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Proc. R. Soc. A (2012) 468, 2383–2398
doi:10.1098/rspa.2011.0671
Published online 14 March 2012
On the anomalous laminar heat transfer
intensification in developing region of nanofluid
flow in channels or tubes
BY J. T. C. LIU1,2, *
1 School
of Engineering and the Center for Fluid Mechanics, Institute for
Molecular and Nanoscale Innovation, Brown University, Providence,
Rhode Island 02912, USA
2 Department of Mathematics, Fluid Dynamics Group, Imperial College,
London SW7 2AZ, UK
The present work theoretically addresses the experimental observations of nanofluid flow
exhibiting highly intensified laminar heat transfer rates at the leading edge of channels or
tubes. The basis for this study is the continuum conservation equations for nanofluids.
The Rayleigh–Stokes approximation is applied to the nonlinear advective effects and
a perturbation scheme, in ascending powers of the nanoparticle volume fraction, is
applied. The disparate thicknesses of momentum, heat and volume fraction is exploited to
advantage in securing analytical, similar solutions. The volume fraction layer is ‘infinitely’
thin in that its effect on the momentum and thermal transport is essentially its bulk value
far from the wall. The composite resulting zeroth- and first-order perturbations show that
an increasing modification in the velocity and temperature profiles occur with increasing
volume fraction and that this is caused, and quantitatively assessed, by inertial effects of
advection and enhanced nanofluid transport properties. Some satisfactory explanations
of experiments are made for aluminium oxide nanoparticles in water, in terms of the ratio
of nanofluid to base fluid heat transfer coefficients, local heat transfer coefficient and the
Nusselt number.
Keywords: nanofluids; heat transfer intensification; nanoparticles
1. Introduction
Nanofluid refers to fluids containing dispersed nano-sized particles, particularly
that of metallic particles. The purpose of this mixture is to enhance the
thermal conductivity of the effective medium and provide heat transfer rates
surpassing that of the base fluid. This has a wide range of engineering applications
discussed, for instance, in the book by Das et al. (2008). The effective-thermal
conductivity was first developed by Maxwell (1873) and Lord Rayleigh (1892) for
dispersed spherical particles of much larger size. The formalism for calculating
the effective-thermal conductivity, nevertheless, is adaptable to a nanofluid. The
effective thermal conductivity is expressed as a function of the nanofluid volume
*[email protected]
Received 9 November 2011
Accepted 6 February 2012
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J. T. C. Liu
fraction, the conductivity of the nanoparticle material and that of the base fluid.
The formalism is developed further to incorporate effects of particle geometry and
particle-fluid interfacial resistance by Nan et al. (1997). The theoretical results
are found to be in good agreement with the recent experimental measurements
at over 30 international laboratories, as reported in Buongiorno et al. (2009).
In general, for small nanoparticle volume concentrations, the ratio of nanofluid
conductivity to that of the base fluid is a linear function of the nanoparticle
volume fraction.
The theoretical description of convective transport in nanofluids is given
impetus by the recent paper of Buongiorno (2006). While the effective-medium
description has certain features of a single phase fluid, the nanoparticles affect
the convective heat transfer via the thermophysical properties. The latter are
dependent on the nanoparticle volume fraction, which in turn is subject to
convective diffusion. Even in the presence of small nanoparticle volume fraction,
where the thermophysical properties are expressible as linear functions of the
volume fraction, there is sufficient coupling to render the convective transport
problem nonlinear. The monograph on nanofluids (Das et al. 2008) addressed
advances to include most of the 2006 year, including controversies of the
effective-thermal conductivity prior to Buongiorno et al. (2009).
Experimental measurements in micro and non-micro channels and tubes (e.g.
Wen & Ding 2004; Jung et al. 2009) indicate that there is strong dependence of
the heat transfer rates on nanoparticle concentration near the channel entrance
region and that this is accentuated closer to the leading edge particularly for
the higher Reynolds number ranges. A small increase in the volume fraction is
accompanied by large increases in the surface heat transfer rates not explainable
on basis of the thermal conductivity alone. In fact, Wen & Ding (2004) propose a
‘smart channel’ for enhanced heat transfer, consisting a series of entrance regions
rather than letting the flow become fully developed in a single long channel.
In the entrance region, the flow is mimicked by the leading edge of a laminar
boundary layer. The behaviour of large heat transfer rates in this region is referred
to as ‘anomalous heat transfer enhancement’ (e.g. Prabhat et al. 2010). Rea
et al. (2009) presented experimental evidence that no abnormal enhancement
occurs, including the entrance region. Ding et al. (2007) also stressed that
in some cases no enhancement takes place. The present contribution hopes
to supplement these observations by considering transport effects from the
conservation equations. It also appears that a more rational ‘equation of state’
for nanofluid density/heat capacity, other than from ideal mixtures relation, is
most likely to help complete the enhancement explanation puzzle as the present
efforts show.
2. Basic equations of the dynamics and thermodynamics of nanofluid flow
With Bird et al. (2001) as a guide (see also, Probtein 1994), Buongiorno (2006)
generalized the conservation equations for the continuum description of the
dynamics and thermodynamics of nanofluid flow. The composite fluid consists
of a mixture of base fluid and dispersed embedded nanoparticles; the latter
phase is in macroscopic momentum and thermodynamic equilibrium with the
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base fluid. The microscopic deviation of nanoparticle velocity from the massaveraged plane manifests itself in the mass conservation for the nanoparticles
and is modelled, in absence of imposed external forces, by Brownian and thermal
diffusion (Buongiorno 2006). Such a continuum formulation is also discussed in
Tzou (2008) and Pfautsch (2008). The resulting conservation equations, in the
case of steady nanofluid flow in the leading edge region of a channel, are essentially
the boundary layer equations (Pfautsch 2008). The global continuity is
vru vrv
+
= 0,
vx
vy
(2.1)
the streamwise momentum equation for plane flow is
ru
vu
vu
vpe vty
+ rv
=−
+
,
vx
vy
vx
vy
(2.2)
where the shear stress is
ty = m
vu
,
vy
the normal momentum equation in the boundary layer approximation is
vp
= 0,
vy
(2.3)
where u, v are the streamwise and normal velocity components in the x, ydirections, respectively; the nanofluid properties are: r is the density, p the
pressure and m the viscosity. The leading edge problem, far from the developing
region, is considered here so that the pressure at the edge of the boundary layer
pe is set equal to constant subsequently. The thermodynamic energy equation in
boundary layer form and for (very) low Mach number flows is
rcu
vqy
vjp,y
vT
vT
+ rcv
=−
+ hp
,
vx
vy
vy
vy
(2.4)
where c is the nanofluid heat capacity, T is the temperature; the normal
component of heat flux qy is accomplished by thermal conduction via the
nanofluid thermal conductivity k and the transport of nanoparticle-specific
enthalpy hp by the normal diffusion flux jp,y ,
qy = −k
vT
+ hp jp,y .
vy
It is very similar to the heat flux in a binary reacting gas including the diffusiontransport of thermal energy elucidated by Lees (1956). The diffusion flux is
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J. T. C. Liu
composed of Brownian diffusion flux and thermal diffusion (Buongiorno 2006)
jp,y = −rp DB
vf
DT vT
− rp
.
vy
T vy
The nanophase continuity is written in terms of the nanophase volume fraction
f and its diffusion flux:
rp u
vjp,y
vf
vf
+ rp v
=−
,
vx
vy
vy
(2.5)
subscript p denotes properties of the nanoparticle phase (subscript f denotes that
of the base fluid phase). For low Mach number flow, the rate of viscous dissipation
and the work done by the pressure gradients are neglected from (2.4).
The boundary conditions are
and
y =0:
u = 0, T = TW ,
y =∞:
u = U , T = T∞ ,
jp,y=0 = 0
f = f∞ ,
where the subscript W denotes condition at the wall and ∞ that in the
bulk fluid far away from the wall. The zero-flux boundary condition jp,y=0 = 0
is first suggested by Buongiorno (2006), as the natural one for non-porous,
inert-solid wall.
The thermophysical properties of the composite fluid are expressed,
following Buongiorno (2006), in terms of the constituent components. On
the basis of mixtures, the nanofluid density is expressed as r = frp +
(1 − f)rf , the heat capacity as c = f(rp /r)cp + (1 − f)(rf /r)cf . On the basis
of correlations, the nanofluid viscosity is expressed as m = mf (1 + a1 f +
a2 f2 ), and thermal conductivity k = kf (1 + b1 f + b2 f2 ), where a1 , . . . , b1 , . . . are
nanoparticle-dependent correlation constants. The diffusion coefficients are: for
Brownian diffusion, Stokes–Einstein relation: DB = kB T /3pmdp , where kB is the
Boltzmann constant, dp the particle diameter. The thermal diffusion coefficient
is DT = mbf/r, where b = b(k, kp ) = ck k/(2k + kp ), ck ∼
= 0.26, kp is the thermal
conductivity for the nanoparticle material, is a tentative correlation for nanofluids
as it is an extension of the correlation for micrometre-size particles. Nanofluid
viscosity, measured by Pak & Cho (1998) showed no non-Newtonian effects. This
is plausible as the nanoparticles essentially follow the motion of the base fluid
(water), which is here Newtonian.
3. Approximate description of the leading edge of channel entrance region
The entrance region length xE of a channel (or tube, when the viscous layer
thickness is small relative to the tube diameter) can be estimated from that
of the base fluid in absence of nanoparticles (Schlichting 1955) xE /D ≈ 0.04 Ref
for the velocity entrance length, where D is the channel width (or hydraulic
diameter), Ref = UD/nf is the base fluid Reynolds number, U is the entrance
velocity and nf = mf /rf is the kinematic viscosity of the base fluid. For experiments
conducted in channels and tubes, the nanofluid properties are those of the
bulk fluid. Since the viscous spreading rate involves the nanofluid kinematic
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Nanofluid heat transfer intensification
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viscosity, it is anticipated that the entrance region length of nanofluid to that
of the base fluid is simply the ratio of kinematic viscosities xE /xE,f ≈ [1 + f∞ (rp /
rf − 1)]/[1 + f∞ a1 ]. For instance, for aluminium oxide nanoparticles at about
1 per cent volume concentration, the nanofluid entrance length is shortened by
about 25 per cent.
In Jung et al.’s (2009) experiments (e.g. their fig. 6), their Re = 286, based on an
averaged velocity, the momentum entrance length is very nearly xE /D ≈ 11. The
thermal entrance length is estimated as xE,th /D ≈ 0.04 RePr, which is lengthier by
a Prandtl number factor. The reduction of thermal entrance length in a nanofluid
is approximated by xE,th /xE,th,f ≈ (1 + f∞ b1 )/[1 + (rp cp /rf cf − 1)]. But still, for
order of magnitude estimates, xE,th /xE ≈ Pr which ranges 5 to 10 for liquids with
small concentration of nanoparticles as in the experiments. Within this leading
edge region, the heat transfer rates are spectacular and increases many fold from
that of the base fluid for a small increase in nanoparticle volume fraction. But
for their Re = 60 case, the measured heat transfer rates are already in the fully
developed region and are thus very nearly flat. We therefore direct our studies
towards the leading edge region, for which simplifying approximations are made
and discussed in the following.
In the leading edge region, the left sides of (2.2, 2.4, 2.5) and the advective part
of the continuity equation (2.1), the nonlinear advective operators are modelled
by the Rayleigh–Stokes approximation,
u
v
v
v
+v ≈U ,
vx
vy
vx
(3.1)
where the nonlinear advection velocities are replaced by entrance velocity U , but
with the respective right sides remaining the same; this approximation is also
known as the ‘plug flow’ model. In this case, the time-like effect of advection
becomes obvious (advection time ≈ x/U ). In the leading edge of the entrance
region, the boundary layers on the wall are not yet interacting with the bulk
fluid, thus vpe /vx = 0 in equation (2.2).
The base fluid, with properties denoted by the subscript f, is taken as
incompressible, so are the nanoparticle properties with subscript p, but not
implying that the nanofluid density r is necessarily ‘incompressible’ because of
its involvement with the volume fraction and its diffusion equation. However,
a common assumption is that the composite nanofluid is taken as r ≈ rf
incompressible (Buongiorno 2006; Tzou 2008), because of the smallness of the
volume fraction in practice. However, this argument can be equally put forth
for other thermophysical properties, and in so doing, the nanofluid would then
be altogether assumed away. In a stratified flow, the incompressible assumption
implies that the nanofluid density remains constant along each streamline as
it is advected by the fluid. In the present approximation, the global continuity
equation becomes, with the Rayleigh–Stokes approximation
vu
vv
vr
+r
+
= 0,
(3.2)
U
vx
vx
vy
retaining the compressible effect that r = rf , consistent with other nanofluid
thermophysical properties (r, c, m, k, . . .) remaining ‘compressible’ because of the
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J. T. C. Liu
presence of the nanoparticle volume fraction f, which causes the nonlinearity.
Properties of the nanoparticle phase (such as rp , cp , . . .) and of the base fluid
(rf , cf , . . .) are individually ‘incompressible’.
The continuity equation, within our considerations, would be used, if desired,
to calculate the normal velocity v from r, which is expressed as a function of f,
the latter, u and T are calculated from the closed set of volume concentration,
energy and x-momentum equations,
y v ln r
vu
+U
dy.
v − v(0) = −
vx
vx
0
The following entrance region characteristic quantities are used to render the
simplified conservation equations dimensionless: U , D, rf , mf , cf , kf , DB,ref , DT,ref ,
f∞ , T∞ , TW –T∞ . The dimensionless quantities
r∗ =
r ∗ m ∗ k ∗ c
DB
DT
, m = , k = , c = , DB∗ =
, DT∗ =
rf
mf
kf
cf
DB,ref
DT,ref
and
u∗ =
ǔ
T
TW − T
f ∗ x
, T∗ =
,q=
,F=
,x = .
U
T∞
TW − T ∞
f∞
D
The thermophysical properties in dimensionless form become
r∗ = 1 + f∞ F(r∗p − 1),
r∗p =
m∗ = 1 + a1 f∞ F + a2 f2∞ F2 ,
rp
,
rf
r∗ c ∗ = 1 + f∞ F(r∗p cp∗ − 1),
cp∗ =
cp
,
cf
k ∗ = 1 + b1 f∞ F + b2 f2∞ F2 ,
T∗
, DT∗ = n∗ b∗ f∞ F
m∗
∗
∗
(2
+
k
)
k
kp
p
and b∗ =
, kp∗ =
∗
∗
(2k + kp )
kf
DB∗ =
with
DB,ref =
k B T∞
,
3pmf dd
DT,ref = nf bf
and
bf = bf (kf , kp ).
A few of the resulting characteristic parameters that appear in the
dimensionless conservation equations differ slightly from that of Buongiorno
(2006) because of the anticipated perturbation in terms of the smallness of f∞ ,
which is extracted out of these parameters; but similar symbols and similar
dimensionless parameters are defined: The Reynolds number is Re = UD/nf ,
defined with the base fluid kinematic viscosity; Prf = nf /af is the base fluid
Prandtl number, where af = kf /rf cf is the base fluid thermal diffusivity; the
Lewis number is defined similarly as Buongiorno (2006), Lef = kf /rp cp DB,ref , but
without the perturbation parameter f∞ in the denominator, as is the Brownian
to thermal diffusion coefficient ratio NBT,ref = T∞ DB,ref /(TW − T∞ )DT,ref in which
f∞ is absent from the numerator; the Schmidt number is defined as Scf = nf /DB,ref .
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Nanofluid heat transfer intensification
4. Perturbation for small nanofluid volume fraction
The intermediate forms of dimensionless conservation equations are similar in
to the dimensional ones and thus will not be stated here. Even to first order,
as the thermophysical properties are expressed as linear functions of the volume
fraction, the flow structure is nonlinear involving products of the volume fraction
and flow quantities. The Rayleigh–Stokes approximation simplifies the nonlinear
effects of advection, but not the nonlinearities owing to ‘compressibility effects’
through the volume fraction.
In order to systematically study the effect of small nanoparticle volume
fraction on the flow structure, a perturbation procedure is devised by expressing
flow quantities and thermophysical properties in ascending powers of the
volume fraction. This is a linearization procedure about the flow of the base
fluid for small volume fraction. In practice f∞ ≈ w(10−2 ), so that any flow
quantity or theromophysical property is systematically represented as a truncated
perturbation series
g ∗ (x∗ ) =
m
fn∞ g (n) (x),
(4.1)
n=0
the zeroth-order quantity is that of the base fluid, the effect of nanoparticle
volume fraction appears as perturbations in ascending series in powers of f∞ ,
in which only the n = 0, 1 terms are retained. As indicated by the right side
of (4.1), all dimensionless asterisks are dropped in the perturbed quantities.
For f∞ 1:
u ∗ = u (0) + f∞ u (1) + · · · ,
q = q(0) + f∞ q(1) + · · · ,
f = f∞ F(1) + · · · .
(4.2)
The thermophysical properties are already in convenient perturbation form:
r∗ = 1 + f∞ (r∗p − 1)F(1) , r∗ c ∗ = 1 + f∞ (r∗p cp∗ − 1)F(1)
m∗ = 1 + f∞ a1 F(1)
and
k ∗ = 1 + f∞ b1 F(1) .
(4.3)
The diffusion coefficients are expanded as
⎫
T∗
⎪
(0)
(1)
(0) (1)
⎬
=
T
+
f
[T
−
a
T
F
]
+
·
·
·
,
∞
1
m∗
⎪
⎭
DT∗ = n∗ b∗ f∞ F = f∞ F(1) + · · · ,
DB∗ =
(4.4)
∗
∗
∗
where in more compact notation, T (0) = TW
− (TW
− 1)q(0) , T (1) = −(TW
− 1)q(1) ,
(0)
(0)
and n = 1, b = 1.
Substituting the expansions (4.1)–(4.4) into the conservation equations
(3.7–3.9) and equating terms of like order in the perturbation parameter f∞
and omitting the asterisk on the dimensionless independent variables, a series of
problems are obtained and individually described in the following.
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J. T. C. Liu
(a) The zeroth-order problem
The zeroth-order problem in the Rayleigh–Stokes approximation reduces to
the familiar ‘heat equation’
⎫
1 v2 u (0) ⎪
vu (0)
⎪
⎪
=
⎬
vx
Re vy 2
(4.5)
vq(0)
1 v2 q(0) ⎪
⎪
⎪
⎭
and
=
vx
RePrf vy 2
with boundary conditions
and
u (0) (0) = 0,
(0)
u (∞) = 1
q(0) (0) = 0,
(0)
q (∞) = 1.
The similarity variables are defined in terms of the respective
estimates
of
√
√
∝
x/Re,
d
∝
x/RePr
momentum and thermal
diffusion
layer
thicknesses
d
T
f
u
√
√
so that hu = y/2 x/Re and hT = y/2 x/RePrf . The zeroth-order Rayleigh–
Stokes problems for the velocity and temperature in similarity form are then
⎫
d2 u (0)
du (0)
⎪
⎪
+
2h
=
0
⎪
u
⎬
dh2u
dhu
(4.6)
⎪
dq(0)
d2 q(0)
⎪
⎪
and
+ 2hT
= 0,⎭
dhT
dh2T
which has solutions in terms of the error function
u (0) (hu ) = erf (hu )
q(0) (hT ) = erf (hT ).
and
(4.7)
The volume fraction enters only into the first-order perturbation, where
df ∝ x/ReScf and its similarity variable, in comparison, is defined as hf =
y/2 x/ReScf .
(b) The scale of diffusion layers and the first-order problems
The nanofluids studied by Buongiorno (2006), Pfautsch (2008) and Tzou (2008)
theoretically and experimentally by Jung et al. (2009), Wen & Ding (2004),
is typified by Al2 O3 of about 10 nm diameter in water at standard conditions.
√
Scf ≈ 2 ×√
104 , Lef ≈ 3.7 × 103 , √
thus du /dT ≈ Prf =
The latter gives
√ Prf ≈ 5.9,
w(1), du /df ≈ Scf = w(102 ), dT /df ≈ Lef = w(102 ). Also, Lef NBT,ref 1 from
similar estimates. In the following, the disparate relative thicknesses with respect
to the volume fraction diffusion thickness are significant and are used to advantage
in simplifying the system of perturbation conservation equations.
Within finite hu , hT respective regions of interest to the momentum and heat
transfer problems, because du , dT df , the perturbation volume fraction F(1) (hf )
and its derivatives take on their free-stream or bulk values so that F(1) (hf ) → 1,
dF(1) (hf )/dhf → 0. In this simplification, the momentum and thermodynamic
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Nanofluid heat transfer intensification
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problems become uncoupled from the concentration problem and are thus
uncoupled from one another. On the other hand, within finite hf region of interest
to the volume fraction diffusion layer, the temperature
problem takes on its wall
√
∗
, q(0) → 0, dq(0) /dhT → 2/ p.
values T (0) (hT ) → TW
(c) The first-order temperature distribution
The first-order conservations are systemically obtained in physical coordinates
and transformed into the respective similarity variables and incorporating the
discussions of scale effects. The effect of order (Lef NBT,ref )−1 1 is neglected in
energy equation so that heat transport between the free stream and the wall is due
to thermal conduction alone
√ (Buongiorno 2006). Substituting the zeroth-order
functions dn q(0) /dhnT = (2/ p)(−2hT )n−1 exp(−h2T ), n = 1, 2 into the first-order
energy equation, the resulting simplified energy equation in similarity form is an
inhomogeneous, source-like second-order heat equation with the same differential
operator as the zero-order equation,
vq(1)
4hT
v2 q(1)
+
2h
= −[(r∗p cp∗ − 1) − b1 ] √ exp(−h2T ).
T
2
vhT
p
vhT
(4.8)
The boundary conditions are homogeneous, q(1) (0) = q(1) (∞) = 0, since the
physical boundary conditions are already satisfied by the zeroth-order problem.
The right side of (4.8) is a source- or sink-like effect if the sign of the bracketed
terms [(r∗p cp∗ − 1) − b1 ] is positive or negative, respectively. The enhanced
nanofluid conductivity has the effect of smoothing the temperature profile, thus
working against the enhanced conductivity in the calculation of heat conduction
at the wall.
Integrating equation (4.8) once as a first-order inhomogeneous differential
equation for dq(1) /dhT yields the general solution (Murphy 1960, p. 14),
2
dq(1)
∗ ∗
2
= C1 + √
(4.9)
(1 − rp cp + b1 )hT exp(−h2T ),
dhT
p
where C1 is an integration constant and is the contribution to the enhanced
surface heat transfer rate from the conservation equations, dq(1) (0)/dhT =
C1 . A straightforward integration of equation (4.9) then gives the first-order
perturbation temperature profile
√ p
1
(1)
erf (hT ) + √
(1 − r∗p cp∗ + b1 )
q (hT ) = C1
2
p
√ p
erf (hT ) + C2 .
(4.10)
× −hT exp(−h2T ) +
2
The homogeneous condition at the wall gives C2 = 0. The condition at
infinity gives
dq(1) (0)
1
= √
(4.11)
C1 =
(r∗p cp∗ − 1 − b1 ) > 0,
dhT
p
which provides a positive (enhanced) contribution to the surface heat transfer rate
in nanofluids, if the parameters in the bracket remains positive. The error function
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J. T. C. Liu
terms in the profile (4.10) cancel and the first-order perturbation temperature
profile becomes
1
(1)
(4.12)
q (hT ) = √
(r∗p cp∗ − 1 − b1 )hT exp(−h2T ).
p
2
The profile function
T ) in the solution (4.12) is weighted by
√ hT∗ exp(−h
∗
the coefficient (1/ p)(rp cp − 1 − b1 ), produces a modification in the overall
temperature profile. This modification depends linearly on the volume fraction
according to the perturbation expansion. The profile modification could
conceivably be experimentally detected against variations of nanofluid properties
and volume fraction (private communication, Buongiorno 2012).
(d) The first-order velocity distribution
The (Lef NBT,ref )−1 1 approximation brings similarity between the energy and
momentum equations, the first-order velocity distribution can thus be inferred
from the steps taken in solving the temperature problem (4.12). Thus, from (4.9)
2
du (1)
∗
2
= C1,u + √
(1 − rp + a1 )hu exp(−h2u ),
dhu
p
where the integration constant, inferred from (4.11) is
du (1) (0)
1
= √
(r∗p − 1 − a1 ).
C1,u =
dhu
p
The first-order perturbation velocity distribution is
1
(1)
u (hu ) = √
(r∗p − 1 − a1 )hu exp(−h2u ).
p
(4.13)
(4.14)
(4.15)
In (4.15), the factor (r∗p − 1 − a1 ), when multiplied by f∞ , contributes to a
modification of the velocity distribution in much the same way as the temperature
distribution. Similarly, the velocity profile modifications could be measured
experimentally against variations of nanofluid properties.
(e) Solution for the volume fraction
The volume fraction diffusion layer is deeply embedded inside the thermal and
momentum boundary layers according to the scale-effects already discussed. The
volume fraction conservation equation (2.5) is recast into the form:
vF(1)
v2 F(1)
+
2b)
= 0,
+
(2ah
f
vhf
vh2f
(4.16)
with the zero-wall flux boundary condition (Buongiorno 2006) and that in the
free stream
dF(1) (0)
+ 2bF(1) (0) = 0, F(1) (∞) = 1,
dhf
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Nanofluid heat transfer intensification
2393
where, for compactness, defined are
∗−1
a = TW
√ ∗
b = (TW
p Lef NBT,ref )−1 .
and
(4.17)
The general solution is obtained systematically, similar as that for the energy and
momentum equations,
hf
(1)
exp[−(ah2f + 2bhf )] + C2,f ,
(4.18)
F (hf ) = C1,f
0
where C1,f , C2,f are integration constants. Re-define the independent variable as
y(hf ) =
√
ahf + y(0),
and
b
y(0) = √ ,
a
which further simplifies the solution representation. The general solution is then,
from Gröbner & Hofreiter (1949, p. 109) for the integral in (4.18),
2
b
b
p
(1)
F (hf ) = C1,f
exp
erf (y) − erf √
(4.19)
+ C2,f ,
4a
a
a
where the integration constants are identified with the volume fraction F(1) (0) =
C2,f and its slope dF(1) (0)/dhf = C1,f at the wall. The boundary conditions for
(4.16) give
2
b
b
p
C2,f = 1 − b
exp
1 − erf √
, C1,f = −2bC2,f .
a
a
a
It can be deduced from the representation for a, b in (4.17) that F(1) (0) =
−1/2 −1
−1/2 −1
) and that dF(1) (0)/dhf = −w(Lef NBT,ref
), so that for
1 + w(Lef NBT,ref
−1/2
−1
Lef NBT,ref
1, the solution for zero-wall flux condition in presence of a uniform
free-stream volume fraction is very nearly approximated by the ‘insulated wall’
condition which renders, at the wall, a slightly negative but nearly zero slope of
the volume fraction and a wall volume fraction very nearly that of the free stream.
−1/2 −1
1 were directly applied
These are anticipated if the approximation Lef NBT,ref
to the volume fraction differential equation and wall boundary condition.
Other possible wall boundary conditions include an equivalent constant wall
condition by setting F(1) (0) = 0. The constant volume fraction wall boundary
condition is used by Tzou (2008) in his work on natural convection in nanofluids
and by Pfautsch (2008) in the direct numerical integration of the forced convection
boundary layer problem with free stream at zero volume fraction. In both cases,
the constant volume fraction wall condition is interpreted physically as that
controlled by mass exchange through a porous wall matrix (Buongiorno 2010,
personal communication). The solution for F(1) (0) = 0, F(1) (∞) = 1 is found from
(4.19) to be
√
erf (y) − erf (b/ a)
(1)
F (hf ) =
,
√
1 − erf (b/ a)
√
−1/2 −1
∗
) 1, reduces to F(1) (hf ) ≈ erf (hf /TW
)
which, again for b/ a = w(Lef NBT,ref
as also anticipated directly from the differential equation and its boundary
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J. T. C. Liu
conditions. In this case, in the overall region of interest to convective heat and
momentum transfer, where df du , dT , by taking F(1) (hf ) → 1 in finite regions
of hu , hT earlier is shown to be a good approximation for either of the volume
fraction wall boundary conditions.
5. Heat transfer
The surface heat transfer rate for boundary layer-type nanofluid flows is,
vT
q0 = − k
+ (hp jp,y )0 ,
vy 0
which includes the mechanisms of heat conduction and diffusion transport of
nanoparticle enthalpy to the wall. For a non-porous wall, the wall boundary
condition advanced by Buongiorno (2006) is that the diffusion flux at the wall
is zero, (jp,y )0 = 0. In this case, heat transfer at the wall is accomplished by
conduction alone,
vT
.
q0 = − k
vy 0
In terms of the dimensionless quantities and similarity variable already defined,
directly from the outcome of the perturbation scheme, to first order
RePrf 1/2 (0)
∗ TW − T ∞
[q (0) + f∞ q(1) (0)],
(5.1)
q0 = kf k0
2D
x/D
√
√
where q(0) (0) = 2/ p, q(1) (0) = (r∗p cp∗ − 1 + b1 )/ p, k0∗ = 1 + f∞ − b1 F(1) (0) and
from the volume fraction solution with Buongiorno’s wall condition, F(1) (0) =
−1/2 −1
). In this case, the local heat transfer coefficient formed from
1 + w(Lef NBT,ref
(5.1) is
r∗p cp∗ − 1 + b1
q0
,
(5.2)
= hx,f 1 + f∞
hx =
TW − T ∞
2
for the same wall and free-stream temperatures for the base and nanofluids. The
base fluid heat transfer coefficient from the Rayleigh–Stokes approximation is
q0,f
kf
RePrf 1/2
=
.
(5.3)
hx,f =
TW − T ∞
D
px/D
The present perturbation scheme brings out the explicit mechanisms contributing
to nanofluid heat transfer. The nanofluid contribution to the heat transfer
coefficient (5.2) is accounted for through the thermal conductivity in the definition
of heat conduction at the wall which give rise to +b1 ; −b1 /2 comes from the
effect of increased heat conductivity in the energy equation. The inertia effect
of the nanofluid density/heat capacity contributes to the factor (r∗p cp∗ − 1)/2.
If only the nanofluid thermal conductivity in the definition of heat transfer at
the wall is taken into account, but nanofluid effects on the fluid temperature
profile according to the conservation equations are not, then the ratio hx /hx,f
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2395
Nanofluid heat transfer intensification
reduces to the ratio of thermal conductivities at the wall, hx /hx,f ≈ 1 + f∞ b1 ≈ k0∗ ,
which is essentially the same as the free-stream nanofluid thermal conductivity
∗
.
ratio k∞
In the developing region, the definition of the local Nusselt number consistent
with the local heat transfer coefficient is Nux = hx x/k0 , where the streamwise
development variable x is appropriately used instead of D and that the thermal
conductivity used is appropriately the same as that in the definition of heat
∗
as shown. The local nanofluid
conduction relation at the wall, but that k0∗ ≈ k∞
Nusselt number is
(r∗p cp∗ − 1 − b1 )
hx,f x
RePrf x 1/2
Nux = Nux,f 1 + f∞
=
.
, where Nux,f =
kf
pD
2
(5.4)
As the base fluid is assumed incompressible in the very beginning, the
conductivity kf is taken as a constant. The nanofluid thermal conductivity enters
as a normalizing factor in the definition of the Nusselt number thus only the
temperature profile modification coefficient of (4.12) appears in (5.4).
Of importance in convective heat transfer is the Stanton number, in which the
surface heat transfer rate is normalized by the free-stream nanofluid convection.
In this case, it is consistent to define the local Stanton number as
q0
,
(5.5)
stx =
r∞ c∞ U (TW − T∞ )
resulting in
stx = stx,f 1 + f∞
b1 − r∗p cp∗ + 1
2
,
where
RePrf px
stx,f =
D
−1/2
.
(5.6)
The free-stream convective/inertia factor in the definition of the Stanton number
incurs a correction to order f∞ .
The present perturbation considerations brought out three aforementioned
mechanisms contributing to heat transfer modification from that of the base
fluid: (i) transport property effect of nanofluid thermal conductivity, as coefficient
of the temperature gradient, in evaluating the heat conduction at the wall,
(ii) nanofluid thermal conductivity effect on the temperature profile through
the energy equation, and (iii) inertial effects of convective transport in terms of
the density/heat capacity on the temperature profile via the energy equation. The
heat transfer coefficient exhibits all three effects explicitly, the Nusselt number
involves only effects (ii) and (iii), whereas the Stanton number involves thermal
conductivity effects (i) and (ii), modified by free-stream nanofluid convection.
6. Skin friction and the nanofluid Reynolds analogy
The skin friction is defined with the nanofluid viscosity coefficient evaluated at the
wall t0 = (mvu/vy)0 . The velocity profile is computed via the momentum problem
in terms of it similar solution; the nanofluid viscosity at the wall is m0 = mf [1 +
f∞ a1 F(1) (0)], in which F(1) (0) ≈ 1 as shown earlier. The local skin friction in
terms of the corresponding base fluid skin friction is t0 = t0,f [1 + (a1 + r∗p − 1)/2],
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2396
J. T. C. Liu
where t0,f = (mf U /D)(Re/px/D)1/2 . The skin friction coefficient is obtained by
consistently normalizing the skin friction by the free-stream dynamic pressure,
where r∞ = rf [1 + f∞ (r∗p − 1)], then the ratio of local nanofluid skin friction
coefficient is cfx = cfx,f [1 + f∞ (a1 − r∗p + 1)/2], where cfx.f = 2(Repx/D)−1/2 .
The Reynolds analogy between heat transfer and skin friction, which follows
from that of the base fluid, cfx,f /2 = stx,f (Prf )1/2 , to first order in f∞ is
a1 − r∗p − b1 + r∗p cp∗ − 3b1 + r∗p cp∗ − 1
cfx
1/2
= stx (Prf )
.
1 + f∞
2
2
Another form is obtained in terms of ratio to the base fluid for the same volume
fraction
cfx
stx
∗
(a1 + rp − 1) =
(b1 + r∗p cp∗ − 1).
stx,f − 1
cfx,f − 1
The largeness of the (measured) constant a1 limits the validity of the perturbation
range of f∞ and the usefulness of the analogy is thus limited, but it illustrates
and assesses the skin friction drag penalty accompanying nanofluid heat transfer
enhancement.
7. Discussion of the results
The experimental results on heat transfer enhancement in nanofluids and its
possible anomaly is recently extensively analysed and discussed in Prabhat et al.
(2010, 2011). These authors compared experimental measurements with standard
laminar heat transfer correlations but using as properties those of nanofluids,
without appealing to the nanofluid transport effect via the conservations
equations. This comparison is valuable and has shown that, especially in the
leading edge of channels, large enhanced hx were measured for small increases
in f∞ . The enhancement experiments are typified by Wen & Ding (2004),
Jung et al. (2009) and Lai et al. (2008) in the entrance region at higher
Reynolds numbers.
On the other hand, Rea et al. (2009) presented convective heat transfer
data for alumina and zirconia nanofluids in laminar flow, showing no abnormal
enhancement with respect to the traditional Shah’s correlation, including the
entrance region. Ding et al. (2007) also noticed that in some cases the
convective heat transfer coefficient is significantly larger than the thermal
conductivity enhancement and not so in other cases. The present calculations,
from the conservation equations, as approximate as it is, could provide possible
explanations of some of the apparent incongruities and not in some other cases.
The heat transfer coefficient derived in (5.2) is focused upon.
In the spirit of Rea et al. (2009), the heat transfer coefficient of the
base fluid, (5.3), is extended empirically to hold for nanofluids provided
that nanofluid properties are used as in the perturbation form from (4.3).
Systematic substitution and expansion for small volume fractions recover the
same relation (5.2). This shows that empirically one can obtain the same
heat transfer relation, without addressing the mechanisms, as obtained from
the differential equation derivation which accounts for the temperature profile
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Nanofluid heat transfer intensification
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modifications. Thus, the derived nanofluid heat transfer relation is embedded
in the otherwise empirically used base fluid ‘correlation’ in the simplified
present example.
In order to explicitly obtain the extent of enhancement, the enhancement
relation from (5.2), (5.3) hx /hx,f − 1 − f∞ (r∗p cp∗ − 1 + b1 )/2 should be examined.
For enhancement to exceed merely the thermal conductivity effect, then r∗p cp∗ >
1 + b1 . However, for examples of nanofluids under discussion, r∗p cp∗ < 1. It is
recalled that presently the ideal mixture relations is used for both the nanofluid
density and heat capacity per unit volume. Most recently, Puliti et al. (2011),
on the basis of molecular dynamics computations, found that for gold-water
nanolayer mixtures, the molecular dynamics-obtained heat capacity is nearly
twice that from ideal mixture results, (rc)MD ≈ 1.85(rc)ideal/mix . The implication
to explaining heat transfer enhancement is enormous in that inertia effects, in
addition to thermal conductivity, give rise to enhanced heat transfer.
The temperature and velocity profile modifications, (4.12) and (4.15),
respectively, could conceivably be measured in the laboratory. It can be seen
that awaited is also improvements to the ideal mixture relations (Puliti et al.
2011), in the form of a similar equation of state, that can be incorporated into
the explanation of heat transfer enhancement in nanofluids within the continuum
transport description.
I am indebted to E. Pfautsch and J. Buongiorno for my introduction to nanofluids (Buongiorno
2006; Pfautsch 2008). The preliminary versions of this work were presented at the Int. Conf.
Nanofluids: Fundamentals and Applications II, 15–19 August 2010 Montréal, 63rd DFD/APS
Meeting, 19–21 November 2010, San Diego and at the 64th DFD/APS Meeting, 20–22 November
2011, Baltimore.
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