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Research Papers

# Evolution of Nanofluid Rayleigh–Bénard Flows Between Two Parallel Plates: A Mesoscopic Modeling StudyOPEN ACCESS

[+] Author and Article Information
Gui Lu

Key Laboratory for Thermal Science
and Power Engineering of MOE,
Tsinghua University,
Beijing 100084, China;
Institute of Thermal Engineering,
Beijing Jiaotong University,
Beijing 100044, China

Yuan-Yuan Duan

Key Laboratory for Thermal Science and
Power Engineering of MOE,
Tsinghua University,
Beijing 100084, China
e-mail: yyduan@tsinghua.edu.cn

Xiao-Dong Wang

State Key Laboratory of Alternate Electrical Power
System with Renewable Energy Sources,
North China Electric Power University,
Beijing 102206, China;
Beijing Key Laboratory of Multiphase Flow and
Heat Transfer for Low Grade Energy,
North China Electric Power University,
Beijing 102206, China
e-mail: wangxd99@gmail.com

1Corresponding authors

Manuscript received January 17, 2014; final manuscript received July 1, 2014; published online August 8, 2014. Assoc. Editor: Calvin Li.

J. Nanotechnol. Eng. Med 4(4), 040905 (Aug 08, 2014) (9 pages) Paper No: NANO-14-1003; doi: 10.1115/1.4027987 History: Received January 17, 2014; Revised July 01, 2014

## Abstract

The developing and developed nanofluid Rayleigh–Bénard flows between two parallel plates was simulated using the mesoscopic thermal lattice-Boltzmann method (LBM). The coupled effects of the thermal conductivity and the dynamic viscosity on the evolution of Rayleigh–Bénard flows were examined using different particle volume fractions (1–4%), while the individual effects of the thermal conductivity and the dynamic viscosity were tested using various particle sizes (11 nm, 20 nm, and 30 nm) and nanoparticle types (Al2O3, Cu, and CuO2). Two different heating modes were also considered. The results show that Rayleigh–Bénard cell in nanofluids is significantly different from that in pure fluids. The stable convection cells in nanofluids come from the expansion and shedding of an initial vortex pair, while the flow begins suddenly in pure water when the Rayleigh number reaches a critical value. Therefore, the average Nusselt number increases gradually for nanofluids but sharply for pure liquids. Uniform fully developed flow cells with fewer but larger vortex pairs are generated with the bottom heating with nanofluids than with pure liquid, with extremely tiny vortexes confined near the top heating plate for top heating. The number of vortex pairs decreases with increasing nanoparticle volume fraction and particle diameter due to the increasing of dynamic viscosity. The average Nusselt number increases with the increasing Rayleigh number, while decreases with the increasing nanoparticle diameters. The nanoparticle types have little effect on the Rayleigh–Bénard flow patterns. The Rayleigh–Bénard flows are more sensitive with the dynamic viscosity than the thermal conductivity of nanofluids.

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## Introduction

Rayleigh–Bénard convection is an orderly convection phenomenon in a thin layer of liquid that was set on a horizontal plane heated from below. The initial movement, due to the buoyancy and gravity, results from the upwelling of warmer liquid from the heated bottom layer, and then the upwelling spontaneously organized into a regular pattern of cells [1-4], as shown in Fig. 1. Rayleigh–Bénard convection is one of the most commonly studied convection phenomena [5-9].

Nanofluids have been investigated quite intensively in recent decades. Nanoparticles added to pure fluids significantly alter the thermal conductivity, thermal diffusivity, viscosity, and other thermophysical parameters of the base fluids [10-16]. Then high thermal conductivity makes nanofluids promising working fluids for a variety of heat transfer enhancement technologies. The interactions between the base fluid and nanoparticles, such as the chaotic movements of the nanoparticles, the flow and heat transfer characteristics of nanofluids significantly different from that of pure fluids, so nanofluids have attracted much research interest.

The natural convection of nanofluids has been widely studied experimentally [17-21], numerically [22-29] and theoretically [30]. In the experimental works of Putra et al. [17], Wen and Ding [19], Li and Peterson [21], the heat transfer rate decreases with increasing nanoparticle loading concentrations. However, Ho et al. [18] and Nnanna et al. [20] reported different results using the same nanofluids (Al2O3-water), in which the heat transfer rates significantly increase at low nanoparticle concentrations, but decrease at high nanoparticle concentrations. Although the heat transfer rate increases with increasing nanoparticle loading concentrations in most of numerical results, as reviewed by Haddad et al. [22], there are still some numerical simulations using single-phase models reported the deterioration behavior as reported in the Putra et al.'s experiments, for example, Abouali and Falahatpisheh [23], Abu-nada [24,25], Li and Violi [26], Santra et al. [27], Rashmi et al. [28], and Ho et al. [29]. In the work of Putra et al. [17], the deterioration of heat transfer rate was attributed to the nanoparticle sedimentation or particle–fluid slip. However, the increasing of the dynamic viscosity was considered to be the main reason of the deterioration of nanofluid heat transfer rate [24,25]. The Rayleigh–Bénard flows are very sensitive with the thermophysical properties, especially with the thermal conductivity and viscosity. The thermal conductivity enhancement is not the only factor that affects the flow and thermal behavior of natural convection of nanofluids [19,20]. Therefore, the heat transfer rates can be found to be enhanced or deteriorated with respect to the base fluid using different effective dynamic viscosity formulas in the single-phase models [24]. Ho et al. [29] thought the usage of two viscosity formulas might contribute to explaining the disparate findings among the numerical predictions and experimental results. The enhancement of heat transfer rates by increasing the thermal conductivity competes with the deterioration by increasing the dynamic viscosity, leading to the diversity of heat transfer performances in nanofluid natural convections.

Nanofluids contain a large number of nanoscale particles with intensive random movements and interactions between the nanoparticles and the fluid molecules. The LBM is based on mesoscopic kinetic equations (the Boltzmann equation). The Navier–Stokes equations can be derived from the Boltzmann equation at small Knudsen numbers using the Chapman–Enskog expansion. Since LBM analysis can model microscopic motions, the LBM has been regarded as a very promising method to predict the flow and heat transfer in nanofluids. Recently, a number of studies have used the LBM to analyze the flow and heat transfer of nanofluids using a single-component single-phase model [31-39] and a multicomponent single-phase model [40-43]. These studies focused on natural convection within enclosures [31,32,34-38], the lid-driven cavity [39] or forced convection in microchannels [31,33,40-43]. However, no studies have analyzed Rayleigh-Bénard flows nanofluids using LBM.

The main purpose of this work is to examine the coupled and the individual effects of the thermal conductivity and the dynamic viscosity on the evolution of Rayleigh–Bénard flows using LBM. In the experimental works, it is difficult to examine one single parameter without change others. The couple effects of the thermal conductivity and the dynamic viscosity were examined by changing the volume concentrations. The individual effect of dynamic viscosity was examined by changing the nanoparticle diameters, while the individual effect of thermal conductivity was tested by changing the nanoparticle materials.

## Model

###### Thermal LBM.

The Rayleigh–Bénard cells of nanofluids between two parallel plates are modeled using a 3000 × 300 lattice space as shown in Fig. 2. Since the flow is between two parallel plates, there is no free surface and natural convection is the dominant mode driving the Rayleigh–Bénard flow. The driving force for the Rayleigh–Bénard flow is from the temperature difference between the two plates, which leads to liquid expansion in hot regions that generates the buoyancy. The present thermal LBM uses two different distribution functions to describe the flow field and the temperature field in the nanofluids.

The nanofluids are assumed to be single-phase fluids as in the previous LBM models [31-35]. The single-phase assumption was also validated by Cianfrini et al. [30]. The density distribution function is calculated by solving the lattice Boltzmann equation, which is a temporal and spatial discretization of the kinetic Boltzmann equation. The general form of the lattice Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) approximation can be written as [44]Display Formula

(1)$fα(x+eαΔt,t+Δt)-fα(x,t)=-Δtτf[fα(x,t)-fαeq(x,t)]$

where the left side is the streaming part, and the right side is the collision term, fα(x, t) is the density distribution function along the α direction at location x and time t, $fαeq$ is the equilibrium distribution function, τf is the collision–relaxation time for the flow field, and eα is the lattice velocity vector. A D2Q9 model is used here, so eα can be expressed asDisplay Formula

(2)$eα=[(0,0),(1,0),(0,1),(−1,0),(0,−1),(1,1),(−1,1),(−1,−1),(1,−1)]$

The equilibrium distribution function isDisplay Formula

(3)$fαeq(x,t)=wαρ[1+3eα·uc2+9(eα·u)22c4-3(u)22c2]$

where $u$ is the macroscopic velocity used to compute $fαeq$ with its initial value assumed to be 0, ρ is the fluid density, wα are the weights with wα = 4/9 for α = 0, wα = 1/9 for α = 1,2,3,4, and 1/36 for α = 5,6,7,8, and c is the lattice speed.

The distribution function ga(x, t) used to model the temperature field is [25]Display Formula

(4)$gα(x+eαΔt,t+Δt)-gα(x,t)=-ΔtτT[gα(x,t)-gαeq(x,t)]$

where τT is the temperature relaxation time. The equilibrium distribution function for the temperature isDisplay Formula

(5)$gαeq(x,t)=wαT[1+3eα·u+4.5(eα·u)2-1.5(u)2]$

The macroscopic parameters, density, velocity vector, and temperature, can be calculated asDisplay Formula

(6)$ρ=∑α=08fαu=1ρ∑α=08fαeαT=∑α=08gα$

The kinematic viscosity and thermal diffusivity are defined asDisplay Formula

(7)$ν=c23(τf-12)a=c23(τT-12)$

For natural convection, the flow field and the temperature field are coupled; thus, Eqs. (1) and (4) must be solved simultaneously. In the present thermal LBM, the equilibrium distribution function for temperature is calculated using the macroscopic velocity obtained by solving Eqs. (1) and (6). The flow is driven by the buoyancy generated from the temperature gradient. Applying the Boussinesq approximation yieldsDisplay Formula

(8)$F=3ρgβ(T-Tref)$

where F is the buoyancy force, g is the gravitational acceleration, β is the thermal expansion coefficient, and Tref is the reference temperature. The buoyancy term is used to modify the macroscopic velocity in Eqs. (3) and (5) asDisplay Formula

(9)$u'=u+τfFρ$

The Rayleigh number (Ra = βgΔT/) is introduced to characterize the ratio of the buoyancy force to the viscous force in the buoyancy driven flow. At Ra lower than the critical Rac, the heat transfer is due to Fourier heat conduction in the quiescent fluid, while for Ra > Rac, the heat transfer is primarily due to convection.

The boundary conditions are shown in Fig. 2. For the flow field calculation, the upper and lower plates are set to be bounce back conditions, while the left and right sides are set to be periodic. For the temperature field, the top and bottom plates are set to constant temperatures, Tt and Tb, while the left and right sides are periodic. Two heating modes are considered with: (i) top heating for Tt > Tb and (ii) bottom heating for Tt < Tb.

###### Nanofluids Properties.

The nanofluids in the present work were assumed to be Newtonian, incompressible fluids with the flow as laminar. The liquid and solid were assumed to be in thermal equilibrium. The temperature difference between the two plates triggering the natural convection within the nanofluids was always small, so the nanofluids properties were assumed to be independent of temperature.

The effective density of the nanofluid was given by [45]Display Formula

(10)$ρnf=(1-φ)ρf+φρn$

where $φ$ is the volume fraction of the nanoparticles and subscripts f, nf, and n stand for the base fluid, nanofluids, and nanoparticles, respectively.

The effective heat capacity of the nanofluid was given by [46]Display Formula

(11)$ρnfcnf=(1-φ)ρfcf+φρncn$

The effective thermal conductivity of the nanofluids with spherical nanoparticles was introduced by Chon et al. [47] asDisplay Formula

(12)$knf=kf[1+64.7φ0.746(dfdn)0.369(kfkn)0.7476PrT0.9955ReT1.2321]$

where dn and df are the diameters of the nanoparticles and the fluid molecules. knf, kf, and kn are the thermal conductivities of the nanofluids, the base fluids, and the nanoparticles, respectively. PrT and ReT are given asDisplay Formula

(13)$PrT=μfρfaf, ReT=ρfkbT3πμf2lf$

where kb is the Boltzmann constant and lf is the mean path of a fluid molecule, which is assumed to be 17 nm in the present work [48].

The effective dynamic viscosity of the nanofluids was calculated the empirical correlation developed by Corcione [49] on the basis of 14 sets of experimental data reported by 10 independent research groupsDisplay Formula

(14)$μnf=μf/(1−34.87(dndf)−0.3ϕ1.03)$

where dn is the nanoparticle diameter and df is the equivalent diameter of a base fluid molecule.

In the thermal LBM, the nanofluids density is updated using Eq. (10) as initial condition for simulating nanofluids. The thermal diffusivity was calculated as a = λ/ρc from Eqs. (10) and (11). The nanofluid viscosity was calculated from Eq. (14). Hence, the system has two modified relaxation times based on the nanofluids properties to calculate the collection term in the lattice Boltzmann equation from Eq. (7).

The nanoparticles used in present work were Al2O3, Cu, and CuO2, various particle volume fractions and diameters added into pure water to form the nanofluids. The thermophysical properties for the base fluid and nanoparticles are given in Table 1. For the nanofluids used in the present work, Eqs. (10)–(14) have been shown to accurately fit experimental data [45-48].

###### Model Validations.

The thermal LBM was validated for nanofluids by simulating the natural convection of nanofluids in a square cavity, as shown in Fig. 3, with the same geometry and thermal conditions as in Ho et al. [18]. Comparison of the numerical results with the experimental data in Table 2 shows satisfactory agreement, which indicates that the present model can be used to accurately predict the natural convection of nanofluids.

## Results and Discussion

###### Rayleigh–Bénard Flows for Nanofluids With Different Heating Modes.

The flow field was produced for the water-based Al2O3 nanofluids with particle diameters of 11 nm and a volume fraction of 1.5%. The reference temperature Tref was 293.12 K. The excess temperature of the cold plate was 0 K, θc = TcTref = 0 K, while the excess temperature of the hot plate θh = Th − Tref varied from 0.025 K to 10 K with = 1.9047 × 10−6 in the LBM system. Thus, the effect of heating mode on the Rayleigh–Bénard cell flows for nanofluids was analyzed for a large Ra range of 1 × 104−4 × 106.

The evolution of the stream function for different Ra top heating is shown in Fig. 4. For Ra < 3 × 105 (Figs. 4(a) and 4(b)), thermal conduction is the dominant heat transfer mode between the parallel plates, with only one vortex pair flowing opposite directions in the calculational region. The right vortex is rotating clockwise with the minimum stream function located in the vortex center while the left one is rotating counterclockwise with the maximum located in the vortex center. As the Rayleigh number increases, the vortex pair gradually expands outward with increasing vortex radius (Figs. 4(c) and 4(d)). For instance, the vortex radius is about 6 times larger for Ra = 5 × 105 than for Ra = 1 × 105. For Ra = 7 × 105, it is about 9 times larger. The convection is enhanced as Ra increases, which causes the initial vortex pair to become unstable, leading to formation of multiple pairs as shown in Fig. 4(e). Thus, there exists a critical transition Rayleigh number (Rac) above which the stable Rayleigh–Bénard cells develop. For the present case, Rac is about 1 × 106. Increasing Ra further to 3 × 106 causes the flow to transfer to turbulent regime.

The stream function and dimensionless temperature ($θ=(T-Tref/Th-Tref)$) distribution at y/y0 = 0.8 for different Ra are shown in Fig. 5. Note again that for Ra ≤ 1 × 105, the convection is quite weak and thermal conduction is the dominant heat transfer mode within the fluid, so the temperature decreases linearly in the y direction with no fluctuations in the stream function and temperature along the x direction. With increasing Ra, the fluid between the plates becomes more unstable and convection begins. For Ra from 1 × 105 to 7 × 105, a vortex pair develops on each side of the channel, leading to changes in the stream function and temperature near the sides. As the number of vortex pairs increases, the variation of the stream function and temperature increases until the stable cell flow develops, for which the stream function has a sinusoidal form. The average temperature at y/y0 = 0.5 also increases with increasing Ra, due to the enhanced convection.

The Rayleigh–Bénard flows for the two heating modes are compared in Fig. 6. There are two key differences between the two heating modes. First, the vortex pairs for the top heating always appear near the upper plate as shown in Fig. 6(a), while they are uniformly distributed between the plates for the bottom heating as shown in Fig. 6(b). Second, there are more vortex pairs and they are smaller for the top heating than for the bottom heating. A smaller Rac of 5.5 × 105 is then observed for the bottom heating. Thus, at the Rac (5.5 × 105 for bottom heating and 1 × 106 for top heating), the maximum stream function for the bottom heating mode is almost twice that for the top heating, as shown in Fig. 6(e), indicating more intensive convection. The buoyancy due to the liquid expansion is the dominant driving force lifting the liquid. For the bottom heating mode, the heated liquid near the bottom expands and flows upward to the upper plate. There is enough space for recirculation, so vortexes develop. However, for the top heating mode, the heated liquid is confined in the upper region due to the buoyancy, leading to much smaller vortices.

###### Comparison of Rayleigh–Bénard Flows for Water and Nanofluids.

The thermal effects for the water-based Al2O3 nanofluids (particle diameter of 11 nm and volume fraction of 1.5%) and pure water are compared in Fig. 7. Since the critical Rayleigh number, Rac, for nanofluids is much smaller than for pure water, Ra', defined as the ratio of Ra to Rac is used in Fig. 7. For the same Ra', the average Nusselt number for nanofluids is much higher than for pure water, indicating that a better heat transfer in the nanofluids. The evolution of the average Nusselt number for the nanofluid can be divided into four stages: (i) at low Ra', the average Nusselt number is quite small and increases slightly with increasing Ra'; (ii) as Ra' further increases, more and more vortex pairs develop and the average Nusselt number increases almost linearly; (iii) after the stable Rayleigh–Bénard cells develop, the Nusselt number remains almost constant; and (iv) when turbulence occurs, the average Nusselt number increases sharply. However, for pure water, the second stage is very short with a sharply increased Nusselt number at Ra' higher than 1.0, which indicates that vortex pairs suddenly develop for pure water, rather than developing slowly for nanofluids.

###### Coupled Effects of Thermal Conductivity and Viscosity Modifications.

The coupled effects of thermal conductivity and viscosity modifications on the nanofluid Rayleigh–Bénard flows were examined using various nanoparticle volume fractions, as shown in Fig. 8. Both the thermal conductivity and viscosity increase with increasing nanoparticle volume fractions. Rac is not the same for the different volume fractions with Rac equal to 8 × 105 for the 1% volume fraction and decreasing to 2 × 105 for the 4% volume fraction. When stable cells develop (Ra' = 1), the stream function and temperature variations are greater for lower volume fractions due to more, smaller vortices at lower volume fractions. For instance, there are five pairs for the 1% volume fraction in the calculation region, but only four pairs for the 4% volume fraction. Consequently, the average dimensionless temperature at the same y/y0 increases with increasing volume fraction. The average dimensionless temperature difference between the bottom plate and y/y0 = 0.5 for the 1% volume fraction is 0.218, but increases to 0.225 for the 4% volume fraction.

###### Individual Effect of Viscosity.

To examine the individual effects of dynamic viscosity on the R–B flows, the additional simulations were conducted using the Maxwell-Garnett model [11] in the calculation of the effective thermal conductivity, in which the nanoparticle diameters do not affect the nanofluid thermal conductivity. The empirical correlation developed by Corcione [49] was used to calculate the effective dynamic viscosity, in which the nanoparticle diameters affect the nanofluid thermal conductivity. Both the thermal conductivity and viscosity decrease with increasing nanoparticle diameters using the Chon and Corcione's model. The reduction of thermal conductivity degrades the heat transfer rate of nanofluid natural convection, while the decrease of viscosity facilitates the heat transfer rate. However, the average Nusselt number increases with increasing nanoparticle diameters, as shown in Fig. 9, indicating the dominated role of viscosity reduction in the nanofluid Rayleigh–Bénard flows. The results of Maxwell-Garnett and Corcione's models are very close to the Chon's and Corcione's models predict, indicating the thermal conductivity has few effects on the heat transfer rate. The nanoparticles size greatly affects the flow patterns in the Rayleigh–Bénard flows. The stream function and dimensionless temperature distributions at y/y0 = 0.5 are shown in Fig. 10. For larger particles, the stream function is more sinusoidal with larger and more uniform convection cells. The original driving force for the Rayleigh–Bénard flow is the temperature difference between the parallel plates, which leads to a reduced liquid density and the buoyancy. The improved heat transfer and the enhanced chaotic motion with the larger particles reduce the temperature variation which reduces the convection and the development of the Rayleigh–Bénard cells. The average dimensionless temperature of the nanofluids at y/y0 = 0.5 is 0.370 for 30 nm particles, but decreases to 0.347 for 20 nm particles and 0.302 for 10 nm particles.

###### Individual Effect of Thermal Conductivity.

The individual effects of thermal conductivity modifications on the nanofluid Rayleigh–Bénard flows were examined using various types of nanoparticles (Al2O3, Cu and CuO, d = 11 nm, ϕ = 2%), as shown in Fig. 11. Rac is not the same for the three nanoparticles with Rac equal to 2 × 105 for Cu, 5 × 105 for Al2O3, and 3 × 105 for CuO. Figure 11 shows that for the stable Rayleigh–Bénard flows (Ra' = 1), all these types of nanofluids generate the same five vortex pairs in the calculation region. These three nanofluids have different thermal conductivities, but the same viscosity because the viscosity is almost independent of the nanoparticle type when the volume fraction, diameter and particle shape are fixed. Thus, the present results imply that the thermal conductivity of the nanofluids has relatively smaller effect on the flow cell structure than the viscosity, which was also reported in the theoretical prediction of Cianfrini et al. [30].

## Conclusions

The thermal LBM was used to investigate Rayleigh–Bénard flows between parallel plates. The results illustrate the effect of heating mode, nanoparticle types, particle volume fraction, and particle size on the Rayleigh–Bénard cell patterns and heat transfer variations. The main conclusions are:

1. (1)Rayleigh–Bénard flow patterns of nanofluids different significantly from those of pure fluids with fewer vortex pairs in the nanofluids than in the pure liquid. The cell structure in the nanofluids is more uniform with almost sinusoidal stream function variation at the center plane between the plates. The stable convection cells in the nanofluids develop from the expansion of two initial vortices at each side but occur suddenly in pure water when the Rayleigh number reaches a critical value. Thus, the average Nusselt number increases gradually for nanofluids, while sharply for pure liquids.
2. (2)The development of the Rayleigh–Bénard flow is affected by the heating mode. More uniform and fully developed flow cells develop for the bottom heating with fewer vortex pairs and larger vortices, while only very small vortexes develop near the heated plated for top heating.
3. (3)The nanoparticles significantly change the thermophysical properties of the pure liquid. The coupled and individual effects of nanofluid thermal conductivity and viscosity were examined using various nanoparticle volume fractions, particle diameters, and nanoparticle materials. The Rayleigh–Bénard flow patterns and heat transfer rates are more sensitive with the modification of effective viscosity by adding nanoparticles than the effects of thermal conductivity modification.

## Acknowledgements

The authors acknowledge the financial support from the National Natural Science Foundation of China (Nos. 21176133, 51276060, and 51321002).

Nomenclature
• a =

thermal diffusivity

• cs =

sound speed

• cnf =

effective heat capacity of a nanofluid

• cα =

lattice speed along the α direction

• d =

diameter

• eα =

lattice velocity vector

• fα =

density distribution function along the α direction

• feq=

equilibrium density distribution function

• g =

gravitational acceleration

• gα =

temperature distribution function along the α direction

• geq =

equilibrium temperature distribution function

• Gr =

Grashoff number

• k =

thermal conductivity

• kb =

Boltzmann constant

• L =

characteristic length of system

• lf =

mean path of a fluid molecule

• Nu =

Nusselt number

• Pr =

Prandtl number

• Ra =

Rayleigh number

• Ra' =

ratio of Ra/Rac

• T =

temperature

• u =

velocity used to calculate the macroscopic distribution function

• w =

weighting factor

• x, y =

x- and y-coordinates

Greek Symbols
• $β$ =

liquid volume expansion coefficient

• Δ =

difference

• ϕ =

nanoparticle volume fraction

• θ =

dimensionless temperature

• μ =

dynamic viscosity (N·s·m−2)

• ν =

kinetic viscosity (m2·s−1)

• ρ =

density

• τ =

collision-relaxation time

Subscripts
• α =

lattice direction

• c =

critical, cooler

• f =

pure fluid, flow field

• h =

heated

• n =

nanoparticles

• nf =

nanofluids

• ref =

reference

• T =

thermal

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Rashmi, W., Ismail, A. F., and Khalid, M., 2011, “CFD Studies on Natural Convection Heat Transfer of Al2O3–Water Nanofluids,” Heat Mass Transfer, 47, pp. 1301–1310.
Ho, C. J., Chen, M. W., and Li, Z. W., 2008, “Numerical Simulation of Natural Convection of Nanofluid in a Square Enclosure: Effects Due to Uncertainties of Viscosity and Thermal Conductivity,” Int. J. Heat Mass Transfer, 51, pp. 4506–4516.
Cianfrini, M., Corcione, M., and Quintino, A., 2011, “Natural Convection Heat Transfer of Nanofluids in Annular Spaces Between Horizontal Concentric Cylinders,” Appl. Therm. Eng., 31, pp. 4055–4063.
Fattahi, E., Farhadi, M., and Sedighi, K., 2011, “Lattice Boltzmann Simulation of Natural Convection Heat Transfer in Nanofluids,” Int. J. Therm. Sci., 52, pp. 137–144.
Lai, F. H., and Yang, Y. T., 2011, “Lattice Boltzmann Simulation of Natural Convection Heat Transfer of Al2O3/Water Nanofluids in a Square Enclosure,” Int. J. Therm. Sci., 50, pp. 1930–1941.
Yang, Y. T., and Lai, F. H., 2011, “Numerical Study of Flow and Heat Transfer Characteristics of Alumina-Water Nanofluids in a Microchannel Using the Lattice Boltzmann Method,” Int. Commun. Heat Mass Transfer, 38, pp. 607–614.
Nabavitabatabayi, M., Shirani, E., and Rahimian, M. H., 2011, “Investigation of Heat Transfer Enhancement in an Enclosure Filled With Nanofluids Using Multiple Relaxation Time Lattice Boltzmann Modeling,” Int. Commun. Heat Mass Transfer, 38, pp. 128–138.
Bararnia, H., Hooman, K., and Ganji, D. D., 2011, “Natural Convection in a Nanofluids-Filled Portioned Cavity: The Lattice-Boltzmann Method,” Numer. Heat Transfer, Part A, 59, pp. 487–502.
Kefayati, Gh. R., Hosseinizadeh, S. F., and Gorji, M., 2012, “Lattice Boltzmann Simulation of Natural Convection in an Open Enclosure Subjugated to Water/Copper Nanofluid,” Int. J. Therm. Sci., 52, pp. 91–101.
Kefayati, Gh. R., Hosseinizadeh, S. F., and Gorji, M., 2011, “Lattice Boltzmann Simulation of Natural Convection in Tall Enclosures Using Water/SiO2 Nanofluid,” Int. Commun. Heat Mass Transfer, 38, pp. 798–805.
He, Y. R., Qi, C., and Hu, Y. W., 2011, “Lattice Boltzmann Simulation of Alumina-Water Nanofluid in a Square Cavity,” Nanoscale Res. Lett., 6.
Nemati, H., Farhadi, M., and Sedighi, K., 2010, “Lattice Boltzmann Simulation of Nanofluid in Lid-Driven Cavity,” Int. Commun. Heat Mass Transfer, 37, pp. 1528–1534.
Xuan, Y. M., Yu, K., and Li, Q., 2005, “Investigation on Flow and Heat Transfer of Nanofluids by the Thermal Lattice Boltzmann Model,” Prog. Comput. Fluid Dyn., 5, pp. 13–19.
Xuan, Y. M., Li, Q., and Yao, Z. P., 2004, “Application of Lattice Boltzmann Scheme to Nanofluids,” Sci. China, Ser. E, 47, pp. 129–140.
Xuan, Y. M., and Yao, Z. P., 2005, “Lattice Boltzmann Model for Nanofluids,” Heat Mass Transfer, 41(3), pp. 199–205.
Zhou, L. J., Xuan, Y. M., and Li, Q., 2010, “Multiscale Simulation of Flow and Heat Transfer of Nanofluid With Lattice Boltzmann Method,” Int. J. Multiphase Flow, 36, pp. 364–374.
Zou, Q., Hou, S., and Chen, S., 1995, “An Improved Incompressible Lattice Boltzmann Model for Time-Independent Flows,” J. Stat. Phys., 81, pp. 35–48.
Pak, B. C., and Cho, Y., 1998, “Hydrodynamic and Heat Transfer Study of Dispersed Fluids With Submicron Metallic Oxide Particle,” Exp. Heat Transfer, 11, pp. 151–170.
Xuan, Y., and Roetzel, W., 2004, “Conceptions for Heat Transfer Correlation of Nanofluids,” Int. J. Heat Mass Transfer, 43, pp. 3701–3707.
Chon, C. H., Kihm, K. D., Lee, S. P., and Choi, S. U., 2005, “Empirical Correlation Finding the Role of Temperature and Particle Size for Nanofluid (Al2O3) Thermal Conductivity Enhancement,” Appl. Phys. Lett., 87, p. 153107.
Saha, L. K., Hossain, M. A., and Gorla, R. S. R., 2007, “Effect of Hall Current on the MHD Laminar Natural Convection Flow From a Vertical Permeable Flat Plate With Uniform Surface Temperature,” Int. J. Therm. Sci., 46, pp. 790–801.
Corcione, M., 2011, “Empirical Correlating Equations for Predicting the Effective Thermal Conductivity and Dynamic Viscosity of Nanofluids,” Energy Convers. Manage., 52, pp. 789–793.
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Ho, C. J., Liu, W. K., Chang, Y. S., and Lin, C. C., 2010, “Natural Convection Heat Transfer of Alumina–Water Nanofluid in Vertical Square Enclosures: An Experimental Study,” Int. J. Therm. Sci., 49, pp. 1345–1353.
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Santra, A. K., Sen, S., and Chakraborty, N., 2008, “ Study of Heat Transfer Augmentation in a Differentially Heated Square Cavity Using Copper-Water Nanofluid,” Int. J. Therm. Sci., 47, pp. 1113–1122.
Rashmi, W., Ismail, A. F., and Khalid, M., 2011, “CFD Studies on Natural Convection Heat Transfer of Al2O3–Water Nanofluids,” Heat Mass Transfer, 47, pp. 1301–1310.
Ho, C. J., Chen, M. W., and Li, Z. W., 2008, “Numerical Simulation of Natural Convection of Nanofluid in a Square Enclosure: Effects Due to Uncertainties of Viscosity and Thermal Conductivity,” Int. J. Heat Mass Transfer, 51, pp. 4506–4516.
Cianfrini, M., Corcione, M., and Quintino, A., 2011, “Natural Convection Heat Transfer of Nanofluids in Annular Spaces Between Horizontal Concentric Cylinders,” Appl. Therm. Eng., 31, pp. 4055–4063.
Fattahi, E., Farhadi, M., and Sedighi, K., 2011, “Lattice Boltzmann Simulation of Natural Convection Heat Transfer in Nanofluids,” Int. J. Therm. Sci., 52, pp. 137–144.
Lai, F. H., and Yang, Y. T., 2011, “Lattice Boltzmann Simulation of Natural Convection Heat Transfer of Al2O3/Water Nanofluids in a Square Enclosure,” Int. J. Therm. Sci., 50, pp. 1930–1941.
Yang, Y. T., and Lai, F. H., 2011, “Numerical Study of Flow and Heat Transfer Characteristics of Alumina-Water Nanofluids in a Microchannel Using the Lattice Boltzmann Method,” Int. Commun. Heat Mass Transfer, 38, pp. 607–614.
Nabavitabatabayi, M., Shirani, E., and Rahimian, M. H., 2011, “Investigation of Heat Transfer Enhancement in an Enclosure Filled With Nanofluids Using Multiple Relaxation Time Lattice Boltzmann Modeling,” Int. Commun. Heat Mass Transfer, 38, pp. 128–138.
Bararnia, H., Hooman, K., and Ganji, D. D., 2011, “Natural Convection in a Nanofluids-Filled Portioned Cavity: The Lattice-Boltzmann Method,” Numer. Heat Transfer, Part A, 59, pp. 487–502.
Kefayati, Gh. R., Hosseinizadeh, S. F., and Gorji, M., 2012, “Lattice Boltzmann Simulation of Natural Convection in an Open Enclosure Subjugated to Water/Copper Nanofluid,” Int. J. Therm. Sci., 52, pp. 91–101.
Kefayati, Gh. R., Hosseinizadeh, S. F., and Gorji, M., 2011, “Lattice Boltzmann Simulation of Natural Convection in Tall Enclosures Using Water/SiO2 Nanofluid,” Int. Commun. Heat Mass Transfer, 38, pp. 798–805.
He, Y. R., Qi, C., and Hu, Y. W., 2011, “Lattice Boltzmann Simulation of Alumina-Water Nanofluid in a Square Cavity,” Nanoscale Res. Lett., 6.
Nemati, H., Farhadi, M., and Sedighi, K., 2010, “Lattice Boltzmann Simulation of Nanofluid in Lid-Driven Cavity,” Int. Commun. Heat Mass Transfer, 37, pp. 1528–1534.
Xuan, Y. M., Yu, K., and Li, Q., 2005, “Investigation on Flow and Heat Transfer of Nanofluids by the Thermal Lattice Boltzmann Model,” Prog. Comput. Fluid Dyn., 5, pp. 13–19.
Xuan, Y. M., Li, Q., and Yao, Z. P., 2004, “Application of Lattice Boltzmann Scheme to Nanofluids,” Sci. China, Ser. E, 47, pp. 129–140.
Xuan, Y. M., and Yao, Z. P., 2005, “Lattice Boltzmann Model for Nanofluids,” Heat Mass Transfer, 41(3), pp. 199–205.
Zhou, L. J., Xuan, Y. M., and Li, Q., 2010, “Multiscale Simulation of Flow and Heat Transfer of Nanofluid With Lattice Boltzmann Method,” Int. J. Multiphase Flow, 36, pp. 364–374.
Zou, Q., Hou, S., and Chen, S., 1995, “An Improved Incompressible Lattice Boltzmann Model for Time-Independent Flows,” J. Stat. Phys., 81, pp. 35–48.
Pak, B. C., and Cho, Y., 1998, “Hydrodynamic and Heat Transfer Study of Dispersed Fluids With Submicron Metallic Oxide Particle,” Exp. Heat Transfer, 11, pp. 151–170.
Xuan, Y., and Roetzel, W., 2004, “Conceptions for Heat Transfer Correlation of Nanofluids,” Int. J. Heat Mass Transfer, 43, pp. 3701–3707.
Chon, C. H., Kihm, K. D., Lee, S. P., and Choi, S. U., 2005, “Empirical Correlation Finding the Role of Temperature and Particle Size for Nanofluid (Al2O3) Thermal Conductivity Enhancement,” Appl. Phys. Lett., 87, p. 153107.
Saha, L. K., Hossain, M. A., and Gorla, R. S. R., 2007, “Effect of Hall Current on the MHD Laminar Natural Convection Flow From a Vertical Permeable Flat Plate With Uniform Surface Temperature,” Int. J. Therm. Sci., 46, pp. 790–801.
Corcione, M., 2011, “Empirical Correlating Equations for Predicting the Effective Thermal Conductivity and Dynamic Viscosity of Nanofluids,” Energy Convers. Manage., 52, pp. 789–793.

## Figures

Fig. 1

Schematic of a Rayleigh–Bénard cell

Fig. 2

Computational grid and boundary conditions

Fig. 3

Natural convection of water-based Al2O3 nanofluids with ϕ = 0.3% in a square cavity: (a) Stream function and (b) temperature

Fig. 4

Evolution of Rayleigh–Bénard cells in a water-based Al2O3 nanofluid (ϕ = 1.5%, 10 nm, top heating)

Fig. 5

Stream function and temperature fields at y/y0 = 0.5 for Al2O3 nanofluid (ϕ = 1.5%, 10 nm, top heating): (a) Stream function and (b) temperature difference

Fig. 6

Effects of heating mode on the Rayleigh–Bénard flows for the water-based Al2O3 nanofluid (ϕ = 1.5%, 10 nm, bottom heating): (a) Stream function for top heating at Rac = 1 × 106; (b) stream function for bottom heating at Rac = 5.5 × 105; (c) temperature for top heating at Rac = 1 × 106; (d) temperature for bottom heating at Rac = 5.5 × 105; and (e) stream function through the cell centers at Rac.

Fig. 7

Average Nusselt number for pure water and the water-based Al2O3 nanofluid (ϕ = 1.5%, 10 nm, bottom heating)

Fig. 8

Effects of nanoparticle volume fraction (10 nm Al2O3, bottom heating) on the Rayleigh–Bénard flows at y/y0 = 0.5: (a) Stream function and (b) temperature difference

Fig. 9

Effects of particle size on the average Nusselt number of nanofluid Rayleigh–Bénard flows (Al2O3, ϕ = 2%, bottom heating)

Fig. 10

Effect of nanoparticle size (Al2O3, ϕ = 2%, bottom heating): (a) Stream function at y/y0 = 0.5 and (b) temperature difference at y/y0 = 0.5

Fig. 11

Effect of nanoparticle material (ϕ = 2%, 10 nm, bottom heating): (a) Stream function at y/y0 = 0.5 and (b) temperature difference at y/y0 = 0.5

## Tables

Table 1 Thermophysical properties of water and nanoparticles
Table 2 Comparisons of the predictions of the present model with experimental data

## Discussions

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