NANO EXPRESS Open Access

Lattice Boltzmann simulation of alumina-waternanofluid in a square cavityYurong He1*, Cong Qi1*, Yanwei Hu1, Bin Qin1, Fengchen Li1, Yulong Ding2

Abstract

A lattice Boltzmann model is developed by coupling the density (D2Q9) and the temperature distribution functionswith 9-speed to simulate the convection heat transfer utilizing Al2O3-water nanofluids in a square cavity. Thismodel is validated by comparing numerical simulation and experimental results over a wide range of Rayleighnumbers. Numerical results show a satisfactory agreement between them. The effects of Rayleigh number andnanoparticle volume fraction on natural convection heat transfer of nanofluid are investigated in this study.Numerical results indicate that the flow and heat transfer characteristics of Al2O3-water nanofluid in the squarecavity are more sensitive to viscosity than to thermal conductivity.

List of symbolsc Reference lattice velocitycs Lattice sound velocitycp Specific heat capacity (J/kg K)ea Lattice velocity vectorfa Density distribution function

feq Local equilibrium density distribution function

Fa External force in direction of lattice velocityg Gravitational acceleration (m/s2)G Effective external forcek Thermal conductivity coefficient (Wm/K)L Dimensionless characteristic length of the square

cavityMa Mach numberPr Prandtl numberr Position vectorRa Rayleigh numbert Time (s)Ta Temperature distribution function

Taeq Local equilibrium temperature distribution

functionT Dimensionless temperatureT0 Dimensionless average temperature (T0 = (TH + TC)/2)TH Dimensionless hot temperatureTC Dimensionless cold temperature

u Dimensionless macrovelocityuc Dimensionless characteristic velocity of natural

convectionwa Weight coefficientx, y Dimensionless coordinates

Greek symbolsb Thermal expansion coefficient (K-1)r Density (kg/m3)ν Kinematic viscosity coefficient (m2/s)c Thermal diffusion coefficient (m2/s)μ Kinematic viscosity (Ns/m2)� Nanoparticle volume fractionδx Lattice stepδt Time step tτf Dimensionless collision-relaxation time for the flow

fieldτT Dimensionless collision-relaxation time for the tem-

perature fieldΔT Dimensionless temperature difference (ΔT = TH -

TC)Error1 Maximal relative error of velocities between

two adjacent time layersError2 Maximal relative error of temperatures between

two adjacent time layers

Subscriptsa Lattice velocity directionavg AverageC Cold

* Correspondence: rong@hit.edu.cn; qicongkevin@163.com1School of Energy Science & Engineering, Harbin Institute of Technology,Harbin 150001, ChinaFull list of author information is available at the end of the article

He et al. Nanoscale Research Letters 2011, 6:184http://www.nanoscalereslett.com/content/6/1/184

© 2011 He et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons AttributionLicense (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium,provided the original work is properly cited.

f FluidH Hotnf Nanofluidp Particle

IntroductionThe most common fluids such as water, oil, and ethylene-glycol mixture have a primary limitation in enhancing theperformance of conventional heat transfer due to low ther-mal conductivities. Nanofluids, using nanoscale particlesdispersed in a base fluid, are proposed to overcome thisdrawback. Nanotechnology has been widely studied inrecent years. Wang and Fan [1] reviewed the nanofluidresearch in the last 10 years. Choi and Eastman [2] are thefirst author to have proposed the term nanofluids to referto the fluids with suspended nanoparticles. Yang and Liu[3] prepared a kind of functionalized nanofluid with amethod of surface functionalization of silica nanoparticles,and this nanofluid with functionalized nanoparticles havemerits including long-term stability and good dispersing.Pinilla et al. [4] used a plasma-gas-condensation-type clus-ter deposition apparatus to produce nanometer size-selected Cu clusters in a size range of 1-5 nm. With thismethod, it is possible to produce nanoparticles with astrict control on size by controlling the experimental con-ditions. Using the covalent interaction between the fattyacid-binding domains of BSA molecule with stearic acid-capped nanoparticles, Bora and Deb [5] proposed a novelbioconjugate of stearic acid-capped maghemite nanoparti-cle with BSA molecule, which will give a huge boost to thedevelopment of non-toxic iron oxide nanoparticles usingBSA as a biocompatible passivating agent. Wang et al. [6]showed the method of synthesizing stimuli-responsivemagnetic nanoparticles and analyzed the influence of glu-tathione concentration on its cleavage efficiency. Huangand Wang [7] produced ε-Fe3N-magnetic fluid by chemi-cal reaction of iron carbonyl and ammonia gas. Guo et al.[8] investigated the thermal transport properties of thehomogeneous and stable magnetic nanofluids containingg-Fe2O3 nanoparticles.Many experiments and common numerical simulation

methods have been carried out to investigate the nano-fluids. Teng et al. [9] examined the influence of weightfraction, temperature, and particle size on the thermalconductivity ratio of alumina-water nanofluids. Nada et al.[10] investigated the heat transfer enhancement in a hori-zontal annuli of nanofluid containing various volume frac-tions of Cu, Ag, Al2O3, and TiO2 nanoparticles. Jou andTzeng [11] studied the natural convection heat transferenhancements of nanofluid containing various volumefractions, Grashof numbers, and aspect ratios in a two-dimensional enclosure. Heris et al. [12] investigatedexperimentally the laminar flow-forced convection heattransfer of Al2O3-water nanofluid inside a circular tube

with a constant wall temperature. Ghasemi and Aminossa-dati [13] showed the numerical study on natural convec-tion heat transfer of CuO-water nanofluid in an inclinedenclosure. Hwang et al. [14] theoretically investigated thenatural convection thermal characteristics of Al2O3-waternanofluid in a rectangular cavity heated from below.Tiwari and Das [15] numerically investigated the behaviorof Cu-water nanofluids inside a two-sided lid-driven differ-entially heated square cavity and analyzed the convectiverecirculation and flow processes induced by the nanofluid.Putra et al. [16] investigated the natural convection heattransfer characteristics of CuO-water nanofluids inside ahorizontal cylinder heated and cooled from both of ends,respectively. Bianco et al. [17] showed the developing lami-nar forced convection flow of a water-Al2O3 nanofluid in acircular tube with a constant and uniform heat flux at thewall. Polidori et al. [18] investigated the flow and heattransfer of Al2O3-water nanofluids under a laminar-freeconvection condition. It has been found that two factors,thermal conductivity and viscosity, play a key role on theheat transfer behavior. Oztop and Nada [19] investigatedthe heat transfer and fluid flow characteristic of differenttypes of nanoparticles in a partially heated enclosure. Hoet al. [20] carried out an experimental study to show thenatural convection heat transfer of Al2O3-water nanofluidsin square enclosures of different sizes.The lattice Boltzmann method applied to investigate the

nanofluid flow and heat transfer characteristic has beenstudied in recent years. Hao and Cheng [21] simulatedwater invasion in an initially gas-filled gas diffusion layerusing lattice Boltzmann method to investigate the effect ofwettability on water transport dynamics in gas diffusionlayer. Xuan and Yao [22] developed a lattice Boltzmannmodel to simulate flow and energy transport processesinside the nanofluids. Xuan et al. [23] also proposedanother lattice Boltzmann model by considering the exter-nal and internal forces acting on the suspended nanoparti-cles as well as mechanical and thermal interactions amongthe nanoparticles and fluid particles. Arcidiacono andMantzaras [24] developed a lattice Boltzmann model forsimulating finite-rate catalytic surface chemistry. Barrioset al. [25] analyzed natural convective flows in two dimen-sions using the lattice Boltzmann equation method. Penget al. [26] proposed a simplified thermal energy distribu-tion model whose numerical results have a good agree-ment with the original thermal energy distribution model.He et al. [27] proposed a novel lattice Boltzmann thermalmodel to study thermo-hydrodynamics in incompressiblelimit by introducing an internal energy density distributionfunction to simulate the temperature field.In this study, a lattice Boltzmann model is developed by

coupling the density (D2Q9) and the temperature distribu-tion functions with 9-speed to simulate the convectionheat transfer utilizing nanofluids in a square cavity.

He et al. Nanoscale Research Letters 2011, 6:184http://www.nanoscalereslett.com/content/6/1/184

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Lattice Boltzmann methodIn this study, the Al2O3-water nanofluid of single phaseis considered. The macroscopic density and velocityfields are still simulated using the density distributionfunction.

f t f t f t f t Ft t t

r e r r r+ +( ) − ( ) = − ( ) − ( )⎡⎣

⎤⎦ +, , , ,

1

f

eq (1)

Fp

faa

a= ⋅−( )

Ge u eq (2)

where τf is the dimensionless collision-relaxation timefor the flow field; ea is the lattice velocity vector; thesubscript a represents the lattice velocity direction; fa(r,t) is the population of the nanofluid with velocity ea

(along the direction a) at lattice r and time t; f teq r ,( )

is the local equilibrium distribution function; δt is thetime step t; Fa is the external force term in the directionof lattice velocity; G = -b(Tnf-T0)g is the effective exter-nal force, where g is the gravity acceleration; b is thethermal expansion coefficient; T is the temperature ofnanofluid; and T0 is the mean value of the high and lowtemperatures of the walls.For the two-dimensional 9-velocity LB model (D2Q9)

considered herein, the discrete velocity set for eachcomponent a is

e

=

( ) =

−( )⎡⎣⎢

⎤⎦⎥

−( )⎡⎣⎢

⎤⎦⎥

⎛

⎝⎜

⎞

⎠⎟ =

0 0 0

12

12

1 2 3

,

cos , sin , ,c ,,

cos , sin , , ,

4

2 2 14

2 14

5 6 7 8c

−( )⎡⎣⎢

⎤⎦⎥

−( )⎡⎣⎢

⎤⎦⎥

⎛

⎝⎜

⎞

⎠⎟ =

⎧

⎨

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

(3)

where c = δx / δt is the reference lattice velocity, δx isthe lattice step, and the order numbers a = 1, ..., 4 anda = 5, ..., 8, respectively, represent the rectangular direc-tions and the diagonal directions of a lattice.The density equilibrium distribution function is cho-

sen as follows:

f wc c

u

c

eq

s s s

= + ⋅ +⋅( ) −

⎡

⎣⎢⎢

⎤

⎦⎥⎥

12 22

2

4

2

2

e u e u(4)

wa

a

a

a

=

=

=

=

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

49

0

19

1 4

136

5 8

, ,

, ,

(5)

where c cs2 2

3= is the lattice sound velocity, and wal-

pha is the weight coefficient.The macroscopic temperature field is simulated using

the temperature distribution function:

T t T t T t T tt t

r e r r r+ +( ) − ( ) = − ( ) − ( )⎡⎣

⎤⎦, , , ,

1

T

eq (6)

where τT is the dimensionless collision-relaxation timefor the temperature field.The temperature equilibrium distribution function is

chosen as follows:

T w Tc c c

a aa a

2

2. 1.52

eq = + × +×( ) −

⎡

⎣⎢⎢

⎤

⎦⎥⎥

1 3 4 522

2

4

e u e u u(7)

The macroscopic temperature, density, and velocityare, respectively, calculated as follows:

T T==

∑ 0

8

(8)

==

∑ f0

8

(9)

u e==

∑1

0

8

f (10)

The corresponding kinematic viscosity and thermaldiffusion coefficients are, respectively, defined as follows:

= −⎛⎝⎜

⎞⎠⎟

13

12

2c tf (11)

= −⎛⎝⎜

⎞⎠⎟

13

12

2c tT (12)

For natural convection, the important dimensionlessparameters are Prandtl number Pr and Rayleigh numberRa defined by

Pr =

(13)

Rag TL Pr=

Δ 3

2(14)

where ΔT is the temperature difference between thehigh temperature wall and the low temperature wall,and L is the characteristic length of the square cavity.

He et al. Nanoscale Research Letters 2011, 6:184http://www.nanoscalereslett.com/content/6/1/184

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Another dimensionless parameter Mach number Mais defined by

Mau

c= c

s(15)

where u g TLc = Δ is the characteristic velocity of

natural convection. For natural convection, the Boussi-nesq approximation is applied; to ensure that the codeworks in near incompressible regime, the characteristicvelocity must be small compared with the fluid speed ofsound. In this study, the characteristic velocity isselected as 0.1 times of speed of the sound.The dimensionless collision-relaxation times τf and τT

are, respectively, given as follows:

f = +0 5

32.

MaL Pr

c t Ra(16)

T = +0 5

32.

Prc t(17)

Lattice Boltzmann model for nanofluidThe fluid in the enclosure is Al2O3-water nanofluid.Thermo-physical properties of water and Al2O3 are given inTable 1. The nanofluid is assumed incompressible and noslip occurs between the two media, and it is idealized thatthe Al2O3-water nanofluid is a single phase fluid. Hence, theequations of physical parameters of nanofluid are as follows:Density equation:

nf f p= − +( )1 (18)

where rnf is the density of nanofluid, � is the volumefraction of Al2O3 nanoparticles, rbf is the density ofwater, and rp is the density of Al2O3 nanoparticles.Heat capacity equation:

c c cpnf pf pp= − +( )1 (19)

where Cpnf is the heat capacity of nanofluid, Cpf is theheat capacity of water, and Cpp is the heat capacity ofAl2O3 nanoparticles.Dynamic viscosity equation [28]:

nff=

−( ) .1 2 5 (20)

where μnf is the viscosity of nanofluid, and μf is theviscosity of water.Thermal conductivity equation [28]:

k kk k k k

k k k knf fp f f p

p f f p

=+ − −+ + −

⎡

⎣⎢⎢

⎤

⎦⎥⎥

( ) ( )

( ) ( )

2 2

2

(21)

where knf is the thermal conductivity of nanofluid, andkf is the thermal conductivity of water.The Nusselt number can be expressed as

Nunf

= hH

k(22)

The heat transfer coefficient is computed from

hq

T Tw=−H L

(23)

The thermal conductivity of the nanofluid is definedby

kq

T xw

nf = −∂ ∂/

(24)

Substituting Equations (23) and (24) into Equation(22), the local Nusselt number along the left wall can bewritten as

NuT

x

H

T T= − ∂

∂⎛⎝⎜

⎞⎠⎟

⋅−H L

(25)

The average Nusselt number is determined from

Nu Nu y dyavg = ∫ ( )

0

1

(26)

Results and discussionThe square cavity used in the simulation is shown inFigure 1. In the simulation, all the units are all latticeunits. The height and the width of the enclosure areall given by L. The left wall is heated and maintainedat a constant temperature (TH) higher than the tem-perature (TC) of the right cold wall. The boundaryconditions of the top and bottom walls are all adia-batic. The initialization conditions of the four walls aregiven as follows:

x T x T

y T y y T y

= = = = = == = ∂ ∂ = = = ∂ ∂ =

⎧⎨⎩

0 0 1 1 0 0

0 0 0 1 0 0

u u

u u

, ; ,

, / ; , /(27)

In the simulation, a non-equilibrium extrapolationscheme is adopted to deal with the boundary, and the

Table 1 Thermo-physical properties of water and Al2O3 [29]

Physical properties Fluid phase (water) Nanoparticles (Al2O3)

r (kg/m3) 997.1 3970

cp (J/kg K) 4179 765

μ (m2/s) 0.001004 /

k (Wm/K) 0.613 25

He et al. Nanoscale Research Letters 2011, 6:184http://www.nanoscalereslett.com/content/6/1/184

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standards of the program convergence for flow field andtemperature field are respectively given as follows:

Error1

2

=+( ) − ( )⎡⎣ ⎤⎦ + +( ) − ( )⎡u i j t u i j t u i j t u i j tx t x y t y, , , , , , , , ⎣⎣ ⎤⎦{ }

+( ) + +( )⎡⎣⎢

⎤⎦⎥

<∑

∑

2

2 2 1i j

x t y ti j

u i j t u i j t

,

,, , , ,

(28)

Error2

2

2 2=+( ) − ( )⎡⎣ ⎤⎦

+( )<

∑∑T i j t T i j t

T i j t

x t xi j

x ti j

, , , ,

, ,

,

,

(29)

where ε is a small number, for example, for Ra = 8 ×104, ε1 = 10-7, and ε2 = 10-7; for Ra = 8 × 105, ε1 = 10-8,and ε2 = 10-8.In the lattice Boltzmann method, the time step t = 1.0,

the lattice step δ = 1.0, the total computational time ofthe numerical simulation is 100 s, and the data of equili-brium state is chosen in the simulation.As shown in Table 2, the grid independence test is

performed using successively sized grids, 192 × 192, 256× 256, and 300 × 300 at Ra = 8 × 105, j = 0.00 (water).From Table 2, it can be seen that the numerical resultswith grids 256 × 256 and 300 × 300 are more close tothose in the literature [20] than with grid 192 × 192,and there is little change in the result as the gridchanges from 256 × 256 to 300 × 300. In order to

accelerate the numerical simulation, a grid size of 256 ×256 is chosen as the suitable one which can guarantee agrid-independent solution.To estimate the validity of above proposed lattice

Boltzmann model for incompressible fluid, the modelis also applied to a nanofluid with nanoparticle volumefraction j = 0.00 in a square cavity, and the researchobject and conditions of numerical simulation are setthe same as those proposed in the literature [20]. Fig-ure 2 compares the numerical results with the experi-mental ones, and a satisfactory agreement is obtained,which indicates that it is feasible to apply the model toincompressible liquids with good accuracy. In Figure 2,there are a few differences because the nanofluid inthe simulation is supposed as a single phase, while thereal nanofluid is a two-phase fluid. Therefore, thesmall differences are accepted in the simulation, andthe model is appropriate for the simulation ofnanofluid.Figure 3 illustrates the velocity vectors and isotherms

of the Al2O3-water nanofluid at different Rayleigh num-bers with a certain volume fraction of Al2O3 nanoparti-cles (j = 0.00). It is observed that there are two bigvortices in the square cavity at Ra = 8 × 105; as the Ray-leigh number increases, they are less likely to beobserved compared with the condition at smaller Ray-leigh numbers. This may be because of the graduallyincreasing Rayleigh number (corresponding to theincrease of the velocity), which causes the nanofluid torotate mainly around the inside wall of the square cav-ity. In addition, it can be seen that the temperature iso-therms become more and more crooked as Ra increases,which illustrates that the heat transfer characteristicstransform from conduction to convection.Figures 4 and 5 present the velocity vectors and iso-

therms at Ra = 8 × 104 and Ra = 8 × 105 for variousvolume fractions of Al2O3 nanoparticles, respectively.

Figure 1 Schematic of the square cavity.

Table 2 Comparison of the mean Nusselt number withdifferent grids

Physicalproperties

192 × 192 256 × 256 300 × 300 Literature[20]

Nuavg 8.367 8.048 7.915 7.704Figure 2 Comparison of the mean Nusselt number at differentRayleigh numbers.

He et al. Nanoscale Research Letters 2011, 6:184http://www.nanoscalereslett.com/content/6/1/184

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Figure 3 Velocity vectors (on the left, ®0.002) and isotherms (on the right) for Al2O3-water nanofluid at different Rayleigh numbers.� = 0.01 (a) Ra = 8 × 105, (b) Ra = 1.4 × 106, (c) Ra = 1.9 × 106, (d) Ra = 2.6 × 106, (e) Ra = 3.3 × 106.

Figure 4 Velocity vectors (on the left, ®0.002) and isotherms (on the right) for Al2O3-water nanofluid at Ra = 8 × 104 with differentvolume fractions. (a) � = 0.00, (b) � = 0.01, (c) � = 0.03, (d) � = 0.05.

He et al. Nanoscale Research Letters 2011, 6:184http://www.nanoscalereslett.com/content/6/1/184

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There are no obvious differences for velocity vectors andisotherms with different volume fractions of nanoparti-cles, which is because the volume fractions are so small,it is not significant in this case on comparing with Ray-leigh number, and the effect of those volume fractions isnegligible. However, it can be seen that there is a littledifference on local part of the isotherms, for example, asthe volume fraction of Al2O3 nanoparticles increases,the lowest isotherm in Figure 4 and the second lowestisotherm in Figure 5 become less and less crooked,which indicates that high values of � cause the fluid tobecome more viscous which causes the velocity todecrease accordingly resulting in a reduced convection.It is more sensitive to the viscosity than to the thermalconductivity for nanofluids heat transfer in a square cav-ity. This phenomenon can also be observed in Figure 6.Figure 6 illustrates the relation between the average

Nusselt number and the volume fraction of nanoparticlesat two different Rayleigh numbers. It is observed that theaverage Nusselt number decreases with the increase ofthe volume fraction of nanoparticles for Ra = 8 × 104 andRa = 8 × 105. In addition, it can be seen that the averageNusselt number decreases less at a low Rayleigh number.For the case of Ra = 8 × 104 and Ra = 8 × 105, it is indi-cated that the high values of � cause the fluid to becomemore viscous which causes reduced convection effectaccordingly resulting in a decreasing average Nusseltnumber, and the flow and heat transfer characteristics ofnanofluids are more sensitive to the viscosity than to thethermal conductivity at a high Ra.

ConclusionA lattice Boltzmann model for single phase fluids isdeveloped by coupling the density and temperature dis-tribution functions. A satisfactory agreement betweenthe numerical results and experimental results isobserved.In addition, the heat transfer and flow characteristics

of Al2O3-water nanofluid in a square cavity are investi-gated using the lattice Boltzmann model. It is foundthat the heat transfer characteristics transform fromconduction to convection as the Rayleigh numberincreases, the average Nusselt number is reduced withincreasing volume fraction of nanoparticles, especially at

Figure 5 Velocity vectors (on the left, ®0.002) and isotherms (on the right) for Al2O3-water nanofluid at Ra = 8 × 105 with differentvolume fractions. (a) � = 0.00, (b) � = 0.01, (c) � = 0.03, (d) � = 0.05.

Figure 6 Average Nusselt numbers at different Rayleighnumbers.

He et al. Nanoscale Research Letters 2011, 6:184http://www.nanoscalereslett.com/content/6/1/184

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a high Rayleigh number. The flow and heat transfercharacteristics of Al2O3-water nanofluid in a square cav-ity are demonstrated to be more sensitive to viscositythan to thermal conductivity.

AcknowledgementsThis study is financially supported by Natural Science Foundation of Chinathrough Grant No. 51076036, the Program for New Century Excellent Talentsin University NCET-08-0159, the Scientific and Technological foundation fordistinguished returned overseas Chinese scholars, and the Key LaboratoryOpening Funding (HIT.KLOF.2009039).

Author details1School of Energy Science & Engineering, Harbin Institute of Technology,Harbin 150001, China 2Institute of Particle Science and Engineering,University of Leeds, Leeds LS2 9JT, UK

Authors’ contributionsYRH conceived of the study, participated in the design of the programdesign, checked the grammar of the manuscript and revised it. CQparticipated in the design of the program, carried out the numericalsimulation of nanofluid, and drafted the manuscript. YWH participated in thedesign of the program and dealed with the figures. BQ participated in thedesign of the program. FCL and YLD guided the program design. Allauthors read and approved the final manuscript.

Competing interestsThe authors declare that they have no competing interests.

Received: 30 October 2010 Accepted: 28 February 2011Published: 28 February 2011

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doi:10.1186/1556-276X-6-184Cite this article as: He et al.: Lattice Boltzmann simulation of alumina-water nanofluid in a square cavity. Nanoscale Research Letters 2011 6:184.

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