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In this paper, a numerical study of natural convection in a square enclosure with non-uniform temperature distribution maintained at the bottom wall and filled with nanofluids is carried out using different types of nanoparticles. The remaining walls of the enclosure are kept at a lower temperature. Calculations are performed for Rayleigh numbers in the range 5 × 10^{3} ≤ Ra ≤ 10^{6} and different solid volume fraction of nanoparticles 0 ≤ χ ≤ 0.2. An enhancement in heat transfer rate is observed with the increase of nanoparticles volume fraction for the whole range of Rayleigh numbers. It is also observed that the heat transfer enhancement strongly depends on the type of nanofluids. For Ra = 10^{6}, the pure water flow becomes unsteady. It is observed that the increase of the volume fraction of nanoparticles makes the flow return to steady state.

Enhancement of heat transfer is an important goal in heat exchanger systems. Many different ways of heat transfer improvement such as placement of fins, use of porous media and others have been utilized. This is due to the fact that typical process or conventional base fluids have low thermal conductivities. An innovative way of heat transfer enhancement is the use of nano-particles of relatively higher thermal conductivities suspended in the base fluids (see Eastman et al. [_{2} particle inclusions and observed the percolation pattern of particle clustering by scanning tunnel microscopic (STM) photos. It was believed that clustering could affect the enhancement prominently. Wang et al. [

Several studies of convective heat transfer in nanofluids have been reported in recent years. Khanafer et al. [

Non-uniform heating of surfaces in buoyancy-driven flow in a cavity has significant effect of the flow and heat transfer characteristics and finds applications in various areas such as crystal growth in liquids, energy storage, geophysics, solar distillers and others. In a relatively recent study, Sarris et al. [

The objective of this work is to study natural convection in a square enclosure filled with a water-based nanofluid (water with Ag, Cu, Al_{2}O_{3} or TiO_{2} nanoparticles) with non-uniform (sinusoidal) temperature distribution maintained at the bottom wall. An accurate finite volume scheme along with a multi-grid technique is devised for the purpose of solution of the governing equations.

Consider laminar natural convection of a water-based nanofluid in a square enclosure of side length L (

. The remaining walls are cooled at. The nanofluids used in the model development are assumed to be Newtonian and the flow is assumed incompressible. The nanoparticles are assumed to have uniform shape and size. Also, it is assumed that both the fluid phase (water) and nanoparticles (Ag, Cu, Al_{2}O_{3} and TiO_{2}) are in thermal equilibrium and they flow at the same velocity. The physical properties of the nanofluids are considered to be constant except the density variation in the body force term of the momentum equation which is determined based on the Boussinesq approximation. Under the above assumptions, the system of equations governing the two dimensional motion of a nanofluid in the enclosure can be written in dimensional form as

Here, is the effective density of the nanofluid defined as:

According to Brinkman’s formula [

The thermal expansion coefficient of the nanofluid can be obtained by:

The thermal diffusivity of the nanofluid is given by:

where is the thermal conductivity of the nanofluid, which for spherical nanoparticles is given by Maxwell [

and where the heat capacitance of the nanofluid is given by:

In the previous expressions, subscripts f and p are related to pure fluid and dispersed nanoparticles, respectively. The thermo-physical properties of the base fluid and the considered nanoparticles in this study are given in

In order to obtain non-dimensional equations, the following dimensionless parameters have been introduced:

, , , , , ,

and

The enclosure boundary conditions consist of no-slip and no penetration walls, i.e., U = V = 0 on all four walls. The thermal boundary conditions on the bottom wall is such that. The left and right vertical walls are at the cold temperature and the bottom wall at.

The local and averaged heat transfer rates at the bottom hot wall of the cavity are presented by means of the local and averaged Nusselt numbers, Nu and Nu_{m}, which are, respectively, determined as follows:

The unsteady Navier-Stokes and energy equations are discretized using staggered, non-uniform control volumes. A projection method (Achdou and Guermond [

A finite-volume method (Patankar [

Our code has been tested for natural convection fluid flows in differentially heated cavities and in RayleighBénard configuration and gave excellent results (see ref. [35,36]. In order to validate the nanofluid version, numerical calculations were carried on an uniform grid containing 80 × 80 nodes and for a test case considered recently by Aminossadati and Ghasemi [_{C}. In order to validate our numerical results, we chose the case Pr = 6.2 (pure water), Ra = 10^{5}, χ = 0.1 and three different nanoparticles, i.e., Cu, Ag and Al_{2}O_{3}. _{m} and are the averaged Nusselt number through the hot source and the maximum value of temperature on the heated source, respectively. The relative errors in Nu_{m} and are denoted by and, respectively.

In the present grid independence test, the Prandtl number is set to Pr = 6.2 (pure water). The nanoparticles are chosen to be copper (Cu) with a solid volume fraction χ = 0.1 and a Rayleigh number Ra = 10^{5}. Numerical computations have been carried on six different grid sizes, i.e., 32 × 32, 48 × 48, 64 × 64, 80 × 80 and 96 × 96 grid sizes.

In this section, the nanofluid-filled enclosure is studied for a range of solid volume fraction 0% ≤ χ ≤ 20% and

^{5} and χ = 0.1.

the Rayleigh number varies from 5 × 10^{3} to 5 × 10^{5}. Four kinds of nanoparticles are envisaged, i.e., silver (Ag), copper (Cu), alumina (Al_{2}O_{3}) and titanium (TiO_{2}). For all simulations the considered base fluid is water (Pr = 6.2). The dimensionless time step varies from ∆τ = 2 × 10^{−3} for Ra = 5 × 10^{3} to ∆τ = 10^{−5} for Ra = 5 × 10^{5}. The steady state is considered as achieved according to the following criterion:

Here, G represents the variable U, V or θ, the superscript r refers to the iteration number and (i, j) refer to the space coordinates. Note that this criterion was only applied for Ra ≤ 5 × 10^{5}. At Ra = 10^{6} where unsteady flow did not develop, the maximum dimensionless time to which the solution was continued was increased by up to 100% to ensure that unsteady flow did not in fact occur at larger dimensionless times. Note that this method is inspired from the one used by Oosthuizen and Paul [

In this section, we consider the case where the cavity is filled with water and alumina particles. The solid volume fraction varies from 0% to 20%. It should be noted that tests were conducted for Ra = 10^{6} and the flow then passes to an unsteady state with the appearance of several bifurcations. The results for Ra = 10^{6} will be the subject of the next section.

In ^{3} ≤ Ra ≤ 5 × 10^{5}, for the case of a water-Al_{2}O_{3} nanofluid (- - - -) and pure water (——). The value of solid volume fraction is set to χ = 0.05. ^{5} and the cases χ = 0.05 and χ = 0.1 for which the maximal intensities of the rotating cells increases by comparison to the case χ = 0. That means that for these combinations, the buoyancy force is greater than the inertia force due to the presence of nanoparticles.

^{3}, the deviation (relative to χ = 0) between the maximum values of velocity is for and for χ = 0.2. By increasing the Rayleigh number, these deviations decrease. For Ra = 5 × 10^{4} and Ra = 5 × 10^{5} these deviations are, , and, respectively. As far as the temperature distribution is concerned, clear differences are observed in the isotherm contour plots compared to the case χ = 0. These differences are accentuated as the solid volume fraction increases. These differences mean that the presence of nanoparticles affect especially the heat transfer rate through the enclosure.

The heat transfer distribution through the hot wall is displayed in ^{3}) and χ = 0, the transfer of heat through the hot wall is relatively low with a slight curvature at X = 0.5. This curvature is due to relatively higher intensity of the counterrotating cells represented by the highest value of when χ = 0. When χ increases to χ = 0.2, the curvature at

Ra = 5 × 10^{3} Ra = 5 × 10^{4} Ra = 5 × 10^{5}

Ra = 5 × 10^{3} Ra = 5 × 10^{4} Ra = 5 × 10^{5}

the center disappears because the fluid velocity decreases. The heat transfer in this case is maximum at X = 0.5 and is higher due to the presence of nanoparticles whose thermal conductivity is much greater than that of water. The same phenomena are observed almost on the curves related to Ra = 5 × 10^{4} and Ra = 5 × 10^{5} with a maximum heat transfer in the vicinity of X = 0.25 and X = 0.75. For example, for Ra = 5 × 10^{4}, the maximum Nusselt number value is Nu_{max} = 7.266 and is situated at both locations X = 0.314 and X = 0.686 for χ = 0. For χ = 0.2, Nu_{max} = 8.370 and is located at X = 0.295 and X = 0.705.

In this part of the study, the solid volume fraction is fixed at 0.1 and three other solid nanoparticles, copper (Cu), silver (Ag), and titanium (TiO_{2}) are studied. The physical properties of these metals are listed in

In ^{4} are reported for Cu, Ag, TiO_{2} and Al_{2}O_{3}. It is observed that with nanoparticles of silver or copper, the velocity of fluid particles is higher than with nanoparticles of alumina or titanium. This fact should result in a heat transfer rate higher with nanoparticles of silver or copper than with nanoparticles of Al_{2}O_{3} or TiO_{2}. Indeed, as shown in

where the values of constant coefficients c and d corresponding to the different types of nanofluid are given in

Finally, we conclude this study by presenting some benchmark results according to the mean Nusselt number reported in ^{3} ≤ Ra ≤ 5 × 10^{5}, the maximum percentage increase in heat transfer rate is obtained for the combination (Ra = 5 × 10^{3}; χ = 0.2; water-Ag) with a gain of 55% and minimum for the combination (Ra = 5 × 10^{4}; χ = 0.05; water-TiO_{2}) with a gain of 3.5%. Note that, the highest values of percentage increase in heat transfer are observed for low values of Rayleigh numbers and high values of solid volume fractions. This is due to the fact that for low Ra, the mechanism of heat transfer is mainly governed by conduction. However, good gains in heat transfer rate can also be obtained for higher Rayleigh number through the use of Nanofluids with appropriate percentage of solid volume fraction of nanoparticles.

In this section, the Rayleigh number is fixed to Ra = 10^{6} and three different fluids (pure water, water-Cu and water-TiO_{2}) are considered. _{tot} = 5 was used. After that, simulations were carried on for water-Cu and water-TiO_{2} with volume fractions in the range. The results are regrouped in _{2}. As a first observation, one can see that by increasing the value of the volume fraction, the flow becomes steady. Indeed, for the case water-Cu, the flow is unsteady for χ = 0.05 and χ = 0.1 but steady for χ ≥ 0.15. The same behaviors are seen for water-TiO_{2} combination. Here, the steadiness appears for χ ≥ 0.1. The average Nusselt number in time and space are reported in _{2}).

We would finally point out that the presented results in this section should be considered only qualitatively because we believe that when the flow becomes unsteady, three-dimensional effects cannot be neglected anymore and can affect quantitatively the global flow structures and hence, the values of the Nusselt number. However, the fact that the increase of volume fraction at a fixed Rayleigh number may change the flow from unsteady to steady state will probably remain the same in 3D simulations. Our next work is the extension to a three-dimensional study on the topic and comparisons between the 2D and 3D results.

This study focused on numerical modelling of natural convection in a square enclosure with non-uniform temperature distribution maintained at the bottom wall and filled with nanofluids using different types of nanoparticles. The remaining walls of the enclosure were kept at lower and equal temperatures. The finite volume method along with a multi-grid technique was used numerically investigate the effect of using different nanofluids on the flow structure and heat transfer rate in the enclosure. For 5 × 10^{3} ≤ Ra ≤ 5 × 10^{5} and, the flow structure was characterized by two counter-rotating circulating cells for all type of nanofluids. For relatively moderate Rayleigh numbers, the intensity of these cells decreased with the increase of solid volume fraction. By increasing the value of Rayleigh number and keeping

^{6}.

other parameters fixed, the heat transfer was enhanced. Comparing the values of averaged Nusselt number obtained with pure water with other nanofluids, it was seen that for low Rayleigh numbers and high solid volume fractions, enhancements up to 55% could be reached. This percentage strongly depended on the type of nanofluids. Indeed, the highest heat transfer enhancements were obtained when using copper or silver nanoparticles. The lowest values of percentage increase in heat transfer were obtained with TiO_{2} nanoparticles but were not negligible as far. It reached 40.8% at high values of solid volume fractions and low Ra. For a higher Rayleigh number, i.e., Ra = 10^{6}, the pure water flow becomes unsteady. Several bifurcations were observed when changing the solid volume fractions values or the kind of the nanofluid used. Also, the increase of the volume fraction of nanoparticles made the flow return to steady state. This could be a good way to delay the transition to unsteadiness if needed. In the near future, this study will be extended for higher Rayleigh numbers, 3D studies and other types of base fluids and nanoparticles.

C_{p} specific heat, J∙kg^{−1}∙K^{−1}

g gravitational acceleration, m∙s^{−2}

k thermal conductivity, W∙m^{−1}∙K^{−1}

L enclosure length, m Nu local Nusselt number Nu_{m} average Nusselt number p pressure, Pa P dimensionless pressure Pr prandtl number,

Ra Rayleigh number,

T temperature, K T_{C} temperature of cold wall, K u, v velocity components in x, y directions, m∙s^{−1}

U, V dimensionless velocity components x, y Cartesian coordinates, m X, Y dimensionless coordinates

thermal diffusivity, m^{2}∙s^{−1}

volumetric thermal expansion coefficient, K^{−1}

solid volume fraction

difference temperature,

dynamic viscosity, N∙s∙m^{−2}

kinematic viscosity, m^{2}∙s^{−1}

non-dimensional temperature,

density, kg∙m^{−3}

C cold f pure fluid H hot nf nanofluid p nanoparticle