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International Journal of Heat and Mass Transfer 73 (2014) 483–491
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ijhmt
Field-synergy analysis of viscous dissipative nanoﬂuid ﬂow in
microchannels
Tiew Wei Ting, Yew Mun Hung ⇑, Ningqun Guo
School of Engineering, Monash University, 46150 Bandar Sunway, Malaysia
a r t i c l e
i n f o
Article history:
Received 28 October 2013
Received in revised form 2 February 2014
Accepted 17 February 2014
Available online 14 March 2014
Keywords:
Field synergy
Microchannel heat sink
Nanoﬂuid
Viscous dissipation
a b s t r a c t
Field-synergy analysis is performed on the viscous dissipative water–alumina nanoﬂuid ﬂow in circular
microchannel heat sinks to scrutinize the synergetic relation between the ﬂow and temperature ﬁelds for
both heating and cooling processes. By varying the Reynolds number and the nanoparticle volume fraction, the effect of viscous dissipation in the convective heat transfer of nanoﬂuid is investigated based on
the ﬁeld synergy principles. For the heating process, under the effect of viscous dissipation, the degree of
synergy between the velocity and temperature ﬁelds of nanoﬂuid ﬂow deteriorates, leading to a dwindled
heat transfer performance of the nanoﬂuid. Due to the presence of two difference heat sources, the synergy angle and ﬁeld synergy number for viscous dissipative ﬂow are deﬁned to characterize the synergetic behavior. The ﬁeld synergy in the nanoﬂuid is greater at low-Reynolds-number ﬂow due to the
reduced effect of viscous dissipation. By reducing the size of the nanoparticle and increasing the diameter
of the microchannel, the degree of synergy between the velocity and temperature ﬁelds of the nanoﬂuid
ﬂow in microchannel can be intensiﬁed, yielding convection heat transfer enhancement. On the other
hand, for nanoﬂuid cooling process, the presence of viscous dissipation would augment the ﬁeld synergy
of the ﬂow.
Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
The enhancement of convective heat transfer has been an everlasting research area aiming to create high thermal performance
and energy efﬁcient devices. Various techniques are utilized to enhance the convective heat transfer such as employing ﬁn attachments, rough surfaces or inserts, by increasing the heat transfer
area or the heat transfer coefﬁcient. However, there is no uniﬁed
theory to understand the essence of heat transfer enhancement
of these techniques. Guo et al. [1] proposed a novel concept known
as the ﬁeld synergy principle [2] to analyze the mechanism of convective heat transfer based on the ﬂow and temperature ﬁelds, as
well as the synergy between them. This principle reveals that the
convective heat transfer can be enhanced by reducing the intersection angle between the velocity and the temperature gradient vectors. The ﬁeld synergy number was later introduced to indicate the
degree of synergy between the velocity and temperature ﬁelds of
the ﬂow [3]. The ﬁeld synergy principle was ﬁrst proposed based
on the parabolic ﬂuid ﬂow [1] and later its application was also validated in elliptical ﬂow condition [2]. Various simulations and
⇑ Corresponding author. Tel.: +60 3 5514 6251; fax: +60 3 5514 6207.
E-mail address: [email protected] (Y.M. Hung).
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.02.041
0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
experimental studies have been conducted to verify the signiﬁcance of the synergetic relation between the ﬂow and temperature
ﬁelds in characterizing the convective heat transfer of the ﬂow
[4–15]. Hence, the ﬁeld synergy principle serves as an effective tool
in the design of heat transfer devices involving convective heat
transfer.
The micro-scale convective heat transfer has been a topical subject due to the miniaturization of the electronic devices. The thermal management in such devices has become a challenging issue.
Due to its large area-to-volume characteristic, microchannel heat
sink appears to be one of the potential solutions to increase the
thermal performance in a conﬁned space. However, the choice of
working ﬂuid is also of great importance. The low thermal conductivity of the ﬂuid poses a primary limitation to the development of
high-performance heat transfer ﬂuid. By suspending ultra-ﬁne
nanoparticle in the conventional ﬂuid, the effective thermal conductivity of ﬂuid is increased tremendously even with a small volume fraction of the nanoparticle, where Choi [16] ﬁrst coined this
kind of ﬂuid as ‘‘nanoﬂuid’’. Nanoﬂuids have been reported to be
able to enhance the convective heat transfer of the ﬂow [17,18].
However, the heat transfer performance of nanoﬂuid ﬂow in the
microchannel heat sink deteriorates when the effect of viscous dissipation is signiﬁcant [19]. In low Peclet number ﬂow, the suspension of nanoparticle increases the effect of streamwise conduction
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T.W. Ting et al. / International Journal of Heat and Mass Transfer 73 (2014) 483–491
Nomenclature
Br0
Ck
Cl
cp
D
dp
Fc
k
Kn
L
Pe
Pr
qw
r
r9
Re
T
u
modiﬁed Brinkman number
thermal conductivity ratio
viscosity ratio
speciﬁc heat (J kg1 K1)
inner diameter of microchannel (m)
diameter of nanoparticle (m)
ﬁeld synergy number
thermal conductivity (W m1 K1)
Knudsen number
length of the microchannel (m)
Peclet number
Prandtl number
heat ﬂux (W m2)
radial coordinate (m)
inner radius of microchannel (m)
Reynolds number
temperature (K)
ﬂuid velocity (m s1)
and intensiﬁes the entropy generation in nanoﬂuid in the presence
of streamwise conduction [20,21]. Mah et al. [22] reported that
both the thermal performance and the exergetic effectiveness of
the viscous dissipative nanoﬂuid ﬂow in microchannel dwindle
with nanoparticle volume fraction. In addition to the heat transferred from the channel wall, viscous dissipation manifests itself
as a heat source in the ﬂuid ﬂow producing appreciable rise in
the ﬂuid temperature due to the conversion of kinetic motion of
the ﬂuid to thermal energy. Such effect is more signiﬁcant in
microchannel due to its large length-to-diameter ratio, leading to
drastic changes in the ﬂow and temperature ﬁelds of the ﬂuid.
The effect of viscous dissipation is further enhanced in ﬂuids of
low speciﬁc heat and high viscosity, such as nanoﬂuids of which
the speciﬁc heat is reduced and the viscosity is increased due to
the suspension of solid nanoparticle in the ﬂuid [23]. Judging from
this, it is instructive to investigate the synergy between the velocity and temperature ﬁeld of nanoﬂuid ﬂow under the effect of viscous dissipation.
The ﬁeld-synergy analysis on nanoﬂuid ﬂow is not available in
the up-to-date literature. The present study, a basic investigation
in ﬁlling this gap, emphasizes the investigation of the synergetic
relationship between the velocity and the temperature gradient
ﬁelds of the nanoﬂuid ﬂow by incorporating the viscous dissipation
effect. The deterioration of the heat transfer performance in viscous dissipative nanoﬂuid ﬂow as reported in [19,22] is analyzed
from the ﬁeld-synergy point of view. An analytical model is developed based on the ﬁrst-law principles for fully developed nanoﬂuid
ﬂow in a circular microchannel heat sink under the uniform wall
heat ﬂux condition by considering the viscous dissipation effect.
The effects of viscous dissipation on forced convection of nanoﬂuid
heating and cooling are scrutinized. In addition, the effects of
nanoparticle suspension and microchannel’s geometry on the ﬁeld
synergy of the ﬂow are analyzed and discussed.
2. Mathematical formulation
x
longitudinal coordinate (m)
Greek symbols
b
synergy angle
/
nanoparticle volume fraction
h
dimensionless temperature
l
dynamic viscosity (N s m2)
q
density (kg m3)
k
molecular mean free path (m)
Subscripts
eff
effective
f
of base ﬂuid
nf
of nanoﬂuid
p
of nanoparticle
vd
of viscous dissipation
1
of Model 1
2
of Model 2
thermal conductivity of water–Al2O3 nanoﬂuid can be estimated
as [24]
keff ¼ C k kf ;
ð1Þ
where Ck is a constant coefﬁcient deﬁned as
1=3
C k ¼ ð1 þ ARem
b Prf /Þ
jð1 þ 2aÞ þ 2 þ 2/½jð1 aÞ 1
:
jð1 þ 2aÞ þ 2 /½jð1 aÞ 1
ð2Þ
The parameter j = kp/kf is the thermal conductivity ratio of the particle thermal conductivity kp to the base ﬂuid thermal conductivity
kf, Prf = cp,flf/kf is the Prandtl number of the ﬂuid, / is the
nanoparticle volume fraction, a = 2Rbkf/dp is the nanoparticle Biot
number, where dp is the nanoparticle diameter and Rb is the interfacial resistance. The Brownian–Reynolds number is deﬁned as
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Reb ¼ 18kb T=pqp dp =mf , with kb as the Boltzmann constant and
mf as the ﬂuid kinematic viscosity. For the suspension of Al2O3 nanoparticles in water, it is reported that Rb = 0.77 108 m2KW1,
m = 2.5 and A = 40,000 [24].
The effective viscosity of water–Al2O3 nanoﬂuid can be modeled
using a modiﬁed Einstein model by taking into account the slip
velocity between the nanoparticles and the base ﬂuid, given by
[25]
leff ¼ C l lf ;
ð3Þ
where lf is the dynamic viscosity of the base ﬂuid and the ratio Cl is
deﬁned as
"
#
2e
dp
2=3ðeþ1Þ
:
C l ¼ ð1 þ 2:5/Þ 1 þ g
/
D
ð4Þ
In Eq. (4), e = 1/4 and g = 280 are the empirical constants for Al2O3
nanoparticles and D is the inner diameter of microchannel. Eqs. (2)
and (4) can be easily replaced with other felicitous correlations
associated with the suspension of nanoparticle in the conventional
ﬂuid. The effective density of nanoﬂuid is taken on the basis of volume average as [26]
2.1. Thermophysical properties of nanoﬂuid
qnf ¼ qf ð1 /Þ þ qp /;
Nanoﬂuid exhibits distinctive thermophysical properties from
the conventional ﬂuid due to the suspension of ultra-ﬁne nanoparticles in the base ﬂuid. Water–Al2O3 nanoﬂuid is selected in the
present study. By taking into account the Brownian motioninduced convection from multiple nanoparticles, the effective
while the effective speciﬁc heat, which is more accurate on the basis
of mass average, is given by Bergman [27]
cp;nf ¼
qf cp;f ð1 /Þ þ qp cp;p /
:
qf ð1 /Þ þ qp /
ð5Þ
ð6Þ
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2
^
1 d
dh
du
^ Br0
¼ ð1 þ 8Br0 Þ u
R
;
R dR
dR
dR
2.2. First-law formulation
The nanoﬂuid ﬂow with low concentration of nanoparticle can
be treated as a continuous medium and the continuum modeling
approach is applicable for channels larger than 1 lm [28]. Viscosity
measurements show that most of the nanoﬂuids manifest themselves as Newtonian ﬂuids [29,30]. For a steady axisymmetric ﬂow
in a microchannel with internal radius ro, the equation of motion
can be written as
1 @
@u
1 @p
¼
:
r
r @r
@r
leff @x
ð7Þ
"
ð8Þ
is the mean velocity of the nanoﬂuid over the cross section
where u
area of the microchannel. The ﬂuid phase and the nanoparticle are
assumed to be in thermal equilibrium with constant thermophysical properties and the energy equation is given by
2
@T k @
@T
@u
þ leff
r
qnf cp;nf u ¼ eff
;
@x
@r
@r
r @r
ð9Þ
where T is the temperature of the nanoﬂuid. The viscous dissipation
term is incorporated in the energy equation as the second term of
the right hand side of Eq. (9). Uniform heat ﬂux qw is applied on
the microchannel wall and the radial thermal boundary conditions
are expressed as
@T keff ¼ qw ;
@r r¼ro
@T ¼ 0;
@r r¼0
ð10Þ
For fully developed ﬂow with isoﬂux boundary condition in the
present study, the temperature gradient along the axial direction
is a constant given by
@T dT
¼
;
@x dx
ð11Þ
where T is the bulk mean temperature deﬁned as
T¼
2
r 20 u
Z
ro
uTr dr:
ð12Þ
dhð0Þ
¼ 0:
dR
hð1Þ ¼ 0;
ð18Þ
Substituting the dimensionless velocity proﬁle into Eq. (17) and
solving it with Eq. (18) yields the closed-form dimensionless temperature proﬁle in the radial direction as
ð19Þ
Eq. (19) is substituted into the non-dimensional form of Eq. (15) to
obtain the dimensionless temperature proﬁle as
h1 ðR; XÞ ¼
2LX 1 2
1
þ R ð1 þ 8Br0 Þ ð1 þ 16Br0 ÞR4
ro Pe 2
8
3
16 0
1þ
Br
8
3
ð20Þ
where X = x/L is the dimensionless axial distance, L is the micro D=leff is
channel length, Pe = Re Pr is the Peclet number, Re ¼ qnf u
the Reynolds number and Pr = cp,nfleff/keff is the Prandtl number.
For the case where the viscous dissipation effect is neglected, i.e.
Br0 = 0, the dimensionless temperature proﬁle becomes
h2 ðR; XÞ ¼
2LX 1 4 1 2 3
R þ R :
ro Pe 8
2
8
ð21Þ
For the purpose of comparison, the case where the viscous dissipation effect is incorporated in the energy equation is denoted as Model
1 (with subscript 1) while the case where the viscous dissipation
effect is neglected is denoted as Model 2 (with subscript 2). The Nusselt number characterizing the heat transfer rate between the microchannel wall and the water–Al2O3 nanoﬂuid can be derived as
hD
48
¼
:
keff 48Br0 þ 11
Nu ¼
ð22Þ
where h ¼ qw =ðT w TÞ is the heat transfer coefﬁcient between the
wall and the nanoﬂuid.
0
In Eq. (12), T is independent of the radial direction. Integrating Eq.
(9) over the cross section of the microchannel yields
@
qnf cp;nf
@x
Z
0
ro
r
Z ro 2
@T o
@u
uTr dr ¼ keff r þ leff
r dr:
@r 0
@r
0
dT
2
2 Þ ¼ w ¼ constant;
¼
ðq r þ 4leff u
r 2o w o
dx qnf cp;nf u
qnf cp;nf
keff ðT T w Þ
h¼
;
qw D
Z
ro
ðV rTÞrdr ¼ keff
Z
0
0
ro
ðr2 TÞrdr þ leff
Z
ro
0
2
@u
rdr:
@r
By utilizing Eqs. (8) and (10), Eq. (23) is reduced to
qnf cp;nf
ð16Þ
where Tw is the temperature of the microchannel wall, Eq. (9) can
be nondimensionalized as
Z
0
ro
2 :
ðV rTÞrdr ¼ r o qw þ 4leff u
ð24Þ
By introducing the dimensionless vectorial variables
V ¼
ð15Þ
where f(r) can be obtained by solving Eq. (9). By utilizing Eq. (14)
and the following dimensionless variables
u
^¼ ;
u
u
The vector form of energy equation, when integrated over the
cross section of microchannel, can be expressed as
ð23Þ
ð14Þ
which demonstrates that the axial temperature gradient is inherently a constant. Following this, it can be deduced from Eqs. (11)
and (14) that the nanoﬂuid temperature distribution takes the form
of
Tðr; xÞ ¼ f ðrÞ þ wx;
2.3. Field-synergy principle
ð13Þ
By utilizing Eqs. (8), (10), and (11), Eq. (13) can be simpliﬁed as
r
R¼ ;
r0
2 =qw D is the modiﬁed Brinkman number based on
and Br0 ¼ leff u
the isoﬂux condition. The dimensionless boundary conditions can
be written as
1
1
3
16 0
1þ
Br :
hðRÞ ¼ ð1 þ 16Br0 ÞR4 þ ð1 þ 8Br0 ÞR2 8
2
8
3
The velocity distribution yields the Hagen–Poiseuille proﬁle as
2 #
r
;
u ¼ 2u 1 ro
ð17Þ
V
;
u
rT ¼
rT
;
ðT w TÞ=r o
ð25Þ
and applying the deﬁnition of Nusselt number as shown in Eq. (22),
Eq. (24) can be written as
Pe
Z
1
ðV rT ÞRdR ¼ Nuð1 þ 8Br0 Þ;
ð26Þ
0
where the viscous dissipation term appears as the second term on
the right hand side of the equation. In Eq. (26), the dot product of
the dimensionless velocity and temperature gradient vectors is
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T.W. Ting et al. / International Journal of Heat and Mass Transfer 73 (2014) 483–491
V rT ¼ jV jjrT j cos b;
ð27Þ
⁄
⁄
where b is the synergy angle between the vectors V and rT . In order to enhance the convective heat transfer in the ﬂow, the convection term on the left hand side of Eq. (26) needs to be as large as
possible. Hence, the synergy angle b has to be as small as possible
in order to obtain cos b 1, which is the ideal case where the velocity and temperature gradient vectors are in full coordination with
each other. The synergy angle can be derived as
b ¼ cos1
0
1
V rT @T=@x
B
C
¼ cos1 @qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃA;
jV jjrT j
ð@T=@xÞ2 þ ð@T=@rÞ2
ð28Þ
which is a function of both the axial and radial temperature gradient of the nanoﬂuid. Eq. (28) can be nondimensionalize as
0
1
0
1 þ 8Br
B
C
b ¼ cos1 @qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃA:
2
2
0 2
ð1 þ 8Br Þ þ ðPe=2Þ ð@h=@RÞ
ð29Þ
Evidently, b varies only along the radial direction due to the fact
that the axial temperature gradient is a constant. The integral in
Eq. (26) is denoted as ﬁeld synergy number which represents the
degree of synergy between the velocity and temperature gradient
ﬁelds for the entire domain, given by
Fc ¼
Z
1
ðV rT ÞRdR ¼
0
Nuð1 þ 8Br0 Þ
:
Pe
ð30Þ
The ﬁeld synergy number which is proportional to the Nusselt number is associated with the enhancement of the convection heat
transfer.
3. Results and discussion
The input parameters of the water–Al2O3 nanoﬂuid used for the
computation are listed in Table 1. We have assumed that the nanoﬂuid in the microchannel is regarded as a continuum. This assumption is validated with the Knudsen number which is deﬁned as
Kn ¼
k
;
D
in water. For Re = 500, the water–Al2O3 nanoﬂuid with / = 4% induces a viscous dissipative heat rate which is 2.7 times of the heat
input from the channel wall. This is more than sufﬁcient to trigger
motivation to investigate the effect of viscous dissipation on the
nanoﬂuid ﬂow ﬁeld synergy due to the alteration of the velocity
and temperature ﬁelds induced by the nanoparticle suspension.
Fig. 1(a) and (b) illustrates the isotherms and streamlines of
water–Al2O3 nanoﬂuid ﬂow with / = 8% and Re = 2000 for Model
1 (with viscous dissipation) and Model 2 (without viscous dissipation), respectively. Ideally, vertical isotherms indicate the best synergy between the velocity and temperature ﬁeld of the ﬂow. In
fully developed ﬂow, the velocity proﬁle is not changing in the
streamwise direction and the streamlines are parallel to the axial
direction of the channel. The radial temperature gradient at the
center of the channel is zero for the case of fully developed ﬂow
with uniform wall heat ﬂux condition. Both models show that
the isotherms are in orthogonal to the streamlines at R = 0, indicating a good synergy of the velocity and temperature ﬁeld at the center of the channel. Furthering away from the proximity of the
center of the channel, the isotherms lose the orthogonality with respect to the streamlines, showing reduced synergy of the ﬂow and
temperature ﬁelds in the region. However, when the effect of viscous dissipation is considered, the isotherms at the channel wall
regain the orthogonality with respect to the streamlines with a
decreasing trend of the synergy angle as illustrated in Fig. 1(a).
Fig. 2 depicts the synergy angle between the velocity vector and
the temperature gradient vector of water–Al2O3 nanoﬂuid ﬂow
with / = 8% and Re = 2000. The angle is relatively small at the center of the channel, implying the synergy between the velocity and
the temperature ﬁelds is optimal at this location. Further away
from the center, the degree of synergy decreases as the angle increases. When R 1, the effect of viscous dissipation reduces the
synergy angle of Model 1 in this region, while this phenomenon
is not observed in Model 2. Fig. 2 concurs with the isotherm and
the streamline plots in Fig. 1 with a lower synergy angle for Model
ð31Þ
where k is the ﬂuid molecule mean free path. The effective size and
the mean free path of the water molecules are both of the same order of 0.3 nm [28]. Similarly, the mean free path of the Al2O3 nanoparticle is regarded to be of the same order with its effective
diameter of 60 nm. For the microchannel of interest with an inner
diameter of 50 lm, the Knudsen number is relatively small with
the value of Kn = 1.2 103, justifying the continuum assumption.
In the present study, the effect of viscous dissipation in the microchannel is augmented due to the suspension of Al2O3 nanoparticles
Table 1
Input parameters used in the analysis.
Parameters
Values
Bulk mean temperature, T
Uniform heat ﬂux
Channel inner diameter
Channel length
Particle diameter
Base ﬂuid density
Base ﬂuid conductivity
Base ﬂuid viscosity
Base ﬂuid speciﬁc heat capacity
Particle density
Particle conductivity
Particle speciﬁc heat capacity
300 K
1 105 W/m2
50 lm
0.02 m
60.4 nm
997 kg/m3
0.613 W/mK
8.55 104 Ns/m2
4179 J/kg K
3975 kg/m3
36 W/m K
778.6 J/kg K
Fig. 1. The isotherms and streamlines for fully developed water–Al2O3 nanoﬂuid
ﬂow with / = 8% and Re = 2000 for (a) Model 1, and (b) Model 2.
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excluded the viscous dissipation. Following this, a synergy angle
for viscous dissipative ﬂow is derived as
0
1
1
B
C
b0 ¼ cos1 @qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃA:
1 þ ðPe=2Þ2 ð@h=@RÞ2
ð34Þ
The corresponding average synergy angle over the cross section is
given by
b0 ¼
Fig. 2. Local synergy angle of water–Al2O3 nanoﬂuid ﬂow with / = 8% and
Re = 2000.
1, indicating that the heat transfer rate is enhanced in the viscous
dissipative nanoﬂuid ﬂow. This ﬁnding contradicts the results reported in [19,22], where a lower heat transfer rate is observed with
the incorporation of the viscous dissipation effect. In such case, the
synergy angle of Model 1 is expected be higher than that of Model
2.
This contradiction can be explained from Eq. (26) that the convection term on the left-hand side is associated with the combined
effects of two heat sources, i.e. the wall heat ﬂux and the viscous
dissipation, which are the ﬁrst and the second terms on the
right-hand side of the equation, respectively. It is worth pointing
out that the ﬁeld synergy principle correlates the velocity and
the temperature gradient vectors in the convection term to a heat
source term. Conventionally, without the viscous dissipation, the
heat source is comprised of the wall heat ﬂux transported from
the microchannel wall to the nanoﬂuid. The convection heat transfer and the ﬁeld synergy can be enhanced by raising the strength of
the heat source term [1]. The viscous dissipation acts as an extra
heat source and the magnitude of the heat source term is increased. This leads to a decrease in the synergy angle, contradicting
the ﬁndings of the previous studies [19,22]. Therefore, by incorporating the viscous dissipation effect, we redeﬁne the synergy angle
and the ﬁeld synergy number in the next section that follows.
Z
1
Ac
b0 dAc ¼ 2
Ac
Z
1
b0 RdR;
ð35Þ
0
And it is evaluated numerically using the Simpson’s rule for the
sake of simplicity and accuracy. By recasting Eq. (33), the ﬁeld synergy number for viscous dissipative ﬂow can be obtained as
Fc0 ¼
Z
0
1
½V ðrh rhvd ÞRdR ¼
Nu
:
Pe
ð36Þ
It is evident that when Br0 = 0, the synergy angle and ﬁeld synergy
number for viscous dissipative ﬂow are reduced to their respective
original values in Eqs. (29) and (30), respectively. When Br0 > 0, the
effect of viscous dissipation is incorporated in the temperature distribution and the Nusselt number which are functions of Br0 .
Fig. 3(a) and (b) illustrates the variations of synergy angle for
viscous dissipative ﬂow, b0 , in the radial direction for the ﬂow with
/ = 0% and / = 8, respectively, with Reynolds number as a parameter. For both cases, the b0 values of Model 1 is greater than those
of Model 2, indicating that the presence of viscous dissipation
attenuates the strength of the synergy between the velocity and
temperature ﬁelds as well as the convection heat transfer. This
concurs with the ﬁndings reported in [19,22]. The use of the synergy angle for viscous dissipative ﬂow is appropriate to correlate
the convection term in viscous dissipative ﬂow. It is observed that
the deviation between the two models is more signiﬁcant in the
proximity of the center of the channel. The viscous dissipation effect incurs higher b0 in the region of R 0 due to the superposition
3.1. The synergy angle and ﬁeld synergy number for viscous dissipative
ﬂow
In order to relate the ﬁeld synergy analysis to the Nusselt number, the viscous dissipation term is combined with the convection
term on the left-hand side of Eq. (26). By imposing adiabatic
boundary condition at the wall in Eq. (24), the viscous dissipation
term can be analogous to an equivalent convective term in vectorial form as
qnf cp;nf
Z
0
ro
2 ;
ðV rT vd Þrdr ¼ 4leff u
ð32Þ
where rTvd is the equivalent temperature gradient vector contributed solely by the viscous dissipation in the ﬂow. The left-hand side
of Eq. (32) is the equivalent convective vectorial form of the viscous
dissipation term. Substituting Eq. (32) into Eq. (24) and performing
the nondimensionalization yields
Pe
Z
0
1
V ðrT rT vd Þ RdR ¼ Nu:
ð33Þ
In Eq. (33), the viscous dissipation term in the form of temperature
gradient vector rT vd is now incorporated in the convection term. It
can be observed that the ﬁeld synergy term (the integral term) in
Eq. (33) which characterizes the convection heat transfer has
Fig. 3. Local synergy angle b0 of water–Al2O3 nanoﬂuid ﬂow, the Reynolds number
as a parameter for (a) / = 0%, and (b) / = 8%.
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T.W. Ting et al. / International Journal of Heat and Mass Transfer 73 (2014) 483–491
of the two temperature gradient vectors in Eq. (33). Higher values
of b0 are observed for nanoﬂuid, implying that with the suspension
of nanoparticles in the base ﬂuid, the synergy between the velocity
and temperature ﬁelds declines. The obvious decline in the ﬁeld
synergy as observed in Fig. 3(b) is due to the changes in the thermophysical properties of nanoﬂuid which alters the ﬂow and temperature ﬁelds of the ﬂuid as depicted in Eqs. (7) and (9).
Fig. 4 plots the average synergy angle for viscous dissipative
ﬂow, b0 , as a function of nanoparticle volume fraction /, with Reynolds number as a parameter. The b0 values of Model 1 are higher
than those of Model 2. The deviation between the two models increases with the increasing nanoparticle volume fraction. Therefore, the effect of viscous dissipation in the ﬂow is more
signiﬁcant in the presence of nanoparticle suspension due to the
increase of the nanoﬂuid effective viscosity [31]. It is observed that
b0 increases with the Reynolds number. Therefore, the ﬁeld synergy
of the ﬂow is attenuated by the suspension of nanoparticles, as
well as the increase of the Reynolds number.
Fig. 5 depicts the variations of the ﬁeld synergy number for viscous dissipative ﬂow, Fc0 , with respect to the nanoparticle volume
fraction /, with Reynolds number as a parameter. The ﬁeld synergy
declines with increasing Reynolds number and the addition of
nanoparticle in the ﬂuid. In addition, the viscous dissipation further reduces the synergy of the ﬂow. As depicted in Eq. (36), the
viscous dissipation term induces a decrease in the temperature
gradient vector and hence reduces the ﬁeld synergy of the ﬂow.
3.2. Effect of nanoparticle diameter
As depicted in Eqs. (2) and (4), the thermal characteristics of the
nanoﬂuid are intimately inﬂuenced by the size of the suspended
Fig. 4. Average synergy angle b0 as a function of nanoparticle volume fraction, the
Reynolds number as a parameter.
Fig. 5. Field synergy number Fc0 as a function of nanoparticle volume fraction, the
Reynolds number as a parameter.
nanoparticle. A decrease in the diameter of nanoparticle causes
an increase in the Nusselt number of the nanoﬂuid ﬂow [32]. It is
instructive to investigate the effect of nanoparticle diameter on
the ﬁeld synergy of the nanoﬂuid ﬂow. Fig. 6(a) depicts the
variations of b0 as a function of nanoparticle diameter dp for
water–Al2O3 nanoﬂuid with / = 1% and / = 8%. It is observed that
a decrease in the nanoparticle diameter reduces the synergy angle
of the ﬂow, indicating a better coordination between the velocity
and the temperature ﬁelds. Correspondingly, the ﬁeld synergy
number increases as the nanoparticle diameter decreases as illustrated in Fig. 6(b). The decrease of the nanoparticle diameter results in the changes of the thermophysical properties of the
nanoﬂuid that lead to enhanced ﬁeld synergy. The ﬁeld synergy
number decreases with the Reynolds number and the nanoparticle
volume fraction. The effect of viscous dissipation is more signiﬁcant at high Reynolds number ﬂow, rendering a higher value of
b0 and a lower value of Fc0 . The effect of nanoparticle volume fraction on b0 and Fc0 is less signiﬁcant at high Reynolds number ﬂow.
Fig. 6(c) plots the variations of Nusselt number with respect to the
nanoparticle diameter, with Reynolds number as a parameter. Evidently, the decrease of the nanoparticle diameter increases the
Nusselt number and hence the convection heat transfer. When
the Reynolds number and the nanoparticle volume fraction are increased, the viscous dissipation effect is magniﬁed and consequently the synergy angle increases while the ﬁeld synergy
number and the Nusselt number are reduced. The trends of the
Nusselt number in Fig. 6(c) concurs with those of the synergy number in Fig. 6(b), suggesting that the heat transfer enhancement for
smaller nanoparticles diameter is due to the improved synergy between the ﬂow and temperature ﬁelds.
3.3. Effect of microchannel diameter
The aspect ratio of the channel has signiﬁcant impact on the
thermal performance of the micro-scale heat sink. The decrease
in the microchannel diameter increases the value of Brinkman
number which characterize the signiﬁcance of viscous dissipation
[33]. Higher value of Brinkman number indicates a greater magnitude of the internal heat generation due to the viscous dissipation.
Fig. 7(a) depicts the variations of b0 as a function of microchannel
diameter for water–Al2O3 nanoﬂuid with / = 1% and / = 8%. The increase of the microchannel diameter decreases the synergy angle
and a greater magnitude of decrease is observed at low-Reynolds-number region. As the decrease in synergy angle implies enhanced ﬁeld synergy, Fc0 increases with increasing microchannel
diameter as depicted in Fig. 7(b). When the microchannel diameter
increases, the Brinkman number decreases, indicating the reduced
effect of viscous dissipation. It can be deduced from Eq. (36) that
when the viscous dissipation effect is reduced, the magnitude of
the temperature gradient vector is higher and hence a greater value of Fc0 is obtained. The variations of Nusselt number as a function of microchannel diameter, with Reynolds number as a
parameter are illustrated in Fig. 7(c). It is observed that the Nusselt
number increases with the increasing size of the microchannel,
attesting the fact that the increase of the microchannel diameter
enhances the convection heat transfer which is justiﬁed by the
intensiﬁed synergy between the velocity and temperature ﬁelds
of the ﬂow.
It is noted that for a ﬁxed value of input heat ﬂux and nanoﬂuid
volume fraction, the microchannel diameter, D and the temperature difference between the wall temperature and the bulk mean
temperature, T w T, determine the Nusselt number of the ﬂow
based on Nu ¼ qw D=keff ðT w TÞ. Fig. 8 depicts the temperature
difference T w T as a function of Reynolds number for nanoﬂuid
of / = 1%, with D being a parameter. We can observe that when
the viscous dissipation effect is neglected, the temperature
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T.W. Ting et al. / International Journal of Heat and Mass Transfer 73 (2014) 483–491
Fig. 6. The effect of nanoparticle diameter on the (a) average synergy angle b0 , (b)
ﬁeld synergy number Fc0 , and (c) Nusselt number, the Reynolds number as a
parameter.
difference is insensitive with the variation of Reynolds number but
increases with the microchannel’s diameter. However, when the
effect of viscous dissipation is incorporated in Model 1, the temperature difference increases with Reynolds number. For lowReynolds-number ﬂow, as the effect of viscous dissipation is
insigniﬁcant in this region, the temperature difference increases
with the channel’s diameter, similar to the trend of Model 2 when
viscous dissipation effect is neglected. As the Reynolds number increases, the effect of viscous dissipation intensiﬁes. Viscous dissipation features as a source term in the ﬂuid ﬂow incurring an
appreciable rise in the wall temperature due to the frictional heating of the ﬂowing ﬂuid with the wall. This results in appreciable increase in the temperature difference. For higher Reynolds number,
opposite trend is observed where the temperature difference increases with decreasing channel’s diameter. This result is in accordance with the intensiﬁcation of viscous dissipation in smaller
channel. It has been veriﬁed that the effect of viscous dissipation
is insigniﬁcant when the Brinkman number is sufﬁciently small.
When the Reynolds number and the Prandtl number do not change
signiﬁcantly, small values of Brinkman number do not affect the
Nusselt number considerably [33]. Typically, the viscous dissipation incurs pronounced impact on the convective heat transfer
489
Fig. 7. The effect of microchannel diameter on the (a) average synergy angle b0 , (b)
ﬁeld synergy number Fc0 , and (c) Nusselt number, the Reynolds number as a
parameter.
Fig. 8. Temperature difference between the microchannel wall temperature and
the bulk mean temperature of nanoﬂuid with / = 1% as a function of Reynolds
number, the microchannel diameter being a parameter.
characteristics when the Reynolds number and the Prandtl number
are high [23]. Based on the fact that the viscous dissipation is a
function of multiple parameters, the Nusselt number and the ﬁeld
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T.W. Ting et al. / International Journal of Heat and Mass Transfer 73 (2014) 483–491
synergy number which include most of the pertinent parameter,
manifest themselves as better performance indicators in characterizing the viscous dissipation effect on the convection heat transfer
in microchannels, in comparison with that by unilaterally analyzing the temperature difference between the ﬂuid and wall
temperatures.
3.4. Field synergy analysis of nanoﬂuid cooling
So far, our analysis is restricted to the nanoﬂuid heating process
in microchannel heat sinks. It is instructive to investigate the ﬁeld
synergy between the ﬂow and temperature ﬁelds in the nanoﬂuid
cooling process to distinguish the difference in the synergetic
behavior. The cooling process here comport with the case where
the ﬂuid is being cooled, with a negative heat ﬂux applied at the
wall of the microchannel. From Eq. (19), the negative Brinkman
number corresponds to the cooling process when the sign of the
heat ﬂux is negative. As commonly exhibited in the conventional
heating and cooling processes, the wall temperature is lower than
the ﬂuid temperature for a cooling process. Fig. 9 depicts the variations of synergy angle for viscous dissipative ﬂow, b0 , in the radial
direction for nanoﬂuid cooling ﬂow with / = 1%. When Re = 50,
Re = 100 and Re = 500, the b0 values of Model 1 is smaller than those
of Model 2, indicating that the presence of viscous dissipation enhances the degree of the synergy between the velocity and temperature ﬁelds. The deviation between the two models increases with
increasing Reynolds number due to the intensiﬁed effect of viscous
dissipation. However, exception is observed for Re = 2000 where
the b0 value of Model 1 is larger than that of Model 2. This is because
the viscous dissipation as internal heat generation has balanced out
the output heat ﬂux at the channel wall at high Reynolds number
ﬂow and the role of convection diminishes.
The variations of ﬁeld synergy number as a function of Reynolds
number, with the nanoparticle volume fraction being a parameter,
are depicted in Fig. 10(a). For a ﬁxed nanoparticle volume fraction,
a singularity in the ﬁeld synergy number is noticed at a particular
value of Reynolds number, denoted as critical Reynolds number,
Rec. Similar trend is also observed when the Nusselt number is
plotted as a function of Reynolds number, as shown in Fig. 10(b).
The underlying physical signiﬁcance of Rec indicates the situation
where the heat output from the nanoﬂuid is balanced out by the
internal heat generation due to viscous dissipation. The bulk mean
temperature approaches the wall temperature, leading to no signiﬁcant heat transfer between the nanoﬂuid and the wall, and thus
the ﬁeld synergy number approaches inﬁnity. In this case, the wallﬂuid temperature difference T w T in the denominator of the heat
transfer coefﬁcient is equal to zero. The existence of singularity in
the ﬁeld synergy number (or Nusselt number) is subsequent to a
change of sign of the wall and bulk mean temperature difference
[19,34,35]. The existence of singularity in the ﬁeld synergy number
Fig. 9. Local synergy angle b0 of water–Al2O3 with / = 1% for nanoﬂuid cooling
process.
Fig. 10. (a) Field synergy number Fc0 , and (b) Nusselt number as a function of
Reynolds number for nanoﬂuid cooling process.
(or Nusselt number) is only applicable to cooling process when the
internal heating effect of viscous dissipation is considered. It is noticed that there are two distinct ranges of Fc0 (or Nu) for cooling
process. In the range of 0 < Re 6 Rec, the ﬁeld synergy number
(or Nusselt number) gradually increases with increasing Reynolds
number and in proximity to Rec, the ﬁeld synergy number (or Nusselt number) increases tremendously. This is due to the decrease in
the temperature difference until it comes to a change of sign. In the
range of Re > Rec, the ﬁeld synergy number (or Nusselt number)
approaches zero from a negative inﬁnite value. This is because
the internal heat generation by the viscous dissipation overcomes
the heat output from the wall, and higher Re translates into higher
internal heat generation leading to higher temperature difference
and subsequently lower ﬁeld synergy number (or Nusselt number).
4. Conclusions
Field-synergy principle is used to investigate the convection
heat transfer of viscous dissipative nanoﬂuid ﬂow in microchannel
heat sinks under the uniform wall heat ﬂux condition. Due to the
presence of two heat sources in the ﬂow, we deﬁne the synergy angle and ﬁeld synergy number for viscous dissipative ﬂow to investigate the relationships of the velocity and temperature ﬁelds
associated with the convection heat transfer enhancement. Under
the effects of viscous dissipation, the synergy angle is increased
and the ﬁeld synergy number is decreased, leading to the deterioration of synergy between the velocity and temperature ﬁelds. The
suspension of nanoparticle in the ﬂuid leads to further decrease in
the ﬁeld synergy. By reducing the size of the nanoparticle and
increasing the diameter of the microchannel, the degree of synergy
between the velocity and temperature ﬁelds of the nanoﬂuid ﬂow
in microchannel can be intensiﬁed. On the other hand, the ﬁeld
synergy in the nanoﬂuid cooling improves in the presence of viscous dissipation. Similar to other passive heat transfer enhancement technique, the suspension of nanoparticles alters the
synergy between the velocity and temperature ﬁelds, leading to
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T.W. Ting et al. / International Journal of Heat and Mass Transfer 73 (2014) 483–491
the distinctive heat transfer characteristics of nanoﬂuids compared
to those of conventional ﬂuids.
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