho \beta)_{\mathrm{nf}}}{\rho_{\mathrm{nf}} \beta_{\mathrm{f}}} \operatorname{Ra} \operatorname{Pr} \theta$ (9)
$\mathrm{U} \frac{\partial \theta}{\partial \mathrm{X}}+\mathrm{V} \frac{\partial \theta}{\partial \mathrm{Y}}=\frac{\alpha_{\mathrm{nf}}}{\alpha_{\mathrm{f}}}\left[\frac{\partial^{2} \theta}{\partial \mathrm{X}^{2}}+\frac{\partial^{2} \theta}{\partial \mathrm{Y}^{2}}\right]$ (10)
where Ra (= g β ∆T H^{3}/α_{f} ν_{f}) is the Rayleigh number, Pr (= ν_{f}/α_{f}) is the Prandtl number.
The average rate of heat transfer across the side walls is expressed in dimensionless form by the Nusselt number as shown Eq. 11:
$\left\mathrm{Nu}_{\mathrm{left} \text { wall }}\right=\left\mathrm{Nu}_{\text { right wall }}\right=\left(\frac{\mathrm{k}_{\mathrm{nf}}}{\mathrm{k}_{\mathrm{f}}}\right) \int_{0}^{1}\left.\left(\frac{\partial \theta}{\partial \mathrm{X}}\right)\right_{\text { wall }} \mathrm{d} \mathrm{Y}$ (11)
The governing conservation equations are discretized in space using the finite volume approach, when the convectiondiffusion terms were treated with a PowerLaw scheme. The resulting algebraic equations, with the associated boundary conditions, are then solved using the line by line method. As the momentum equation is formulated in terms of the primitive variables (U, V and P), the iterative procedure includes a pressure correction calculation method, namely SIMPLE modified [22] to solve the pressurevelocity coupling. Noted that the convergence criterion for temperature, pressure, and velocity is given as:
Error $=\frac{\sum_{j=1}^{m} \sum_{i=1}^{n}\left\xi_{i, j}^{k+1}\xi_{i, j}^{k}\right}{\sum_{j=1}^{m} \sum_{i=1}^{n}\left\xi_{i, j}^{k+1}\right} \leq 10^{6}$ (12)
where both m and n are the grid points numbers in X and Y directions, respectively. ξ is any of the computed field variables, k is the iteration number, i and j are nodes according to X and Y directions.
The performance of the using code via the natural convection of nanofluids is established by comparing its predictions with other numerical results, and by verifying the grid independence of the present results. First, the present results are consistent with previous computations, namely those of Ghia et al. [23] in terms of Streamlines within a square enclosure of an upper moving wall.
Figure 2 displays the fluid’ motion within such conditions as a function of Reynolds number such as 100, 400 and 1000 as well. As we can see, the present predictions present an identical solution with Ghia et al. [23].
(Re = 100)
(Re = 400)
(Re = 1000)
Ghia et al. [23] Present predictions
Figure 2. Fluid’ motion validation with Ghia et al. [23]
Figure 3. Mean Nusselt number of the cold square partitioned enclosure, a comparison with the results of Arefmanesh et al., [24]
Table 2. Average Nusselt number of the heat source, comparison with Oztop and AbuNada [15]
Ra 
φ 
Oztop and Abunada [15] 
Present study 
Relative gap (%) 

0.00 
1.004 
1.005 
0.10 

0.05 
1.122 
1.128 
0.53 
10^{3} 
0.10 
1.251 
1.248 
0.24 

0.15 
1.423 
1.419 
0.28 

0.20 
1.627 
1.621 
0.37 

0.00 
2.010 
2.032 
1.08 

0.05 
2.122 
2.109 
0.61 
10^{4} 
0.10 
2.203 
2.199 
0.18 

0.15 
2.283 
2.291 
0.35 

0.20 
2.363 
2.382 
0.80 

0.00 
3.983 
3.992 
0.22 

0.05 
4.271 
4.256 
0.35 
10^{5} 
0.10 
4.440 
4.389 
1.16 

0.15 
4.662 
4.680 
0.38 

0.20 
4.875 
4.861 
0.29 
Going far, and taking the second reference such Oztop and AbuNada [15], by considering the case of natural convection in a partiallyheated square cavity filled with Cuwater nanofluid. Table 2 illustrates the evolution of the average Nusselt number calculated at the heat source with both, the Rayleigh number and the Cuwater nanofluid volume fraction. As we can see, the present results and those of Oztop and AbuNada are in excellent agreement since the maximum gap is about 1.2%.
In order to determine an optimal grid for the present numerical predictions, a grid independence study is conducted. The mean Nusselt number was calculated for several mesh distributions, ranging from 60x60 to 120x120, as displayed in Figure 4. It can be observed that a 90x90 uniform grid is adequate for a grid independent solution. However, to minimize the discretization errors in the numerical simulations, a fine structured mesh of 100x100 is used to avoid roundoff error for all other calculations in this study.
Figure 4. Average Nusselt number for different uniform grids; Y_{p} = 0.15; φ_{Al}_{2}_{O}_{3} = 0.10
The presented results are generated for different pertinent dimensionless groups as: the Rayleigh number (10^{4} ≤ Ra ≤ 10^{6}) and the solid volume fraction (0% to 10%). The default parameters are assigned to the heaters’ width value, (w = 0.10), and the Prandtl number, (Pr = 6.2). Three different types of nanoparticles are investigated as: Ag, CuO and Al_{2}O_{3}. The predicted hydrodynamic and thermal fields’ variables are presented through the streamlines, the isotherms, and the corresponding velocity profiles. The average Nusselt number is also represented in order to supply useful information about the influence of each parameter, quoted above, on the mean transfer enhancement.
The present study is carried out in two phases: in the first one, and for each nanofluid, the effects of both; Rayleigh number and solid volume fraction on the heat transfer performance is investigated, when the heaters location Y_{p} is fixed at 0.15 and 0.75, respectively. In the second part, calculations are performed for a wide range of the heaters position: 0.15 ≤ Y_{p} ≤ 0.75.
5.1 Rayleigh number & solid volume fraction
With each value of the Rayleigh number Ra, the streamlines and the isotherm plots of the (Ag, CuO or Al_{2}O_{3})water nanofluid with a volume fraction φ = 0.10 as well as the base fluid, (φ = 0), are shown in Figure 5. The related stream functions values are displayed in Table 3.
In the absence of the nanoparticles, φ = 0, and with a low value of the Rayleigh number, (which means that conduction heat transfer regime is dominant), the streamlines show two counterrotating eddies established in the enclosure.
By increasing the Rayleigh number, the left counterclockwise cell gets slighter whereas the streamlines become more packed adjacent to the cold wall, transporting the cold liquid towards the bottom. As Ra = 10^{6}, the convection mechanism becomes more pronounced and then, the right cell gets larger near the bottom wall indicating consequently, the strength motion of the cold liquid in this area. The same phenomenon occurs nearer the opposite wall where the left cell gets larger next to the top portion of the enclosure.
In the presence of the nanoparticles, the fluid motion is better than that of the pure fluid. In other words, the velocity components of nanofluids rise as the fluid’s energy transport increases with the increase nanoparticles volume fraction. The increase in the Rayleigh number adds to the flow strength which causes, in turn, the rise in the tendency of the cell to grow.
Regarding the thermal plots in Figure 5, for a Rayleigh number equals to 10^{4}, the isotherms are uniformly distributed inside de square, revealing a dominatedconduction heat transfer regime. As the Rayleigh number increases over than 10^{4}, the natural convection effect turns out to be dominant and then, the isotherms are distributed in a stratified manner with a horizontal profile in the middle of the enclosure, whereas thin thermal boundary layers are shaped around the heaters as well as along the side cold walls, indicating a large temperature gradient along these surfaces.
Table 3. Streamfunction values for various Rayleigh number and different type of nanoparticles

φ = 0 
φ = 0.10 

Ra 
Pure water 
Ag 
CuO 
Al_{2}O_{3} 


Y_{min} 
Y_{max} 
Y_{min} 
Y_{max} 
Y_{min} 
Y_{max} 
Y_{min} 
Y_{max} 
10^{4} 
2.16 
2.12 
3.17 
3.12 
2.61 
2.51 
2.23 
2.20 
10^{5} 
8.55 
9.20 
12.30 
13.05 
10.89 
11.53 
10.00 
10.56 
10^{6} 
19.69 
20.41 
27.75 
28.93 
26.05 
26.46 
25.09 
25.46 
Figure 7. Mean Nusselt number with the nanoparticles volume fraction for various values of the Rayleigh number
Figure 6 illustrates the vertical velocity component profiles (Vvelocity) along the horizontal midplane of the cavity for the inspected types of the nanofluids. The examination of the magnitude of the vertical velocity component for different values of Ra confirms the weakness of the rotating eddies, previously observed in the streamlines, for a low Rayleigh number. Besides, the vertical velocity profiles verify the existence of a clockwise circulating cell at the right cavity side and an anticlockwise circulating cell at the opposite side. The increase of the magnitude of the Vvelocity with the increasing Rayleigh number indicates strong buoyant flows within the enclosure. The heat transfer mechanism is, therefore, expected to be due to convection while conduction is responsible for the heat transfer at low Rayleigh numbers.
It is important to denote here that, the negative velocity values illustrate the downward motion of the cold fluid nearer the side walls, and setting the heaters inside the enclosure as shown in Fig. 1 makes the cold fluid velocity greater on the left side compared to the right one.
Due to the nanoparticles thermophysical properties at φ = 0.10, the nanofluids are associated with a higher absolute magnitude of the velocity compared to that of the pure water. The main fluid motion is noted with Agwater nanofluid compared to CuO or Al_{2}O_{3}water nanofluids.
Figure 7 displays the mean Nusselt number calculated along the cavity sides with respect to the solid volume fraction at various values of the Rayleigh number. In general, the mean Nusselt number is an increasing function of the Rayleigh number as the convective regime becomes dominates and increases with the increase of the nanoparticles volume fraction, due to the improvement of the fluid thermal conductivity. A maximum heat transfer enhancement is computed with the Agwater nanofluid compared to the CuOwater and Al_{2}O_{3}water nanofluids.
Figure 5. Streamlines and isotherms plots for different values of the Rayleigh number, [ ^{_ _ _} φ = 0.10, −−− base fluid].(a) Agwater nanofluid. (b) CuOwater nanofluid. (c) Al_{2}O_{3}water nanofluid
Figure 6. Effect of both the Rayleigh number and the types of nanoparticles on the vertical velocity profiles at Y_{p} = 0.15; [ ^{_ _ _} φ = 0.10, −−− base fluid]
5.2 Heaters position Y_{P}
In this section, the heaters position Y_{p} is ranging from 0.15 to 0.75. The main circulation cells as well as the isotherm plots of the Agwater nanofluid, at Ra = 10^{6}, are presented in Figure. 8 for two different solid volume fractions such 0 and 0.10. Due to the symmetrical plan located at Y_{p} = 0.50, the results are presented between 0.15 and 0.45.
Compared to the plots of the section before, Figure. 5(a), the proposed dispositions of the heaters affect the nanofluid flow motion. For Ra = 10^{6} and when the distance wallpartition is about 0.15, two asymmetrical counterrotating cells are formed inside the enclosure, whereas no secondary vortices appear. By increasing the distance wallpartition, the weak cell (located at the left side of the enclosure) enlarges to be identical to the right one. As a result, two symmetrical counterrotating cells are observed and two secondary vortices grow between the primary ones due to the fluid motion.
Figure 9 illustrates the vertical velocity profiles along the horizontal midplane of the enclosure for different distances Y_{p}. As it is expected, the examination of the magnitude of the Vvelocity at Y_{p} = 0.45 confirms the symmetry of the primary rotating cells, previously observed in analyzing streamlines. In the same way as before, by using Ag nanoparticles (with φ = 0.10), the absolute magnitude of the velocity becomes higher.
The isotherms for each distance Y_{p} are adjusted according to the presented streamlines. Once again, the clustering of these latter towards the heaters and the side walls reveals the high temperature gradient and the development of thin boundary layers nearer these surfaces.
The effect of Y_{p} on the mean Nusselt number for different values of both; the nanofluid volume fraction and the Rayleigh number is displayed in Figure 10. As we can see, the increase in the heaters position Y_{p} has no significant effect on the heat transfer enhancement as the mean Nusselt number occurs at Y_{p} = 0.45 is found to be quietly over 8% to that calculated at 0.15
The analysis of nanofluids natural convection inside a square enclosure, including two heated square bodies mounted nearer the cold walls was numerically realized in this paper. Taking into account the effects of various parameters, such the Rayleigh number, the heaters vertical position, the nanoparticles category and their volume fraction, optimal conditions leading to better heat transfer within this kind of enclosures are expected. The results may resume as follows:
 For the investigated nanofluids, heat transfer is found as an increasing function of the Rayleigh number and the nanoparticles volume fraction.
 For fixed values of both: the Rayleigh number and the nanoparticles volume fraction, the maximum heat transfer enhancement is obtained with the Agwater nanofluid compared to the CuOwater and the Al_{2}O_{3}water nanofluids. Hence, the use of Ag nanoparticles is required.
 A further increase in the heaters distance Y_{p} affects the nanofluid motion but it has no significant effect on heat transfer enhancement.
Figure 8. Streamlines and isotherm plots of the Agwater nanofluid for different values of Y_{p }; [ ^{_ _ _} φ = 0.10, −−− base fluid]
(a) Y_{p} = 0.15
(b) Y_{p} = 0.25
(c) Y_{p} = 0.35
(d) Y_{p} = 0.45
Figure 9. Vertical velocity profiles for different values of the Rayleigh number and the heaters vertical position, [ ^{_ _ _} φ_{Ag} = 0.10, ^{_______} base fluid]
Figure 10. Effect of the heaters position on the mean Nusselt number for various values of Ra and φ_{Ag}
C_{p} 
Specific heat at constant pressure, J.kg^{1}. K^{1} 

g 
Gravitational acceleration, m.s^{2} 

H 
Cavity height, m 

k 
Thermal conductivity, W. m^{1}. K^{1} 

Nu 
Average Nusselt number 

p* 
Pressure, Pa 

P 
Dimensionless pressure, = p* H^{2}/ ρ_{nf} α_{f}^{2} 

Pr 
Prandtl number 

Ra 
Rayleigh number 

T 
Dimensional temperature, K 

u, v 
Velocity components, m. s^{1} 

U, V 
Dimensionless velocity components, = u (or v) H / α_{f} 

W 
Heater’ width, m 

x, y 
Cartesian coordinates, m 

X, Y 
Dimensionless Cartesian coordinates, = x (or y) / H 

X_{p} 
Heater’ distance from the cold wall 

Y_{p} 
Heater’ distance from the bottom wall 

Greek Symbols 

a 
Thermal diffusivity, m^{2}. s^{1} 

b 
Thermal expansion coefficient, K^{1} 

q 
Dimensionless temperature, = (TT_{c})/(T_{h}T_{c}) 

μ 
Dynamic viscosity, kg. m^{1}. s^{1} 

r 
Density, kg. m^{3} 

φ 
Solid volume fraction 

Y 
Stream function 

Subscripts 

c 
Cold 

h 
Hot 

f 
Fluid (pure water) 

nf 
Nanofluid 

s 
Solid particles 
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