1. International Journal of Heat and Mass Transfer 54 (2011) 4376–4388
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ijhmt
Numerical investigation of effective parameters in convective heat transfer
of nanofluids flowing under a laminar flow regime
Ehsan Ebrahimnia-Bajestan a,⇑, Hamid Niazmand a, Weerapun Duangthongsuk b, Somchai Wongwises c,d
a
Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
b
Department of Mechanical Engineering, South-East Asia University, Bangkok, Thailand
c
Fluid Mechanics, Thermal Engineering and Multiphase Flow Research Laboratory (FUTURE), Department of Mechanical Engineering,
King Mongkut’s University of Technology Thonburi, Bangmod, Bangkok 10140, Thailand
d
The Royal Institute of Thailand, Academy of Science, Sanam Sueapa, Dusit, Bangkok 10300, Thailand.
a r t i c l e i n f o a b s t r a c t
Article history: This article presents a numerical investigation on heat transfer performance and pressure drop of nano-
Received 3 December 2010 fluids flows through a straight circular pipe in a laminar flow regime and constant heat flux boundary
Received in revised form 29 April 2011 condition. Al2O3, CuO, carbon nanotube (CNT) and titanate nanotube (TNT) nanoparticles dispersed in
Accepted 29 April 2011
water and ethylene glycol/water with particle concentrations ranging between 0 and 6 vol.% were used
Available online 27 May 2011
as working fluids for simulating the heat transfer and flow behaviours of nanofluids. The proposed model
has been validated with the available experimental data and correlations. The effects of particle concen-
Keywords:
trations, particle diameter, particles Brownian motions, Reynolds number, type of the nanoparticles and
Nanofluids
Heat transfer performance
base fluid on the heat transfer coefficient and pressure drop of nanofluids were determined and discussed
Pressure drop in details. The results indicated that the particle volume concentration, Brownian motion and aspect ratio
Numerical study of nanoparticles similar to flow Reynolds number increase the heat transfer coefficient, while the nano-
Thermal conductivity particle diameter has an opposite effect on the heat transfer coefficient. Finally, the present study pro-
vides some considerations for the appropriate choice of the nanofluids for practical applications.
Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction thermophysical properties and flow characteristics of nanofluids
[3–6]. However, this article is aimed at reviewing only the novel
Common heat transfer fluids such as oil, water and ethylene literature considering convective heat transfer of nanofluids with
glycol have inherently poor thermal conductivity compared to a numerical approach. These studies are briefly described as
most solids. This problem is the primary obstacle to the high follows.
compactness, light in weight and effectiveness of heat exchangers. Mirmasoumi and Behzadmehr [7] reported the effect of nano-
In order to enhance the thermal conductivity of conventional heat particle diameter on convective heat transfer performance of
transfer fluids, it has been tried to develop a new type of modern Al2O3/water nanofluid flowing under a fully developed laminar
heat transfer fluid by suspending ultrafine solid particles in base flow regime numerically. In their study, a two-phase mixture mod-
fluids. In 1993, Masuda et al. [1] studied the heat transfer perfor- el was used. The results demonstrated that the heat transfer coef-
mance of liquids with solid nanoparticles suspension. However, ficient of the nanofluid dramatically increases with decreasing the
the term of ‘‘nanofluid’’ was first named by Choi [2] in 1995, and diameter of nanoparticle. Moreover, the results also indicated that
successively gained popularity. Because of the extensively greater nanoparticle diameter has no significant effect on the skin friction
thermal conductivity and heat transfer performance of the nanofl- coefficient.
uids as compared to the base fluids, they are expected to be ideally Kalteh et al. [8] numerically studied forced convective heat
suited for practical applications. transfer of Cu/water nanofluid inside an isothermally heated
Since a decade ago, research publications related to the use of microchannel under a laminar flow regime. An Eulerian two-fluid
nanofluids as working fluids have been reported both numerically model was used to simulate the heat transfer characteristic of
and experimentally. There are also some review papers that the nanofluid. The results indicated that the heat transfer perfor-
elaborate on the current stage in the thermal behaviours, mance increases with increasing Reynolds number as well as
particle volume fraction. On the contrary, heat transfer enhance-
ment increases with decreasing nanoparticle diameter. Finally,
⇑ Corresponding author. Tel.: +98 915 300 6795; fax: +98 511 876 3304. the results also showed that the pressure drop of nanofluids is
E-mail address: ehsan.ebrahimnia@gmail.com (E. Ebrahimnia-Bajestan). slightly higher than that of base fluids.
0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijheatmasstransfer.2011.05.006

2. E. Ebrahimnia-Bajestan et al. / International Journal of Heat and Mass Transfer 54 (2011) 4376–4388 4377
Mirmasoumi and Behzadmehr [9] investigated the laminar indicated that the average Nusselt number of the nanofluid in-
mixed convection heat transfer of Al2O3/water nanofluid flowing creases with increasing particle concentration. In contrast, the re-
through a horizontal tube numerically. A two-phase mixture mod- sults also showed that the bulk temperature of the nanofluid
el was used to describe the hydrodynamic and thermal behaviour decreases with increasing particle concentration.
of the nanofluid. The numerical results indicated that in the fully Akbarinia and Laur [18] presented the laminar mixed convec-
developed region the particle concentration has insignificant ef- tion heat transfer of Al2O3/water nanofluid flows in a circular
fects on the hydrodynamic parameters, while it has important ef- curved tube numerically. A two-phase mixture model and the
fects on the thermal parameters. Moreover, the results showed control-volume technique were used to investigate the effect of
that nanoparticle concentration is higher at the bottom of the test particle diameter on the hydrodynamic and thermal parameters.
tube and at the near wall region. Their results indicated that the Nusselt number and secondary flow
Akbarinia [10] and Akbarinia and Behzadmehr [11] numerically decrease with increasing the particle diameter and uniform distri-
investigated the fully developed laminar mixed convection of bution of nanoparticles is observed.
Al2O3/water nanofluid flowing through a horizontal curved tube. Zeinali Heris et al. [19] numerically investigated the convective
In their studies, three-dimensional elliptic governing equations heat transfer of nanofluid in a circular tube with constant wall
were used. The effects of the buoyancy force, centrifugal force temperature, employing a dispersion model. Their results showed
and particle concentration on the heat transfer performance were that decreasing nanoparticle size and increasing nanoparticle con-
presented. The results showed that the particle concentration has centration augment the heat transfer coefficient.
no direct effect on the secondary flow, axial velocity and skin fric- Raisi et al. [20] carried out a numerical study on laminar con-
tion coefficient. However, when the buoyancy force is more impor- vective heat transfer of Cu/water nanofluid inside a microchannel
tant than the centrifugal force, the effect of particle concentration with slip and no slip boundary conditions. They investigated the ef-
on the entire fluid temperature can affect the hydrodynamic fect of different parameters such as Reynolds number, particle con-
parameters. Moreover, the results also indicated that the buoyancy centration, and slip velocity coefficient on the nanofluid heat
force decreases the Nusselt number whereas the particle concen- transfer characteristics. The results indicated that the particle con-
tration has a positive effect on the heat transfer enhancement centration and slip velocity coefficient have significant effects on
and on the skin friction reduction. the heat transfer rate at high Reynolds numbers.
Izadi et al. [12] studied the hydrodynamic and thermal behav- Ghasemi and Aminossadati [21] investigated the natural con-
iours of an Al2O3/water nanofluid flowing through an annulus vective heat transfer of CuO/water nanofluid inside an inclined
under a laminar flow regime. In their study, a single-phase model enclosure with top and bottom wall at different temperatures.
was used for nanofluid simulation. The results indicated that the The effects of Rayleigh number, inclination angle, and particle con-
particle volume concentration has no significant effect on the centration on heat transfer performance were studied. The results
dimensionless axial velocity, but affects the temperature field showed that the flow pattern, temperature field and heat transfer
and increases the heat transfer coefficient. rate are affected by inclination angle at high Rayleigh numbers.
He et al. [13] numerically studied the convective heat transfer Furthermore, it was found that the heat transfer rate is maximised
of a nanofluid with TiO2 nanoparticles dispersed in water under at specific particle concentration and inclination angle.
laminar flow conditions. A single-phase model and combined Euler Zhou et al. [22] presented the lattice Boltzman method (LB
and Lagrange methods were used to investigate the effects of vol- method) to study the microscale characteristics of the multicom-
ume concentration, Reynolds number and aggregate size on the ponent flow of nanofluids. In this method, the computation domain
convective heat transfer and flow behaviour of the nanofluid. Their was separated into fine mesh and coarse mesh regions, respec-
results indicated that the nanofluid significantly enhances the Nus- tively. The multicomponent LB method was used in the fine mesh
selt number, especially in the entrance region. Moreover, the region and the single-component LB method was applied in the
numerical results were consistent with experimental data. coarse mesh region. The results indicated that the present model
Bianco et al. [14] investigated the heat transfer performance of can be used to predict the microscopic characteristics of the
an Al2O3/water nanofluid flowing through a circular tube under a nanofluid and the computational efficiency can be significantly
laminar flow regime numerically. A single-phase model and two- improved.
phase model were used to determine the heat transfer coefficient Although numerous papers are currently available on the
of the nanofluid. The results demonstrated that the heat transfer numerical study of laminar convective heat transfer of nanofluids,
performance increases with increasing Reynolds number as well there is no comprehensive study on different effective parameters
as particle volume concentration. Moreover, differences in the in this field. The effect of several parameters such as nanoparticle
average heat transfer coefficient between the single-phase and shape, based fluid type and nanoparticle material are not consid-
two-phase models were observed as approximately 11%. ered in literature. On the other hand, naturally the increase in heat
Kumar et al. [15] used a single-phase thermal dispersion model transfer performance due to the nanofluids is accompanied by
to numerically investigate the thermal properties and flow field several undesirable effects such as an increase in pressure drop.
of a Cu/water nanofluid in a thermally driven cavity. The results Therefore, it needs to find the suitable nanofluid for optimum oper-
indicated that the Grashof number, particle volume fraction and ation. No significant attention is paid to find some criteria for the
particle shape factor augment the average Nusselt number of choice of appropriate nanofluids in different heat transfer applica-
nanofluids. tions. In the present study a modified single-phase model for pre-
Talebi et al. [16] presented the numerical formulation to evalu- dicting the heat transfer performance of nanofluids is proposed
ate the laminar mixed convection heat transfer of Cu/water nano- and a home-made FORTRAN computer program is developed. The
fluid flowing through a square lid-driven cavity. They found that, at model has been validated against the measured data of Kim et al.
a given Reynolds number, the particle concentration affects the [23] and the predicted values from Shah and London [24] for the
thermal behaviour and flow characteristic at larger Rayleigh num- base fluid. The effects of particle concentration, mean particle
bers. Moreover, the effect of particle concentration decreased with diameter, Reynolds number, Brownian motion, nanoparticle mate-
increasing Reynolds number. rial and shape, and type of base fluid on the heat transfer perfor-
Shahi et al. [17] reported a numerical investigation to simulate mance of nanofluids are then investigated in detail. Finally, some
the heat transfer performance of Cu/water nanofluid flowing guidelines related to choice of the appropriate nanofluid for partic-
through a square cavity under a laminar flow regime. Their results ular applications are provided.

3. 4378 E. Ebrahimnia-Bajestan et al. / International Journal of Heat and Mass Transfer 54 (2011) 4376–4388
2. Mathematical modelling thermal conductivity and viscosity of the nanofluids according to
the first method.
Normally, two different approaches can be used in predicting 0:0475745 0:42561
the heat transfer performance of nanofluids. One is the two-phase le ¼ À0:0001953 þ À
T T2
model, where solid particles are treated as a solid phase in the base
fluid, and the other is the single-phase model, where the presence
for 20 C 6 T 6 80 C ð6Þ
of the nanoparticles is accounted for by introducing the effective
thermophysical properties for the nanofluid. A single-phase model ke ¼ 0:65 þ 4:864 Â 10À7 T 3 for 22 C 6 T 6 52 C ð7Þ
is simpler to implement and requires less computational time, and where T is the temperature in Celsius. However, it should be men-
therefore, adopted here to describe the heat transfer characteristics tioned that above correlations are valid only for Al2O3/water nano-
of nanofluids flowing through a straight circular tube under con- fluid containing 3 vol.% of suspended particles.
stant wall heat flux and laminar flow regime, as shown in Fig. 1. As for the second method, the Vajjha et al.’s [25] model is taken
On the basis of this model, the governing equations are as follows. for viscosity with two parameters A and B, which are determined
For continuity equation: according to the experimental data of Kim et al. [23].
ZZ
qe ~ Á d~ ¼ 0:
V A ð1Þ le ¼ lf ðTÞA expðB/Þ ð8Þ
This correlation applies for 20 °C 6 T 6 90 °C and 1% 6 / 6 10%,
For momentum equation:
with A = 0.9 and B = 10.0359.
Z ZZ ZZ ZZ For thermal conductivity, the model presented by Koo and Kle-
@~
V ~q ~ Á d~ ¼ À
qe d8 þ V eV A p~ Á d~ þ
n A le r~ Á d~
~V A ð2Þ instreuer [26] and modified by Vajjha and Das [27] is employed.
8 @t
The model consists of static thermal conductivity based on the
For energy equation: Maxwell’s theory and dynamic thermal conductivity to include
Z ZZ ZZ Brownian motion of nanoparticles. This model is expressed as
@T
ðqC p Þe d8 þ ðqC p Þe T ~ Á d~ ¼
V A ke rT Á d~
~ A ð3Þ follows:
8 @t
kp þ 2kf À 2ðkf À kp Þ/
where ~ p, T, t, , ~ and ~ are the velocity vector, pressure, temper-
V, A n ke ¼ kf þ ð5 Â 104 b/qf C p;f Þ
kp þ 2kf þ ðkf À kp Þ/
ature, time, volume, cross-sectional area vector and normal unit sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
vector, respectively. The effective thermophysical properties of jðT þ 273:15Þ
 f ðT; /Þ ð9Þ
the nanofluid indicated by the subscript e are density (q), thermal qp dp
conductivity (k), dynamic viscosity (l), and heat capacity (Cp). These
effective properties are modelled as temperature dependent vari- ðT þ 273:15Þ
ables using available correlations and experimental data as follows. f ðT; /Þ ¼ ð2:8217 Â 10À2 / þ 3:917 Â 10À3 Þ
ðT 0 þ 273:15Þ
Density:
þ ðÀ3:0669  10À2 / À 3:91123  10À3 Þ ð10Þ
qe ð/; TÞ ¼ ð1 À /Þqf ðTÞ þ /qp ð4Þ
Heat capacity: b ¼ 8:4407ð100/ÞÀ1:07304 ð11Þ
ð1 À /ÞðqðTÞC p ðTÞÞf þ /ðqp C p Þp where j is the Boltzmann constant (1.381 Â 10À23 J/K), dp is the
C p;e ð/; TÞ ¼ ð5Þ nanoparticle diameter (m), and T is the temperature (°C). T0 = 0 °C
ð1 À /Þqf ðTÞ þ /qp
is the reference temperature. The correlation is valid for
where / is the volume fraction of the nanoparticles, and the sub- 25 °C 6 T 6 90 °C and the volume fractions between 1% and 10%
scripts p and f indicate the nanoparticle and base fluid, respectively. for Al2O3 nanofluids [27]. It should be mentioned that all thermo-
Regarding the dynamic viscosity and thermal conductivity of physical properties of the base fluid (lf, kf, Cp,f and qf) are consid-
nanofluids two different methods have been considered. In the first ered as functions of temperature. For water as the base fluid,
method, available measured data of these properties in static con- density, heat capacity and thermal conductivity are obtained from
dition are used. However, in the second method, available models the curve-fits applied to available data [28], while for dynamic vis-
for dynamic viscosity and thermal conductivity of nanofluids, cosity the following correlation is employed [29]:
where different effects such as base fluid type, Brownian motions,
lwater ¼ 0:00002414 Â 10½247:8=ðTþ133ÞŠ ð12Þ
volume concentration, and nanoparticle diameters are also in-
cluded are employed. Clearly, these models are designed to accu- The above methods for effective viscosity (Eqs. (6) and (8)) have
rately predict the heat transfer coefficient. been compared with measured data of Kim et al. [23] for 3.0 vol.%
In the present study, the measured data of Kim et al. [23] is used Al2O3/water nanofluids, where reasonable agreements are
to develop the following correlations for predicting the effective observed (Fig. 2). Similarly, the temperature variations of thermal
L=2m
Flow
D= 4.57 mm
D
x
q = const.
Fig. 1. Flow geometry and numerical grid distributions.

4. E. Ebrahimnia-Bajestan et al. / International Journal of Heat and Mass Transfer 54 (2011) 4376–4388 4379
conductivity according to Eqs. (7) and (9) have been compared walls. For the thermal field, constant heat flux of 60 W (2089.56 W/
with the measured data of Kim et al. [23] as shown in Fig. 3. It m2) and the inlet temperature of 22 °C are employed correspond-
can be seen that Eq. (9) over predicts the thermal conductivity as ing to the data of Kim et al. [23], while constant temperature gra-
compared to the measured data, however it can predict the heat dient is applied at the outlet.
transfer coefficient more accurately as compared to Eq. (7) as will In order to simulate the nanofluid flow, pipe geometry with
be discussed later. 4.57 mm in diameter and 2 m long has been adopted according
to the flow geometry in the experimental work of Kim et al. [23]
as shown in Fig. 1. The problem under investigation is steady, how-
3. Solution methodology ever, in the numerical scheme the steady solution is obtained
through sufficient integrations in time. The governing equations
In order to evaluate the heat transfer coefficient of nanofluids are solved numerically in a body-fitted coordinate system using a
numerically, the following assumptions are required. control-volume technique. The numerical solution is based on a
As for boundary conditions for velocity filed, a uniform velocity projection-type method which solves the flow field in two steps.
profile is assumed at the inlet, while zero gradients are applied to Firstly, an intermediate velocity field is obtained using the avail-
all hydrodynamic variables at the outlet, with no-slip condition at able pressure field. Secondly, velocity and pressure corrections
.0014
Measured data of Kim et al. [23]
Eq. (6)
.0012 Eq. (8)
.0010
Viscosity (Pa.s)
.0008
.0006
.0004
.0002
10 20 30 40 50 60 70 80 90
o
Temperature ( C)
Fig. 2. Comparison of the predicted viscosity using Eqs. (6) and (8) with the measured data.
.90
Measured data of Kim et al. [23]
Eq. (7)
.85 Eq. (9)
Thermal conductivity (W/mK)
.80
.75
.70
.65
.60
10 20 30 40 50 60 70
Temperature (oC)
Fig. 3. Comparison of the predicted thermal conductivity using Eqs. (7) and (9) with the measured data.

5. 4380 E. Ebrahimnia-Bajestan et al. / International Journal of Heat and Mass Transfer 54 (2011) 4376–4388
are calculated from a Poisson equation designed to satisfy the con- 4. Model validation
tinuity equation. The numerical scheme was originally developed
by Chorin [30] and improved further by Dwyer [31] as well as Ren- For validating the present numerical scheme, the axial varia-
ksizbulut and Niazmand [32]. tions of the local Nusselt number have been compared with the
Moreover, extensive computations have been performed to experimental data of Kim et al. [23] and the results of Shah and
identify the number of grid points that produce reasonably grid London [24] as shown in Fig. 5. A laminar water flow in a straight
independent results. In Fig. 4 the grid resolution effects on the axial circular pipe geometry, as mentioned above, has been considered
variations of the centreline velocity (nondimensionalized with the under the constant heat flux condition of 60 W (2089.56 W/m2)
inlet velocity) and the heat transfer coefficient for just four differ- at Reynolds number of 1620. The Shah and London [24] equa-
ent mesh distributions are presented. It is clear that the grid sys- tion for the axial variations of the local Nusselt number is
tem of 41 Â 50 Â 150 points in respective directions of radial, expressed as:
azimuthal, and axial adequately resolve the velocity and thermal 8
3:302xÀ1=3 À 1;
à xà 6 0:00005
fields with reasonable accuracy. Uniform grid spacing is used in
the azimuthal direction, while the expansion ratios of 1.15 and NuðxÞ ¼ 1:302xÀ1=3 À 0:5;
à 0:00005 xà 0:0015
:
1.02 are employed in the radial and axial directions. 4:264 þ 8:68ð103 xà ÞÀ0:506 expðÀ41xà Þ; xà 0:001
ð13Þ
(a) 2.0
1.8
Dimensionless velocity
r θ x
1.6 21 x 30 x 70
41 x 50 x 150
51 x 50 x 170
61 x 50 x 190
1.4
1.2
1.0
0.0 .5 1.0 1.5 2.0
x (m)
(b) 2500
Heat transfer coefficient (W/m K)
2
2000
r θ x
21 x 30 x 70
41 x 50 x 150
1500 51 x 50 x 170
61 x 50 x 190
1000
500
0.0 .5 1.0 1.5 2.0
x (m)
Fig. 4. Grid resolution effects on the axial variations of (a) dimensionless centreline velocity and (b) heat transfer coefficient for 6.0 vol.% Al2O3/water nanofluid at Reynolds
number of 1460.

6. E. Ebrahimnia-Bajestan et al. / International Journal of Heat and Mass Transfer 54 (2011) 4376–4388 4381
x=D
where xà ¼ Re Pr. Excellent agreements between the results can be Other numerical scheme validations for nanofluids can be found
observed in Fig. 5. in Ebrahimnia-Bajestan et al. [33], which are not repeated here for
For the validation of the effective nanofluid properties models, conciseness.
the convective heat transfer of nanofluid in a straight pipe has been
considered. Fig. 6 shows the axial variations of heat transfer coef-
ficient of Al2O3/water nanofluid containing 3.0 vol.% of nanoparti- 5. Results and discussion
cles for Reynolds number of 1460. The results indicate that the
second method (Eqs. (8) and (9)) agrees better with the measured As shown in Fig. 6, the second method of nanofluid properties
data of Kim et al. [23] as compared to the first method (Eqs. (6) and models predicts the heat transfer coefficient more accurately than
(7)). This implies that the changes in thermal conductivity of nano- the first method. In addition, the second method is more general
fluids at static condition cannot justify the enhancement in heat than the first method, which is only valid for 3.0 vol.% Al2O3/water.
transfer coefficient of nanofluids and a model, which adjusts the Therefore, the second method has been employed for the evalua-
thermal conductivity to accurately estimate the heat transfer coef- tions of the effective thermal conductivity and viscosity in all
ficient, is required. numerical computations reported here. Moreover, the effects of
30
Measured data of Kim et al. [23]
Present model
25
Shah and London [24]
20
Nusselt number
15
10
5
0
0.0 .5 1.0 1.5 2.0
x (m)
Fig. 5. Comparison of the axial variations of the Nusselt number with available data in literature at Reynolds number of 1620.
2000
Measured data of Kim et al. [23]
1800 Present model (Using Eqs. 6 7)
Present model (Using Eqs. 8 9)
Heat transfer coefficient (W/m K)
2
1600
1400
1200
1000
800
600
400
0.0 .5 1.0 1.5 2.0
x (m)
Fig. 6. Comparison of the local heat transfer coefficient with measured data of 3.0 vol.% Al2O3/water nanofluid at Reynolds number of 1460.

7. 4382 E. Ebrahimnia-Bajestan et al. / International Journal of Heat and Mass Transfer 54 (2011) 4376–4388
several parameters on the convective heat transfer characteristics 5.2. The effect of nanoparticle diameter
of nanofluids will be examined in detail.
The effect of nanoparticle diameter on the heat transfer perfor-
mance of nanofluid for Reynolds number of 1460 and heat flux of
5.1. The effect of nanoparticle concentration 2089.56 W/m2 is shown in Fig. 8. The results show that the nano-
fluid with smaller nanoparticle diameter slightly increases the heat
Fig. 7 shows the effect of particle volume concentration on the transfer coefficient as compared to the larger particle diameter,
heat transfer coefficient along the pipe for Reynolds number of especially at lower volume concentrations. In general, according
1460 and heat flux of 2089.56 W/m2. The results indicate that to Eq. (9) the nanoparticle diameter has a negative effect on the
the heat transfer coefficient of nanofluid increases with increasing thermal conductivity and consequently on the heat transfer
particle volume concentration. According to Eq. (9), nanofluids coefficient.
with higher particle concentrations have higher static and dynamic
thermal conductivities, which in turn increase the heat transfer 5.3. The effect of Reynolds number
coefficient. For example, in the case of 6.0 vol.%, the local heat
transfer coefficient is about 22% larger than pure water at the Fig. 9 presents the effect of Reynolds number on the heat
end of the pipe. transfer coefficient along the pipe for Al2O3/water nanofluid with
2000
water
1800 2.0 vol.%
Heat transfer coefficient (W/m K)
4.0 vol.%
2
6.0 vol.%
1600
1400
1200
1000
800
600 Particle type : Al2O3
Re = 1460
400
0.0 .5 1.0 1.5 2.0
x (m)
Fig. 7. Axial variations of heat transfer coefficient for different particle volume concentrations of Al2O3/water nanofluid at Reynolds number of 1460.
2000
dp = 20 nm dp = 100 nm
1800 2.0 vol.% 2.0 vol.%
Heat transfer coefficient (W/m K)
6.0 vol.% 6.0 vol.%
2
1600
1400
1200
1000
800
600 Particle type : Al2O3
Re = 1460
400
0.0 .5 1.0 1.5 2.0
x (m)
Fig. 8. Axial variations of heat transfer coefficient for different particle diameters and concentrations of Al2O3/water nanofluid at Reynolds number of 1460.

8. E. Ebrahimnia-Bajestan et al. / International Journal of Heat and Mass Transfer 54 (2011) 4376–4388 4383
particle diameter of 20 nm. The results indicate that the heat trans- with the new nanofluid thermal conductivity model. Fig. 10 com-
fer coefficient of nanofluid increases with increasing Reynolds pares the axial variations of the heat transfer coefficient of Al2O3/
number. This is due to the fact that higher Reynolds numbers lead water nanofluid with and without Brownian motion effects on
to higher velocity and temperature gradients at the pipe wall. thermal conductivity model for various particle concentrations at
Reynolds number of 1460 and particle diameter of 20 nm. It is
clearly seen that the Brownian motion has significant effects on
5.4. The effect of Brownian motion of nanoparticles the heat transfer coefficient of the nanofluids.
According to Eq. (9), the second term is related to the dynamic
thermal conductivity and reflects the Brownian motion effects on 5.5. The effect of nanoparticle material
the thermal characteristics of nanofluids. In order to study the ef-
fects of Brownian motion on the heat transfer performance of The nanoparticle material affects the nanofluid properties and
nanofluids, Eq. (9) without the dynamic thermal conductivity term consequently is an important factor in heat transfer performance.
is used to simulate the convective heat transfer. It means that the In order to study the effect of nanoparticle material, the CuO/water
classical two-phase model of Maxwell is employed and compared nanofluid is selected to compare with the results of the Al2O3/
2000
dp = 20 nm
1800 Particle type : Al2O3
Heat transfer coefficient (W/m K)
φ = 0 vol.% φ = 4.0 vol.%
2
1600
Re = 500 Re = 500
Re = 1000 Re = 1000
1400 Re = 1460 Re = 1460
1200
1000
800
600
400
0.0 .5 1.0 1.5 2.0
x (m)
Fig. 9. Axial variations of heat transfer coefficient for different Reynolds numbers and particle concentrations of Al2O3/water nanofluid.
2200
φ = 2.0 vol.% φ = 6.0 vol.%
2000 with Brownian effect with Brownian effect
without Brownian effect without Brownian effect
Heat transfer coefficient (W/m K)
1800
2
1600
1400
1200
1000
800
Re = 1460
600
Particle type : Al2O3
400
0.0 .5 1.0 1.5 2.0
x (m)
Fig. 10. Effects of Brownian motion on the local heat transfer coefficient as a function of particle concentration for Al2O3/water nanofluid at Reynolds number of 1460.

9. 4384 E. Ebrahimnia-Bajestan et al. / International Journal of Heat and Mass Transfer 54 (2011) 4376–4388
water nanofluid. For the viscosity of the CuO/water nanofluid, Eq. 5.6. The effect of base fluid properties
(8) is employed, where constants A = 0.9197 and B = 22.8539 are
obtained following Vajjha et al. [25]. The thermal conductivity of Another important factor in the heat transfer characteristics of
CuO/water is evaluated based on Eqs. (9) and (10), however, the nanofluids is the type of base fluid. In addition to water, a common
parameter b is determined from [27] as expressed below, which base fluid consists of 60% ethylene glycol and 40% water (60% EG/
is valid for 25 °C 6 T 6 90 °C and 1% 6 / 6 6%. water) by mass has also been considered. This kind of heat transfer
fluid is common in cold regions of the world because of its low
b ¼ 9:881ð100/ÞÀ0:9446 ð14Þ freezing point. The thermophysical properties of 60% EG/water as
functions of temperature are obtained from [34] as follows:
As shown in Fig. 11, CuO/water nanofluids give higher convec-
tive heat transfer coefficients than the Al2O3/water nanofluids. q60%EG=water ¼ À2:475 Â 10À3 T 2 À 0:35T þ 1090:93 ð15Þ
For example, the heat transfer coefficient of 4.0 vol.% CuO/water
nanofluids is greater than that of the 6.0 vol.% Al2O3/water nanofl- C p;60%EG=water ¼ 4:248T þ 3042:74 ð16Þ
uids. This is because of the fact that the thermal conductivity of
CuO nanoparticles is much larger than the thermal conductivity k60%EG=water ¼ À3:196 Â 10À6 T 2 þ 7:54 Â 10À4 T þ 0:34 ð17Þ
of Al2O3 nanoparticles.
2000
CuO Al2O3
2.0 vol.% 2.0 vol.%
1800 4.0 vol.% 4.0 vol.%
Heat transfer coefficient (W/m K)
6.0 vol.% 6.0 vol.%
2
1600
1400
1200
1000
800 dp = 20 nm
Re = 1460
600
0.0 .5 1.0 1.5 2.0
x (m)
Fig. 11. Effects of particle type and particle concentration on the local heat transfer coefficient at Reynolds number of 1460 and particle diameter of 20 nm.
2000
Water
1800 60%EG/Water
Heat transfer coefficient (W/m K)
4.0 vol.% (Al2O3 - Water nanofluid)
2
1600 4.0 vol.% (Al2O3 - 60%EG/Water nanofluid)
1400
1200
1000
800
600
dp = 40 nm
Re = 1460
400
0.0 .5 1.0 1.5 2.0
x (m)
Fig. 12. Axial variations of heat transfer coefficient for different base fluids at Reynolds number of 1460 and particle diameter of 20 nm.

10. E. Ebrahimnia-Bajestan et al. / International Journal of Heat and Mass Transfer 54 (2011) 4376–4388 4385
Table 1
Properties of CNT/water and TNT/water nanofluids.
Nanofluids Nanoparticles aspect ratio Nanoparticles concentration (vol.%) Viscosity (Pa s) Thermal conductivity (W/m K)
k = a + bT + cT2
a b c
CNT/water 100 0.0384 0.00308 51.88156 À0.35487 6.14 Â 10À4
TNT/water %10 0.6 0.0015 0.42548 6.4 Â 10À4 0
2000
0.0384 vol.% (CNT-Water)
1800 0.6 vol.% (TNT-Water)
2.0 vol.% (Al2O3-Water)
Heat transfer coefficient (W/m K)
2
4.0 vol.% (Al2O3-Water)
1600
4.0 vol.% (CuO-Water)
1400
1200
1000
800
600
Re = 1460
400
0.0 .5 1.0 1.5 2.0
x (m)
Fig. 13. Effects of particle shape on the local heat transfer coefficient at Reynolds number of 1460.
3135:6 Table 2
l60%EG=water ¼ 0:001 exp À 8:9367 ð18Þ Different nanofluids flow conditions simulated in the present study and their
T þ 273:15
corresponding temperature increase, pressure drop, and pumping power.
These correlations are valid for 0 °C 6 T 6 97 °C. Type of Re dp (nm) / D T DP Pump
According to Vajjha and Das [27], Eqs. (8)–(11) are still applica- working fluids (vol.%) (°C) (Pa) power (W)
ble for the evaluation of viscosity and thermal conductivity of 60% Al2O3/60% EG/water 1460 20 4 0.56 43240 1.38
EG/water base nanofluids. 60% EG/water 1460 – – 0.68 26240 0.69
CNT/water 1460 Lp/dp 100 0.038 0.9 9625 0.16
In Fig. 12, the heat transfer coefficients of Al2O3 nanoparticles in
CuO/water 1460 20 6 1.05 8838 0.12
both water and 60% EG/water base fluids are compared. The results CuO/water 1460 20 4 1.52 3830 0.04
indicate that the heat transfer characteristics of nanofluids are TNT/water 1460 dp = 10, Lp = 100 0.6 1.8 2241 0.017
strongly influenced by the type of the base fluid. Al2O3/water 1460 100 6 2.05 2020 0.014
Al2O3/water 1460 40 6 2.06 2020 0.014
Al2O3/water 1460 20 6 2.08 2020 0.014
5.7. The effect of nanoparticle shape CuO/water 1460 20 2 2.18 1670 0.011
Al2O3/water 1460 40 4 2.39 1416 0.008
Al2O3/water 1460 20 4 2.39 1417 0.008
It has been shown that nanofluids containing nanoparticles Al2O3/water 1460 20 3 2.52 1188 0.007
with a higher aspect ratio have better thermal properties [35,36]. Water 1620 – – 2.6 974 0.005
For example, the cylindrical nanoparticles give higher thermal con- Al2O3/water 1460 20 2 2.72 996 0.005
ductivity and heat transfer coefficient than the spherical nanopar- Al2O3/water 1460 100 2 2.72 995 0.005
Al2O3/water 1460 40 2 2.73 995 0.005
ticles. To study the effect of nanoparticle shape, two kinds of Water 1460 – – 2.78 869 0.004
cylindrical nanoparticles, which are CNT and TNT are used and Al2O3/water 1000 20 4 3.44 936 0.004
then compared with the nanofluids with spherical nanoparticles. Water 1000 – – 4.08 572 0.002
He et al. [35] studied the heat transfer performance of CNT/water Al2O3/water 500 20 4 6.91 438 0.0009
Water 500 – – 8.19 266 0.0004
nanofluid containing 0.1 wt.% (0.0384 vol.%) and TNT/water nano-
fluid containing 2.5 wt.% (0.6 vol.%) numerically and experimen-
tally. They indicated that, at a given temperature, the shear
viscosity of both nanofluids decreases with increasing shear rate, water and TNT/water nanofluids, respectively. They simulated the
exhibiting a shear thinning behaviour. Moreover, the shear viscos- convective heat transfer of nanofluids as both Newtonian and
ity approaches a constant minimum value at higher shear rate. The non-Newtonian fluids. Their results showed that the heat transfer
measured viscosity did not show an important change versus share coefficients based on the non-Newtonian model agree well with
rate, more than the shear rates of 200 1/s and 500 1/s for the CNT/ the measured data. Furthermore, they indicated that, when the

11. 4386 E. Ebrahimnia-Bajestan et al. / International Journal of Heat and Mass Transfer 54 (2011) 4376–4388
nanofluid is considered as Newtonian with the constant minimum On the other hand, because of the low aspect ratio of TNT (%10)
value of viscosity, the heat transfer coefficient has also reasonable and its lower thermal conductivity as compared with Al2O3, CuO
agreement with the result of non-Newtonian model. and CNT, the heat transfer coefficient of TNT/water nanofluid is
In the present work, the constant minimum value of viscosity lower than those. Yet, it should be mentioned that the concentra-
presented by He et al. [35] was adopted as the effective viscosity. tion of TNT is lower than Al2O3, CuO. The TNT/water nanofluid con-
Similarly, the thermal conductivity of nanofluids as a function of taining 0.6 vol.% nanoparticle shows approximately the same heat
temperature is taken from [35] as listed in Table 1. transfer coefficient as that of Al2O3/water nanofluid containing
Fig. 13 presents the comparison of axial heat transfer coeffi- 2 vol.%, which also indicates the considerable effect of nanoparticle
cients for nanoparticles with different aspect ratios at Reynolds shape on the heat transfer characteristics of nanofluids.
number of 1460. The results show that the heat transfer coefficient Thus far, several influential parameters on the heat transfer of
of the CNT/water nanofluid is significantly greater than that of the nanofluids are investigated; yet the important question is which
other nanofluids, which can be attributed to its large aspect ratio nanofluid is more appropriate for a specific application. In this arti-
(100) as listed in Table 1. In addition to the nanoparticle shape, cle, considering a constant wall heat flux condition, two important
the higher thermal conductivity of CNTs compared with Al2O3 factors are considered as the criteria for the choice of proper nano-
and CuO is the reason of substantial thermal characteristics fluids. These factors are pressure drop and temperature difference
enhancement of CNT/water nanofluids, even at low concentrations. along the pipe. For a constant wall heat flux condition with a given
2000
Particle type : Al2O3
Re = 1460
dP = 20 nm 1800
2.6
1600
ΔP (Pa)
ΔT ( C)
o
ΔT
2.4 1400
ΔP
1200
2.2
1000
0 1 2 3 4 5 6
Particle volume fraction (%)
Fig. 14. Temperature differences and pressure drops for different particle volume concentrations of Al2O3/water nanofluid at Reynolds number of 1460 and particle diameter
of 20 nm.
Particle type : CuO
2.6 8000
Re = 1460
dP = 20 nm
2.4
2.2 6000
ΔP (Pa)
ΔT ( C)
2.0
o
ΔT
1.8 ΔP
4000
1.6
1.4
2000
1.2
0 1 2 3 4 5 6
Particle volume fraction (%)
Fig. 15. Temperature differences and pressure drops for different particle volume concentrations of CuO/water nanofluid at Reynolds number of 1460 and particle diameter
of 20 nm.

12. E. Ebrahimnia-Bajestan et al. / International Journal of Heat and Mass Transfer 54 (2011) 4376–4388 4387
amount of heat transfer to the fluid, the lower temperature differ- motion, particle diameter, particle shape, particle material and
ence along the pipe can be considered as an advantage in some type of base fluid on the heat transfer performance of nanofluids
applications. On the other hand, lower pressure drop reduces the are examined in details. Major findings can be summarised as
pumping cost. follows:
In the view of these two criteria, all cases simulated in the pres-
ent study are listed in Table 2, which have been sorted by pressure – The predicted heat transfer coefficients based on the thermo-
drop. It can be concluded from the table that almost all cases with physical properties of nanofluids in a static condition are lower
lower pressure drops have higher temperature differences, which than the experimental data. This means that the static thermal
also implies that nanofluids should be selected according to the conductivity increase is not the only reason of convective heat
application requirements. transfer enhancement of nanofluids.
From Table 2 it is observed that considerable pressure drop are – A thermal conductivity model which considers the effect of
associated with high particle concentration of nanofluids, while Brownian motion of nanoparticles predicts the heat transfer
much less pressure drop has been reported in numerical studies coefficient of nanofluids more accurately as compared to the
of [37,38] for similar flow conditions. It must be emphasised that models based on the pure static conditions of the nanofluids.
the viscosity model, le ¼ lf ð1 À /ÞÀ2:5 , which has been used in – The particle volume concentration, Brownian motion and aspect
these studies produces much lower values for viscosity as com- ratio of nanoparticles similar to the flow Reynolds number
pared to the present experimental data [23] and consequently increase the heat transfer coefficient of nanofluids, while the
leads to relatively lower pressure drops. However, the applied vis- particle diameter has an opposite effect. Moreover, the heat
cosity model in the present study (Eq. (8)) represents the experi- transfer characteristics of nanofluids are strongly influenced
mental data with reasonable accuracy and therefore, the by the type of both base fluid and nanoparticle.
resulting pressure drops are higher. – There are several parameters to select the proper nanofluids for
Another related issue to the pressure drop is the relative in- convective heat transfer of nanofluids in pipes. These parame-
crease in viscosity with respect to thermal conductivity as the ters are the amount of heat transfer enhancement, pump power,
nanoparticle concentration increases. In general, the advantage of stability, cost, toxicity and chemical corrosion of pipe wall. Also,
the nanofluids for the heat transfer applications is directly related there are some other restrictions in the special applications
to the amount of the thermal conductivity increase, leading to the such as low freezing temperature nanofluids as the anti-refrig-
heat transfer enhancement, to the amount of the viscosity increase erant fluids in cold region.
associated with higher pressure drops. For the nanofluids used in
the present study, the viscosity of the base fluid increases faster
than its thermal conductivity as nanoparticle concentration in- Acknowledgements
creases, and therefore, the heat transfer enhancements come at
the expense of relatively considerable pressure drop. The authors wish to thank Ferdowsi University of Mashhad
Considering Table 2, the temperature increase along the pipe (Iran), South-East Asia University (Thailand), King Mongkut’s Uni-
and the pressure drop behave in an opposite manner. In Figs. 14 versity of Technology Thonburi (Thailand) for the valuable support
and 15 these parameters are plotted versus the particle volume in the present study. The fourth author would like to acknowledge
fraction to see if an optimum nanoparticle concentration for a spe- the Thailand Research Fund and the National Research University
cific application can be identified. Fig. 14 shows the Al2O3 nanopar- Project for financial supporting.
ticles in water with particle diameter of 20 nm and Reynolds
number of 1460. From this figure, the optimum choice for the References
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