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Application of LBM in simulation of natural convection in a nanofluid filledsquare cavity with curve boundaries

M. Sheikholeslami, M. Gorji-Bandpy, S.M. Seyyedi, D.D. Ganji, Houman B.Rokni, Soheil Soleimani

PII: S0032-5910(13)00414-2DOI: doi: 10.1016/j.powtec.2013.06.008Reference: PTEC 9627

To appear in: Powder Technology

Received date: 31 January 2013Revised date: 27 April 2013Accepted date: 5 June 2013

Please cite this article as: M. Sheikholeslami, M. Gorji-Bandpy, S.M. Seyyedi, D.D.Ganji, Houman B. Rokni, Soheil Soleimani, Application of LBM in simulation of naturalconvection in a nanofluid filled square cavity with curve boundaries, Powder Technology(2013), doi: 10.1016/j.powtec.2013.06.008

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Application of LBM in simulation of natural convection in a nanofluid filled

square cavity with curve boundaries

M. Sheikholeslami a, M. Gorji-Bandpy

a, S.M. Seyyedi

a, D.D. Ganji

a, Houman B. Rokni

b,a 1, Soheil

Soleimanic

aDepartment of Mechanical Engineering, Babol University of Technology, Babol, IRAN

bUniversity of Denver, Mechanical and Materials Engineering, 2390 S York St., Denver, CO 80208, USA.

cDepartment of Mechanical and Materials Engineering, Florida International University, Miami, FL 33199

Abstract

A two-dimensional numerical study has been performed to investigate natural convection in a

square cavity with curve boundaries filled with Cu-water nanofluid. Lattice Boltzmann Method (LBM) is

used to simulate this problem. The effective thermal conductivity and viscosity of nanofluid are

calculated by the Maxwell–Garnetts (MG) and Brinkman models, respectively. This investigation

compared with other numerical methods and found to be in excellent agreement. Effects of nanoparticle

volume fraction, Rayleigh numbers and inclination angle on flow and heat transfer are considered. The

results proved that the change of inclination angle has a significant impact on the thermal and

hydrodynamic flow field. Also it can be found that maximum values of enhancement are obtained

at3Ra 10 and

5Ra 10 for 0 and 0 , respectively.

Keywords: Lattice Boltzmann Method; Nanofluid; Curved boundary; Natural convection.

Nomenclature

c Lattice speed Greek symbols

ic Discrete particle speeds Thermal diffusivity[m

2s−1

]

1 Corresponding Author:

Email: Houman.Babazadehrokni@du.edu (Houman B. Rokni) , mohsen.sheikholeslami@yahoo.com (M.

Sheikholeslami)

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pC specific heat at constant pressure Dimensionless temperature

F External forces Dynamic viscosity[Pa s

−1]

f

Density distribution functions Kinematic viscosity [m2 s]

eqf

Equilibrium density distribution functions Angle measured from the up

point of curve boundary

g Internal energy distribution functions

Inclination angle

eqg

Equilibrium internal energy distribution

functions

Fluid density[kgm

−3]

yg

Gravitational acceleration [m s−2

] c

Relaxation time for temperature

k Thermal conductivity v

Relaxation time for flow

L Height or width of the adiabatic wall

Thermal expansion coefficient

[K-1

]

Nu Local Nusselt number stream function

p Pressure[Pa] Subscripts

Pr Prandtl number ( / ) c Cold

Ra Rayleigh number 3( / )g TL h Hot

R

Radius of curve boundary nf Nanofluid

T Fluid temperature f Base fluid

,u v Velocity components in (x,y) directions,

respectively

s Solid particles

,x y Cartesian coordinates ave Average

,X Y

Dimensionless coordinates loc Local

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1. Introduction

Natural convection heat transfer is an important phenomenon in engineering and industry with

widespread applications in diverse fields, such as, geophysics, solar energy, electronic cooling, and

nuclear energy. Davis [1] investigated natural convection of air in a square cavity using numerical

methods. His work became the benchmark solution referred by other researchers. Inaba and Fukuda [2]

studied the problem of steady laminar natural convection of water in an inclined square cavity

experimentally. Their investigations revealed the flow patterns depend strongly on the inclination angle.

Enhancement of heat transfer performance is an essential topic from an energy saving

perspective. The low thermal conductivity of conventional heat transfer fluids such as water and oils is a

primary limitation in enhancing the performance and the compactness of such systems. Solid typically has

a higher thermal conductivity than liquids. For example, copper (Cu) has a thermal conductivity 700 time

greater than water and 3000 greater than engine oil. An innovative and new technique to enhance heat

transfer is by using solid particles in the base fluid (i.e. nanofluids) in the range of sizes 10–50 nm [3].

Khanafer et al. [4] firstly conducted a numerical investigation on the heat transfer enhancement due to

adding nano-particles in a differentially heated enclosure. They found that the suspended nanoparticles

substantially increase the heat transfer rate at any given Grashof number. Sheikholeslami et al. [5] used

CVFEM to simulate the effect of a magnetic field on natural convection in an inclined half-annulus

enclosure filled with Cu–water nanofluid. Their results indicated that Hartmann number and the

inclination angle of the enclosure can be considered as control parameters at different Rayleigh number.

Ghasemi et al. [6] presented the results of a numerical study on natural convection heat transfer in an

inclined enclosure filled with nanofluid. They reported that the heat transfer rate is maximized at a

specific inclination angle depending on Rayleigh number and solid volume fraction. Squeezing unsteady

nanofluid flow and heat transfer has been studied by Sheikholeslami et al. [7].They showed that for the

case in which two plates are moving together, the Nusselt number increases with increase of nanoparticle

volume fraction and Eckert number while it decreases with growth of the squeeze number. Abu-Nada et

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al. [8] investigated natural convection heat transfer enhancement in horizontal concentric annuli field by

nanofluid. They concluded that for low Rayleigh numbers and nanoparticles with higher thermal

conductivity cause more enhancement in heat transfer. A theoretical study on a heated cavity was reported

by Hwang et al. [9]. They observed that the heat transfer coefficient of Al2O3/water nanofluids is reduced

when there is an increase in the size of nanoparticles. Recently, several researchers used different

methods in order to simulate the effect of adding nanoparticle on flow and heat transfer [10-15].

The lattice Boltzmann method (LBM) is a powerful numerical technique based on kinetic theory

for simulating fluid flows and modeling the physics in fluids. Mohamad and Kuzmin [16] used Lattice

Boltzmann Method (LBM) to present a detailed analysis of natural convection problem. They showed the

high efficiency of the LBM in simulating Natural convection. Mohammad et al. [17] studied the natural

convection in an open ended cavity using Lattice Boltzmann Method (LBM). Their work was intended to

address the physics of flow and heat transfer in open end cavities and close end slots. Sheikholeslami et

al. [18] investigated the natural convection in a concentric annulus between a cold outer square and

heated inner circular cylinders in presence of static radial magnetic field using Lattice Boltzmann method.

Their results revealed that average Nusselt number is an increasing function of nanoparticle volume

fraction as well as Rayleigh number, while it is a decreasing function of Hartmann number. Bararnia et al.

[19] considered the natural convection in a nanofluid filled portion cavity with a heated built in plate by

LBM. Their results have been obtained for different inclination angles and lengths of the inner plate.

Effect of static radial magnetic field on natural convection heat transfer in a horizontal cylindrical annulus

enclosure filled with nanofluid was investigated numerically using the Lattice Boltzmann method by

Ashorynejad et al. [20].They found that the average Nusselt number increases as nanoparticle volume

fraction and Rayleigh number increase, while it decreases as Hartmann number increases. Due to

significant multi-scale heterogeneity, understanding sub-grid structures is critical to effective continuum-

based description of gas–solid flow. There were several papers related to this topic [21-25]. Recently

smoothed particle hydrodynamics employed to simulate gas–solid flow [26-29].

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The aim of this present work is to study natural convection of nanofluids in an inclined cavity

with curve boundaries using lattice Boltzmann method. Effects of nanoparticle volume fraction, Rayleigh

numbers and inclination angle on the flow and heat transfer characteristics have been examined.

2. Problem definition and Mathematical model

2.1. Problem statement

The schematic diagram of square cavity with curve boundaries used in the present LBM program

is shown in Fig. 1. The left and Right walls are maintained at constant temperatures hT and cT ,

respectively while the two other walls are thermally insulated. Also is introduced as inclination angle.

Furthermore R and L are radius of curve boundary and length of adiabatic wall, respectively. We

assume that / 0.5R L .

2.2. The Lattice Boltzmann Method

The LB model used here is the same as that employed in [30] and [31]. The thermal LB model

utilizes two distribution functions, f and g, for the flow and temperature fields, respectively. It uses

modeling of movement of fluid particles to capture macroscopic fluid quantities such as velocity, pressure

and temperature. In this approach, the fluid domain discredited to uniform Cartesian cells. Each cell holds

a fixed number of distribution functions, which represent the number of fluid particles moving in these

discrete directions. The D2Q9 model was used and values of 0

4 / 9w for 0

0c (for the static particle),

1 41 / 9w

for

1 41c

and

5 91 / 36w

for

5 92c

are assigned in this model (Fig. 2(a)).

The density and distribution functions i.e. the f and g, are calculated by solving the lattice

Boltzmann equation (LBE), which is a special discretization of the kinetic Boltzmann equation. After

introducing BGK approximation, the general form of lattice Boltzmann equation with external force is as

follow:

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For the flow field:

( , ) ( , ) [ ( , ) ( , )]eq

i i i i i i k

v

tf x c t t t f x t f x t f x t tc F

(1)

For the temperature field:

( , ) ( , ) [ ( , ) ( , )]eq

i i i i i

C

tg x c t t t g x t g x t g x t

(2)

where t denotes lattice time step, i

c is the discrete lattice velocity in direction i , k

F is the external

force in direction of lattice velocity, v

and C denotes the lattice relaxation time for the flow and

temperature fields. The kinetic viscosity and the thermal diffusivity , are defined in terms of their

respective relaxation times, i.e. 2( 1 / 2)

s vc and

2( 1 / 2)

s Cc , respectively. Note that the limitation

0.5 should be satisfied for both relaxation times to ensure that viscosity and thermal diffusivity are

positive. Furthermore, the local equilibrium distribution function determines the type of problem that

needs to be solved. It also models the equilibrium distribution functions, which are calculated with Eqs.

(3) and (4) for flow and temperature fields respectively.

2 2

2 4 2

. ( . )1 11

2 2

eq i i

i i

s s s

c u c u uf w

c c c

(3)

2

.1eq

i i

i

s

g wTc u

c

(4)

where i

w is a weighting factor and is the lattice fluid density.

In order to incorporate buoyancy forces in the model, the force term in the Eq. (1) need to calculate as

below in vertical direction (y):

3i y

F w g (5)

For natural convection, the Boussinesq approximation is applied and radiation heat transfer is negligible.

To ensure that the code works in near incompressible regime, the characteristic velocity of the flow for

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natural ( )ynatural

V g TH regime must be small compared with the fluid speed of sound. In the present

study, the characteristic velocity selected as 0.1 of sound speed.

Finally, macroscopic variables calculate with the following formula:

,

c

.

Flow density :

Momentum : u ,

Temperature :

i ii

ii

fii

T g

f

(6)

2.3. Boundary conditions

2.3.1. Curved boundary treatment for velocity

For treating velocity and temperature fields with curved boundaries, the method proposed in [32]

has been used. An arbitrary curved wall separating solid region from fluid is shown in Fig. 2(b). The link

between the fluid node fx and the wall node wx intersects the physical boundary at bx . The fraction of

the intersected link in the fluid region is /f w f bx x x x . To calculate the post-collision

distribution function ( , )bf x t based upon the surrounding nodes information, a Chapman–Enskog

expansion for the post-collision distribution function on the right-hand side of Eq. (1) is conducted as:

*

2

3( , ) (1 ) ( , ) ( , ) 2 e .ub f b wf x t f x t f x t w

c

(7)

where

*

2

3( , ) ( , ) ( , ) . ,

2 1 1u u ( , ), , 0

2 2

2 11 3 1u 2 3 , , 1

2 2 1 / 2 2

eq

b f f bf f

bf ff ff

bf f w

f x t f x t w x t e u uc

u x t if

u u if

(8)

In the above, e e ; fu is the fluid velocity near the wall; wu is the velocity of solid wall and ubf is

an imaginary velocity for interpolations.

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2.3.2. Curved boundary treatment for temperature

Following the work of Yan and Zu [32] the non equilibrium parts of temperature distribution

function can be defined as:

( , ) ( , ) ( , )eq neq

b b bg t g t g t x x x (9)

Substituting Eq. (9) into Eq. (2) leads to:

1( , ) ( , ) (1 ) ( , )

eq neq

b b b

s

g t t g t g t

x x x

(10)

Obviously, both ( , )eq

bg t x and ( , )neq

bg t

x are needed to calculate the value of ( , )b

g t t

x . In Eq.

(10) the equilibrium part is defined as:

*

2

*3( , ) 1 .

eq

b b bg t w Tc

x e u (11)

where *

bT is defined as a function of

1[ ( 1) ] /

b w fT T T and

2[2 ( 1) ] / (1 )

b w ffT T T

*

1

*

1 2

, if 0.75

(1 ) , if 0.75

b b

b b b

T T

T T T

(12)

and *

bu is defined as a function of

1[ ( 1) ] /

b w fu uu and

2[2 ( 1) ] / (1 )

b w ffu uu

*

1

*

1 2

, if 0.75

(1 ) , if 0.75

b b

b b b

u

u u

u

u

(13)

The non equilibrium part in Eq. (14) is defined as:

( , ) ( , ) (1 ) ( , )neq neq neq

b f ffg t g x t g x t

x (14)

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2.4. The Lattice Boltzmann model for nanofluid

In order to simulate the nanofluid by the lattice Boltzmann method, because of the interparticle

potentials and other forces on the nanoparticles, the nanofluid behaves differently from the pure liquid

from the mesoscopic point of view and is of higher efficiency in energy transport as well as better

stabilization than the common solid-liquid mixture. For pure fluid in absence of nanoparticles in the

enclosures, the governed equations are Eqs. (1-14). However for modeling the nanofluid because of

changing in the fluid thermal conductivity, density, heat capacitance and thermal expansion, some of the

governed equations should change. The fluid is a water based nanofluid containing Cu (copper). The

nanofluid is a two component mixture with the following assumptions: incompressible, no-chemical

reaction, negligible viscous dissipation, negligible radiative heat transfer, nano-solid-particles and the

base fluid are in thermal equilibrium and no slip occurs between them. The thermo physical properties of

the nanofluid are given in Table1 [4]. The effective density n f , the effective heat capacity p n fC and

thermal expansion n f

of the nanofluid are defined as [4]:

(1 )n f f s (15)

(1 )p p pn f f sC C C

(16)

(1 )n f f s

(17)

where is the solid volume fraction of the nanoparticles and subscripts ,f nf and s stand for base fluid,

nanofluid and solid, respectively.

The viscosity of the nanofluid containing a dilute suspension of small rigid spherical particles is

(Brinkman model [33]):

2.5(1 )

f

nf

(18)

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The effective thermal conductivity of the nanofluid can be approximated by the Maxwell–Garnetts (MG)

model as [4]:

2 2 ( )

2 ( )

n f s f f s

f s f f s

k k k k k

k k k k k

(19)

In order to compare total heat transfer rate, Nusselt number is used. The Local and average Nusselt

numbers are defined as follows:

/ 2

0

2n f

loc ave

f

k TNu and Nu Nu d

k r

(20)

3. Grid testing and code validation

To verify the grid independence of the solution scheme, numerical experiments are performed as

shown in Table2. Different mesh sizes were used for the case of resolution

at 510 ,Pr 6.8, 0.06Ra and 0 .The present code is tested for grid independence by calculating

the average Nusselt number on the hot wall. It is found that a grid size of 120 180 ensure the grid

independent solution for the present case. The convergence criterion for the termination of all

computations is:

1 7max 10n n

grid

(21)

where n is the iteration number and stands for the independent variables.

The present numerical solution is validated by comparing the present code results against the results of

Davis [1] for Pr 0.71 .This comparison is presented in Table3. Furthermore, another validation test was

carried for natural convection in an enclosure filled with Cu–water for different Grashof numbers with the

results of Khanafer et al. [4] in Fig. 3. All of the previous comparisons indicate the accuracy of the

present LBM code.

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4. Results and discussion

Natural convection of Cu-water nanofluid in an inclined cavity with curve boundaries is

investigated numerically using LBM. Calculations are made for various values of volume fraction of

nanoparticle ( 0,0.02,0.04 and0.06 ), Rayleigh number ( 3 4Ra 10 ,10 and 510 ) and inclination angle

( 60 , 30 ,0 ,30 to60 ) when Prandtl number is fixed ( Pr 6.8 ) and aspect ratios ( R / L 0.5 ).

Comparison of the isotherms (up) and streamlines (down) contours for different values of

Rayleigh numbers and inclination angles is shown in Fig. 4.The calculated maximum values of absolute

stream function are also given with each graph for reference. Here, as Rayleigh number increases max

increases. It can be seen that the change of inclination angle has a significant impact on the thermal and

hydrodynamic flow field. In case, by increasing the Rayleigh number, streamlines start to penetrate

toward the cavity center. When0 , at 3Ra 10 streamlines have a single vortex which its center is

located at the center of the system. At this Rayleigh number the conduction heat transfer mechanism is

more pronounced and the isotherms are parallel to each other and also take the enclosure shape. As the

Rayleigh number increases ( 4Ra 10 ), the central streamline is distorted into an elliptic shape due to

higher flow velocity. At 5Ra 10 , the central streamline is elongated and two secondary vortices appear

inside it. For case, 0 when the hot surface is left, the isotherms are very dense near the bottom of the

hot wall and the top of the cold wall and temperature gradient near the walls is very large, especially in

higher Rayleigh numbers. According to streamlines in Fig. 4, the double vortex may occur when the hot

wall is up. This fact is due to the existence of gravitational force which prevents fluid to flow from cold

wall to the hot one. Also isotherms are more dens at the middle of enclosure. When the hot wall located at

the bottom, a single vortex is observed in the cavity.

The variation of local Nusselt number along the hot surface at different inclination of cavity for

three Rayleigh numbers is depicted in Fig.5. It displays the influence of the Rayleigh number and the

inclination angles on the local Nusselt number along the heated surface (locNu ).As illustrated in Fig. 5,

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for low value of Rayleigh, Local Nusselt number is approximately independent of inclination angle. The

locNu increases with growth of Rayleigh number which it is due to increase in convective heat transfer. It

is observed that, the Local Nusselt number increases slightly with increase of inclination angle. For lower

Rayleigh numbers, the trend of local Nusselt number variation is the same. In fact at low Rayleigh

numbers, it is independent to the inclination angle variation. In addition, the most increasing rate of local

Nusselt number occurs when increases from 145 to

180 .

Effects of the volume fraction of nanoparticles, Rayleigh number and inclination angle on

average Nusselt number is shown in Fig. 6.The sensitivity of thermal boundary layer thickness to volume

fraction of nanoparticles is related to the increased thermal conductivity of the nanofluid. In fact, higher

values of thermal conductivity are accompanied by higher values of thermal diffusivity. The high value of

thermal diffusivity causes a drop in the temperature gradients and accordingly increases the thermal

boundary layer thickness. This increase in thermal boundary layer thickness reduces the Nusselt number;

however, Nusselt number is a multiplication of temperature gradient and the thermal conductivity ratio

(conductivity of the nanofluid to the conductivity of the base fluid). Since the reduction in temperature

gradient due to the presence of nanoparticles is much smaller than the thermal conductivity ratio,

therefore an enhancement in Nusselt number is taken place by increasing the volume fraction of

nanoparticles. As seen in Fig. 6, increasing Rayleigh number leads to increase in Nusselt number. Also it

can be found that maximum value of Nusselt number occurs at 0 for 3Ra 10 but for other values of

Rayleigh number maximum values of Nusselt number are obtained at 30 . For all values of Rayleigh

number minimum value of Nusselt number is obtained at 60 .

The enhancement of heat transfer between the case of 0.06 and the pure fluid (base fluid)

case is defined as:

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ave ave

ave

Nu 0.06 Nu basefluidEn 100

Nu basefluid

(22)

Heat transfer enhancement due to addition of nanoparticles for different values of Rayleigh number and

inclination angle is shown in Fig. 7. For 0 , the effect of nanoparticles is more pronounced at low

Rayleigh number than at high Rayleigh number because of greater amount of rate of enhancement and

increasing Rayleigh number leads to decrease in enhancement of heat transfer. For 0 maximum

values of enhancement occurs at5Ra 10 .Also it can be seen that at

3Ra 10 , enhancement profile has

one maximum and one minimum point while for other values of Rayleigh number enhancement decreases

with increase of .

5. Conclusions

In this study, natural convection heat transfer in an inclined cavity with curve boundaries filled

with nanofluid is investigated numerically using LBM scheme. Effects of nanoparticle volume fraction,

Rayleigh numbers and inclination angles on the flow and heat transfer characteristics have been

examined. The results show that lattice Boltzmann method based on double-population is a powerful

approach for simulating natural convection in this geometry that include curved boundaries. This method

can simulate the velocity and temperature fields with second order accuracy. Also it can be found that

Nusselt number on the both inner and outer cylinders is an increasing function of nanoparticle volume

fraction and Rayleigh number. Besides, it can be seen that the maximum value of Nusselt number occurs

at 0 for 3Ra 10 but for other values of Rayleigh number maximum values of Nusselt number are

obtained at 30 .

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Fig. 1. Geometry of the problem

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(a) (b)

Fig. 2. (a)Discrete velocity set of two-dimensional nine-velocity (D2Q9) model; (b) Curved

boundary and lattice nodes.

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Fig. 3. Comparison of the temperature on axial midline between the

present results and numerical results by Khanafer et al. [4] 0.1

and Pr 6.8 Cu Water .

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3Ra 10 4Ra 10 5Ra 10

γ =

-6

0˚

max 0.0021 max 0.0077 max 0.0055

γ =

0˚

max 0.0072 max 0.0282 max 0.0439

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γ =

60

˚

max 0.0150 max 0.0683 max 0.129

Fig. 4. Comparison of the isotherms (up) and streamlines (down) contours for different values

of Rayleigh numbers and inclination angles at 0.06 .

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3Ra 10 4Ra 10

5Ra 10

Fig. 5.Effects of the Rayleigh number and inclination angle on local Nusselt number when 0.06 .

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3Ra 10 4Ra 10

5Ra 10

Fig. 6.Effects of the volume fraction of nanoparticles, Rayleigh number and inclination angle on

average Nusselt number when Pr 6.8 .

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Fig. 7.Effects of the Rayleigh number and inclination angle on enhancement heat transfer

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Table1. Thermo physical properties of water and nanoparticles [4]

3( kg / m ) pC ( j / kgk ) k(W / m.k ) 5 110 ( K )

Pure water 997.1 4179 0.613 21

Copper( Cu ) 8933 385 401 1.67

Table2.Comparison of the average Nusselt number along the surface of the hot wall ( aveNu ) for different

grid resolution at 510Ra , Pr 6.8, 0.06 and 0 .

Mesh size 80 120 100 150 120 180 140 210

aveNu 5.489647 5.497227 5.503237 5.517481

Table3. Comparison of the present solution with previous works for different Rayleigh numbers

when Pr=0.7.

Ra Present De Vahl Davis [1]

310 1.1432 1.118

410 2.2749 2.243

510 4.5199 4.519

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Application of LBM in Simulation of natural convection in a nanofluid filled square cavity with curve

boundaries

Graphical abstract

5Ra 10

60 0 60

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Highlights

> Natural convection of nanofluid filled cavity investigated.

> LBM is used to solve this problem.

>Nusselt number increases with increase of and Ra .

> maxE is obtained at

3Ra 10 and5Ra 10 for 0 and 0 , respectively.