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
Received June 10, 2013; revised June 17, 2013; accepted June 27, 2013
Copyright © 2013 Mohammad Abu Taher et al. This is an open access article distributed under the Creative Commons Attribution
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.
C
opyright © 2013 SciRes. OJFD

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]
 
 

,
2
,,
1,
21
2
ii i
eq
ii
m
m
i
mi
,
f
xetttfxt
f
xt fxt
DeF
Ac








 
 




(1)
The function

,
i
f
xt

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:


,,
i
i

x
tmfx

t 

and


,,
ii
i

x
tmfx
 
  uet (2)
where m

is the molecular mass. For two dimensional
D2Q9 model, the equilibrium distribution function can
be defined as

,
2
,,
24 2
39 3
1. .
22
eq
i
,2eq eq
ii i
f
wu
cc c




eq 
(3)


 



eueu

where wi is the lattice weighting factors. Therefore the
equilibrium velocity becomes
2
m
eq
m
 












uF
u (4)
Simultaneously, the lattice Boltzmann energy equation
without viscous dissipation for nanofluid defined as [19]

 
 

,
,,
1,,
ii
ei
eq
ii i
gxttt gxt
g
xt gxttwG







 

(5)
The energy distribution function can be written as

,
2
,,
24 2
39 3
1. .
22
eq
i
,2 eq eq
ii i
g
wu
cc c




eq 
(6)


 



eueu





,
i
,
i
x
tgx

 t

is the internal energy variable
and
p
Gq C


  is the source or force term per unit
time, which is related to the heat generation


q

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
 
22
11
,
22
,,
,
mm
ii
i
p
cCstcCs
mf xtxt
TC
 


 



 



t






 

e
u

(7)
where m c

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. 94
get all information that we interested in our study.
0.25 0.5 0.751
-0.36
-0.18
0
0.18
2.2. Numerical Analysis
boundary conditions for
The ratio of the buoyancy force to the product of vis-
co
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
g
TH v

 , where Gr is the
Grashof ndtl number. The
boundary conditions are defined as:
number and Pr is the Pra
0
T
uv y  
, at y = 0, H and 0  x  L
, at x = 0 and 0  y  H
0
2.3. Code Validation
th the conventional benchmark
,0
h
TTuv
,
c
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
0.25
0.5
0.75
1
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
0.05
0.1
0.15
Present-LBM
Khanaferetal.[3]-Num
(
a
)
x
/
H
V-Velocity
0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1Present-LBM
Khanafer et al.[3]-Num
(
b
)
x
/
H
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
1
x
/
H
y/H
(
a
)
0.2 0.40.6 0.81
0.2
0.4
0.6
0.8
1
x/H
y/H
(
b
)
Figure 4. Velocity vectors and stream lines for (a) pure fluid
(H2O) and (b) nanofluid (H2O-Cu).
pure and nanofluids. It
of
ing the parti-
cl
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
0.15
0.3 dp=5
dp=5
dp=10
dp=10
dp=20
dp=20
x/H
(a)
V-Velocity
0.2 0.4 0.6 0.8 1
0
0.15
0.3
0.45
0.6
0.75
0.9 dp=5
dp=5
dp=10
dp=10
dp=20
dp=20
x
/
H
(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. 96
0.2 0.4 0.6 0.8 1
-0.08
0
0.08
0.16



(a)



x/H
V-Velocity
0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1






x
/
H
(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
g
in
sionless form of the heat transfer coeffi-
ci
solid
e
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:
L
f
hH
Nu k  (8)
The heat transfer coefficient h is defined as
w
wnf
hc
TT x
q
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
na
, 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
 
1
0
d, nf
LL
wall f
k
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
pa
conduc-
as

nfnfp nf
kC   (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
3
6
9
,q=0.0
,q=5.0
,q=0.0
,q=5.0
Nu
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
K
nf
/K
f
Particle diameter
(b)
00.02 0.04 0.06 0.080.10.12
4
6
8
10
(a)
dp=5,q=0.0
dp=5,q=5.0
dp=1 0,q=0.0
dp=1 0,q=5.0
Nu
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)
K
nf
/K
f
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
in
ce
ar
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. 98
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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