Heat Transfer Enhancement of Cu-H<sub>2</sub>O Nanofluid with Internal Heat Generation Using LBM

doi:10.4236/ojfd.2013.32A015

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Open Journal of Fluid Dynamics, 2013, 3, 92-99

http://dx.doi.org/10.4236/ojfd.2013.32A015 Published Online July 2013 (http://www.scirp.org/journal/ojfd)

Heat Transfer Enhancement of Cu-H2O Nanofluid with

Internal Heat Generation Using LBM

Mohammad Abu Taher1, Yeon Won Lee2, Heuy Dong Kim1*

1School of Mechanical Engineering, Andong National University, Andong, South Korea

2School of Mechanical Engineering, Pukyong National University, Busan, South Korea

Email: *[email protected]

Received June 10, 2013; revised June 17, 2013; accepted June 27, 2013

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

Fluid flow and heat transfer analysis of Cu-H2O nanofluid in a square cavity using a Thermal Lattice Boltzmann

Method (TLBM) have been studied in the present work. The LBM has built up on the D2Q9 model and the single re-

laxation time method called the Lattice-BGK (Bhatnagar-Gross-Krook) model. The effect of suspended nanoparticles

on the fluid flow and heat transfer analysis have been investigated for different non dimensional parameters such as

particle volume fraction (φ) and particle diameters (dp) in presence of internal heat generation (q) of nanoparticles. It is

seen that flow behaviors and the average rate of heat transfer in terms of the Nusselt number (Nu) as well as the thermal

conductivity of nanofluid are effectively changed with the different controlling parameters such as particle volume frac-

tion (2% ≤ φ ≤ 10%), particle diameter (dp = 5 nm to 40 nm) with fixed Rayleigh number, Ra = 105. The present results

of the analysis are compared with the previous experimental and numerical results for both pure and nanofluid and it is

seen that the agreement is good indeed among the results.

Keywords: Nanofluid; Lattice-Boltzmann; Volume Fractions; Particle Diameter; Heat Generation

1. Introduction

The term nanofluid is envisioned to describe a solid-liq-

uid mixer which consists of nano-sized solid particles

and a base liquid and this is one of new challenges for

thermo-sciences provided by the nanotechnology. It has

potential applications in the microelectromechanical sys-

tems (MEMS) and electronics cooling industries. There-

fore, in the recent years, the micro and nano systems

have become of great interest due to their important and

promising applications in various fields. As a result, the

research topic of nanofluids has been receiving increased

attention worldwide. In fact, numerous theoretical and

experimental studies of the effective thermal conductiv-

ity of suspensions that contains solid particles have been

conducted since Maxwell’s theoretical work was pub-

lished more than 100 years ago. All of the studies on

thermal conductivity of suspensions have been consid-

ered with millimeter to micrometer sized particles. The

use of the conventional millimeter and micrometer-sized

particles in heat transfer fluids in practical devices is

greatly limited by the tendency of such particles to settle

rapidly and to clog mini and micro channels. Until now,

it is very difficult to solid particles from eventually set-

tling out of suspension because of particle size. However,

nanoparticles appear to be ideally suited for applications

in which fluids flow through small passages, because the

nanoparticles are stable and small enough not to clog

flow passage. Therefore, the main goal of nanofluids is to

achieve the maximum thermal properties with the mini-

mum possibility of particle volume fractions (less than

1%) and size of particles (less than 10 nm) by uniform

dispersion and stable suspension of nanoparticles in base

fluids/liquids. The performance of heat transfer of nan-

ofluid depends on more factors such as shape of particles,

the dimension of particles, the volume fractions of parti-

cle in the suspensions, and the thermal properties of par-

ticle materials [1,2]. However, significant amounts of

experimental and theoretical work have been performed

on buoyancy induced flow in conventional fluid [3,4].

The mechanism of nanoparticles, enhancement of heat

transfer characteristics are described more detailedly in

the books [5,6] and Yu et al. [7].

From the microscopic point of view, classical me-

chanics has no insight into the microstructure of the sub-

stance; however, statistical mechanics calculated the

*Corresponding author.

M. A. TAHER ET AL. 93

properties of state on the basis of molecular motions in a

space, and on the basis of the intermolecular interactions.

The Lattice Boltzmann equation (LBE) is one of the

methods available to deal with such problems by Shan

and Chen [8]. In general case, most of authors considered

only mass and momentum conservation in LBM [9]. The

macroscopic equations of these models correspond to the

Navier-Stokes (NS) equations with ideal gas equation of

state and a constant temperature. However, sometimes it

is important to simulate thermal effects simultaneously

with the fluid flows [10]. The more detailed thermal Lat-

tice-Boltzmann method (TLBM) with some examples is

discussed in the books written by Mohammad [11] and

Succi [12].

A lattice Boltzmann model has been developed by

Shan and Chen [13] to simulate the fluid flows contain-

ing multiple phases and components. It is also known as

Shan-Chen (S-C) multi-component model. Using the

two-component LBM, the Rayleigh-Benard convection

in two and three dimensions for different cases have been

studied by many researchers [14-17]. However, Buick

and Gretaed [18] introduced a body force into LBM by

modifying the collision function. To overcome the limi-

tations of multi-component flows in the existing tradi-

tional computational methods, many researchers were

interested to use multi-component LBM for multi-phase

flows like solid-fluid mixtures [19-21]. In the present

work, multi-component thermal Lattice-Boltzmann method

(TLBM) is used for simulating natural convection H2O-

Cu nanofluid with Boussinesq approximation in a square

cavity considering the internal heat generation effect of

Cu nanoparticles. As far as we know, there is no work on

nanofluid heat transfer under natural convection using

TLBM with considering internal heat generation effect as

well. The results of the analysis are compared with ex-

perimental and numerical data both for pure and nan-

ofluids, and shown a relatively good agreement.

2. Formulation of the Problem

2.1. Mathematical Analysis

To ensure the model satisfies the N-S equations for a

fluid under the influence of body force, multi-component

Lattice-Boltzmann equation (LBE) can be written as [19]



 



ii i

xetttfxt

xt fxt

DeF











 

 







(1)

The function





is the particle distribution

function of



= 1, 2 components with lattice velocity

vectors ei, Ai is the adjustable coefficient, D is the di-

mension, F is applied force, and 1m







 is the re-

laxation parameter that depends on the local macroscopic

variables ρ and



u. These variables should satisfy the

following laws of conservation:







tmfx



t





and







tmfx

 





uet (2)

where m



is the molecular mass. For two dimensional

D2Q9 model, the equilibrium distribution function can

be defined as



24 2

39 3

1. .

,2eq eq

ii i

cc c











(3)





 







eueu



where wi is the lattice weighting factors. Therefore the

equilibrium velocity becomes

 



















u (4)

Simultaneously, the lattice Boltzmann energy equation

without viscous dissipation for nanofluid defined as [19]







 



1,,

ii i

gxttt gxt

xt gxttwG















 



(5)

The energy distribution function can be written as



24 2

39 3

1. .

eq eq

ii i

cc c











(6)





 







eueu











tgx





t



is the internal energy variable

and

Gq C







 is the source or force term per unit

time, which is related to the heat generation







per

unit volume. In the present study, we consider the heat is

generated from nanoparticles only, not from base fluid.

The mean velocity, temperature, viscosity and thermal

diffusivity of the nanofluid can be written as

 

cCstcCs

mf xtxt

 





 







 



















 





(7)

where m



and c





are the concentration of viscosity

and diffusivity of each component respectively. Solving

the Equations (1) and (5) with other approximations, we

M. A. TAHER ET AL.

get all information that we interested in our study.

0.25 0.5 0.751

-0.36

-0.18

0.18

2.2. Numerical Analysis

boundary conditions for

The ratio of the buoyancy force to the product of vis-

The physical configuration with

the present study is shown in Figure 1. A closed square

cavity of length H is considered here. The horizontal

walls are assumed to be insulated whereas the vertical

walls are maintained at constant but different tempera-

tures Th (hot) and Tc (cold). For natural convection, the

momentum and energy equations are coupled and the

flow is driven by temperature or mass gradient, Under

Boussinesq approximation, the force term per unit mass

can be written as



,tFx

 



, ,

ref

tgTtT



xx

us force and heat diffusion rates defines the Rayleigh

number, 3

RaPr Gr

TH v





, where Gr is the

Grashof ndtl number. The

boundary conditions are defined as:

number and Pr is the Pra

T

uv y

 

, at y = 0, H and 0  x  L

, at x = 0 and 0  y  H

2.3. Code Validation

th the conventional benchmark

TTuv

TTuv, at x = L and 0  y  H

To verify our results wi

and with experimental results, first we consider the cav-

ity is filled with pure fluid like air (Pr = 0.71). It is shown

in Figure 2. It can be seen from these figures that our

present numerical simulation is in very good agreement

with numerical results of [3] and experimental solutions

of [4]. Moreover, the present LBM also applied for nan-

ofluids and it is necessary to verify our method with

other published work. A good agreement is obtained be-

0.36

(a)

Present-LBM

Krane &Jessee[4]-Exp.

Khanafer et al.[3]-Num

x/H

V-velocity

00.25 0.5 0.75 1

0.25

0.5

0.75

Present- LBM

Krane & Jessee [4]-Exp.

Khanafer er al.[3]-Num

x/H

Temperature

(b)

Figure 2. Comparison of dimensionless (a) velocity and (b)

temperature profiles for pure fluid (Pr = 0.71, Ra = 1.89

the present solution and bench mark solution of [3]

simultaneously on a

105).

tween

as illustrated in Figure 3. This confirms that our method

is correct both for pure and nanofluids.

3. Results and Discussions

Equations (1) and (5) are solved

uniform 2D grid system along with boundary conditions

and other equations described in the above sections. Each

numerical time step consists of three stages (1) collision,

(2) streaming, and (3) boundary conditions steps fol-

lowed by the LBM approaches. Fluid flow and heat

transfer enhancement of nanofluid for a wide range of

controlling parameters are discussed in this section. It is

assumed that the nanoparticles of Cu are uniformly sus-

pended in water; there is no aggregation of nanoparticles

in the fluid medium.

The computed velocity field in the buoyancy driven

cavity flow for pure

and nanofluids are seen in Figures

4(a) and (b) respectively. The arrows denote the velocity

vectors in both magnitude and direction and the solid

lines represent the stream lines of H2O-Cu nanofluid. The

general magnitude of the velocity can be seen to increase

as the hot fluid, near the hot wall, flow upward and the

Figure 1. The configuration of the problem under cons-

eration. id

M. A. TAHER ET AL. 95

0.2 0.4 0.6 0.8 1

-0.15

-0.1

-0.05

0.05

0.1

0.15

Present-LBM

Khanaferetal.[3]-Num

(

)

V-Velocity

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1Present-LBM

Khanafer et al.[3]-Num

(

)

Temperature

Figure 3. Comparison of velocity and temperature profiles

for nanofluid (Pr = 6.2, Gr = 105,



= 10%).

0.2 0.4 0.6 0.81

0.2

0.4

0.6

0.8

y/H

(

)

0.2 0.40.6 0.81

0.2

0.4

0.6

0.8

x/H

y/H

(

)

Figure 4. Velocity vectors and stream lines for (a) pure fluid

(H2O) and (b) nanofluid (H2O-Cu).

pure and nanofluids. It

ing the parti-

cold fluid, near the cold wall, flow downward due to the

effect of buoyancy force for both

is one of the main criteria of natural convection in a dif-

ferently heated cavity. Thus it is clear that the nanofluids

flow behavior like a pure fluid.

The effect of internal heat generation of nanoparticles

on the velocity and temperature fields at the mid section

the cavity for different particle diameters and volume

fractions are illustrated in Figures 5 and 6.

It is seen from Figure 5(a) that the velocity profiles of

nanofluid decrease remarkably with increas

e diameter near hot wall and increase near the cold wall.

But, in presence of internal heat generation of nanoparti-

cles (dashed lines, Figure 5(a)), the velocity profiles

near the hot and cold walls have shown infinitesimal

changed, decreased (zoom viewed) near hot wall. It is

also seen that the location of the local maximum and

minimum for all cases have tend to moved to right from

hot wall and left from cold wall with increasing the par-

ticle diameter. For dp = 5 nm, the local maximum and

minimum are seen at approximately x/H = 0.06 and x/H =

0.85, whereas for dp = 10 and 20 nm, the locations are

observed at x/H = 0.09 and x/H = 0.93; and x/H = 0.10

and x/H = 0.90 respectively. These represented that the

velocity boundary layer become thicker with increasing

particle diameters.

0.2 0.4 0.6 0.8

-0.3

-0.15

0.15

0.3 dp=5

dp=5

dp=10

dp=20

x/H

(a)

V-Velocity

0.2 0.4 0.6 0.8 1

0.15

0.3

0.45

0.6

0.75

0.9 dp=5

dp=5

dp=10

dp=20

(b)

Temperature

Figure 5. (a) Dimensionless velocity and (b) temperature

profiles for different particle diameters (nm) (solid lineo

heat generation, dashed lines: with heat generation).

s: n

M. A. TAHER ET AL.

0.2 0.4 0.6 0.8 1

-0.08

0.08

0.16





(a)





x/H

V-Velocity

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8







(b)

Temperature

Figure 6. (a) Dimensionless velocity and (b) temperature

profiles for different particle volume fractions (%) (

lines: no heat generation, dashed lines: with heat genra-

s increase slightly near the hot wall but decrease

sionless form of the heat transfer coeffi-

solid

tion).

From Figure 5(b), it is observed that, the temperature

profile

si nificantly throughout the cavity with increasing the

particle diameters and finally satisfied the boundary con-

ditions. Moreover, in presence of heat generation, dased

lines in Figure 5(b), the temperature profiles slightly

increased compare to without heat generation. It means

that, if the nanoparticles generate the heat, the particle

lost some energy to the fluid and consequently the aver-

age temperature of nanofluid increased. From Figure 5

(b), the significant temperature gradient observed in the

vicinity of the heated surface approximately, 0  x/H 

0.15, and unheated surface 0.95  x/H  1 for dp = 5 nm;

0  x/H  0.22 and 0.92  x/H  1 for dp = 10 nm; 0 

x/H  0.28 and 0.88  x/H  1 for dp = 20 nm. Therefore,

it is seen that thermal stratification and increase in tem-

perature in the direction of heat flow in the core region

respectively 0.15 < x/H < 0.95, 0.22 < x/H < 92 and 0.28

< x/H < 0.88 for dp = 5 nm, 10 nm and 20 nm. This indi-

cates that thermal boundary layer increased with in-

creasing particle diameters.

The effects of volume fractions with internal heat gen-

eration of nanoparticles on fluid flow are shown in Fig-

ure 6. It is shown that the velocity profiles of nanofluid

creases remarkably with volume fractions of nanofluid

near hot wall and decreased near cold wall. In more de-

tails, the velocity along the vertical walls of the cavity

show a higher level of activity as predicted by thin layer

of hydrodynamic velocity boundary layers. Whereas, the

variation of the velocity at the center of the cavity for all

volume fractions of nanoparticles are negligible com-

pared with those of boundaries in all cases. This is ex-

pected as the maximum fluid motion at the boundaries

and almost stagnant in the core region. The locations of

the local maximum and minimum velocities for all cases

are approximately at the same position of X(=x/H). The

effect of dimensionless temperature profiles with in-

crease of volume fractions at the centerline of the cavity,

which is perpendicular direction of the heated walls of

the cavity, as shown in Figure 6(b). The variation of

temperature near hot and cold walls versus dimensionless

horizontal length is linear for all volume fractions, which

is the characteristic of heat transfer by convection. For all

values of



, a significant temperature gradient is ob-

served in the vicinity of the heated surface approximately,

0  x/H  0.22, and unheated surface 0.92  x/H  1.

Moreover, it is seen that thermal stratification and in-

crease in temperature in the direction of heat flow in the

core region 0.22 < x/H < 0.92. In physical meaning, the

temperature should decrease in the direction of heat flow,

but for natural convection in a cavity, the rate of cooling

is expected to be higher near the heated and unheated

walls due to the fluid motion and hydrodynamics effects.

This phenomenon is attributed to buoyancy-induced cel-

lular flows in the boundary layer adjoining the heated

and cold walls. The fluid motion increases the rate of

heat transfer.

A usual means of characterizing heat transfer is to cal-

culate the Nusselt number. Actually the Nusselt number

Nu is a dimen

ent. To investigate the heat transfer performance in

terms of effective thermal conductivity of the nanofluid,

the local Nusselt number NuL and the averaged Nusselt

number Nu over the flow channel are respectively de-

fined as follows:

Nu k

 (8)

The heat transfer coefficient h is defined as

wnf

TT x

hqk

T





, (9)

Here T

h and T

c are the temperature of heated and

cooled walls respectively. The heat flux

can be expressed in terms of thermal conductivity Knf of

, qw, of nanofluid

nofluid, kf are the thermal conductivity of the base

fluid and H is the channel height. The Nusselt number

and the average Nusselt number in dimensionless form

on the left heated wall are defined as

M. A. TAHER ET AL. 97

 

d, nf

wall

NuNu YYNu YkX















 (10)

where, θ, is the non-dimensional temperature. On the

basis of the definition [5], the effective thermal

tivity for a two-component mixture (Cu-H2O) is defined

verage Nusselt number and the thermal conduc-

tivity ratios for constant Rayleigh5

100,000 time steps with various

conduc-



nfnfp nf



 (11)

The a

number, Ra = 10 at

volume fractions and

rticle diameters as well as the internal heat generation

of nanoparticles are shown in Figures 7 and 8.

The dimensionless rate of heat transfer called Nusselt

number (Nu) is significantly decreased with increasing

the particle diameters from dp = 5 nm to 40 nm, as de-

scribed in Figure 7(a), though it is increased with parti-

cle volume fractions. Moreover, it is observed that in

presence of nanoparticles heat generation, the Nusselt

number slightly decreases (dashed lines) compare to

without heat generation (solid lines). Similar phenome-

non is seen for thermal conductivity ratio. It is seen

010 20 30 40 50

,q=0.0

,q=5.0

,q=0.0

,q=5.0

Particlediameter

(a)

010 20 30 40 50

1.09

1.092

1.094

1.096

1.098

1.1

,q=0 .0

,q=5 .0

,q=0 .0

,q=5 .0

Particle diameter

(b)

00.02 0.04 0.06 0.080.10.12

(a)

dp=5,q=0.0

dp=5,q=5.0

dp=1 0,q=0.0

dp=1 0,q=5.0

Particle Volume Fractions(%)

00.02 0.040.06 0.080.10.12

1.092

1.096

1.1

1.104 dp=5,q=0.0

dp=5,q=5.0

dp=10,q=0.0

dp=10,q=5.0

Particle Volume Fractions(%)

(b)

Figure 8. (a) The average rate of heat transfer (Nusselt

number, Nu); (b) Thermal conductivity ratios for different

particle volume fraction s (%) (solid lines: no heat genera-

tion, dashed lines: with heat generation).

ity are increased with

highly undesirable for many practical cooling industries.

almost constant for dp > 30 nm for all cases. This is be-

cause constant Rayleigh number effect. In natural con-

ection flow, the thermal conductivity is strongly de-v

pendent on the Rayleigh number.

It can be seen from Figure 8 that both of the measured

dimensionless rate of heat transfer in terms of Nusselt

number and the thermal conductiv

creased of volume fractions. This increased is signifi-

cantly depending on particle size. For increasing particle

diameter both of the heat transfer characteristics are

dramatically decreased. With the heat generation of

nanoparticles, it is also observed that both of heat trans-

fer rate and conductivity ratio decreases very slightly.

Actually, in the above cases, the particle size is im-

portant factor. The large size of particle diameter means

that the less number of particles. Therefore, the surfa

ea of nanoparticle decreases, giving less heat transfer

area between the phases. In natural convection flow, only

buoyancy force is a dominant force, therefore, particle

movement is very important. The main problem for large

size of particle is the rapid settling in the fluid. Other

problems are abrasion and clogging. These problems are

Figure 7. (a) The average rate of heat transfer (Nusselt

number, Nu); (b) Thermal conductivity ratios for different

particle diameters (solid lines: no heat generation, dashed

lines: with heat generation) .

M. A. TAHER ET AL.

4. Conclusions

The Thermal Lattice-Boltzmann Method (TLBM) is suc-

cessfully applied in this work to simulate buoyancy-

driven heat transfer characteristics and flow performance

of Cu-H2O nanoflui

calculation, it is ass

d in a square cavity. Throughout our

umed that constant Rayleigh number,

ith increasing the particle volume fractions

ansfer and thermal con-

d (decreased) with internal heat generation

r, the nanofluid behaves

nal of Heat Transfer, Vol. 121, No. 2,

1999, pp. 280-289.

[2] Y. Xuan and Qhancement of Nan-

Ra = 105. So the fluid flow and heat transfer analysis are

depending on particle size and volume fractions. More-

over, it is considered that the internal heat generation

effect of nanoparticles to the base fluid, and then finally,

we have shown our result of the resultant fluid called

nanofluid. The present results indicate that the following

statements:

 Both of the thermal boundary layer and velocity

boundary layer thickness increased with increasing

the particle diameter. However, the change of thick-

ness of thermal and velocity boundary layers are neg-

ligible w

within the range up to 10%.

 The heat transfer features, the heat transfer rate and

the thermal conductivity ratio, of a nanofluid are in-

creased with increase of the nanoparticle volume frac-

tions.

 However, the rate of heat tr

ductivity ratio are significantly decreased with in-

crease of particle diameter.

 In addition, all of the above features are very slightly

change ef-

fect of nanoparticle due to the constant Rayleigh

number and Prandtl number at the present study.

 For small particle diamete

more like a fluid than the conventional solid-fluid

mixer. The results from our analysis have shown

quite good and this temperature characteristic en-

hancement plays a significant role in engineering ap-

plications such as in the electronic cooling industries

or MEMS devices.

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