Journal of Magnetism and Magnetic Materials 382 (2015) 296–306
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials
journal homepage: www.elsevier.com/locate/jmmm
MHD mixed convection in a vertical annulus filled with Al2O3–water
nanofluid considering nanoparticle migration
A. Malvandi a,n, M.R. Safaei b, M.H. Kaffash a, D.D. Ganji c
a
Department of Mechanical Engineering, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran
Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran
c
Mechanical Engineering Department, Babol University of Technology, Babol, Iran
b
art ic l e i nf o
a b s t r a c t
Article history:
Received 26 November 2014
Received in revised form
16 January 2015
Accepted 23 January 2015
Available online 24 January 2015
In the current study, an MHD mixed convection of alumina/water nanofluid inside a vertical annular pipe
is investigated theoretically. The model used for the nanofluid mixture involves Brownian motion and
thermophoretic diffusivities in order to take into account the effects of nanoparticle migration. Since the
thermophoresis is the main mechanism of the nanoparticle migration, different temperature gradients
have been imposed using the asymmetric heating. Considering hydrodynamically and thermally fully
developed flow, the governing equations have been reduced to two-point ordinary boundary value
differential equations and they have been solved numerically. It is revealed that the imposed thermal
asymmetry would change the direction of nanoparticle migration and distorts the velocity, temperature
and nanoparticle concentration profiles. Moreover, it is shown that the advantage of nanofluids in heat
transfer enhancement is reduced in the presence of a magnetic field.
& 2015 Elsevier B.V. All rights reserved.
Keywords:
Nanofluid
Nanoparticle migration
MHD
Mixed convection
Asymmetric heating
1. Introduction
Economic incentives, energy saving and space considerations
have increased efforts to construct a more efficient heat exchange
equipment. Many techniques have been presented by researchers
to improve heat transfer performance, which is referred to as heat
transfer enhancement, augmentation, or intensification. Bergles
[1] was the first that classified the heat transfer enhancement
techniques to (a) active techniques which require external forces
to maintain the enhancement mechanism such as an electrical
field or vibrating the surface and (b) passive techniques which do
not require external forces, including geometry refinement [2],
special surface geometries [3], or fluid additives.
Active techniques commonly present a higher augmentation
thought they need additional power that increases initial capital
and operational costs of the system. In this class, the study of the
magnetic field has important applications in medicine, physics and
engineering. Many industrial types of equipment, such as MHD
generators, pumps, bearings and boundary layer control are affected by the interaction between the electrically conducting fluid
and a magnetic field. The behavior of the flow strongly depends on
the orientation and intensity of the applied magnetic field. The
exerted magnetic field manipulates the suspended particles and
n
Corresponding author. Fax: þ98 21 65436660.
E-mail address: amirmalvandi@aut.ac.ir (A. Malvandi).
http://dx.doi.org/10.1016/j.jmmm.2015.01.060
0304-8853/& 2015 Elsevier B.V. All rights reserved.
rearranges their concentration in the fluid which strongly changes
heat transfer characteristics of the flow. The seminal study about
MHD flows was conducted by Alfvén who won the Nobel Prize for
his works. Later, Hartmann did a unique investigation on this kind
of flow in a channel. Effects of MHD on nanofluids have been
considered by Sheikholeslami et al. [4–8] on free convection of
nanofluids in enclosures, Rashidi et al. [9,10] for entropy generation of nanofluids over a rotating disk, Uddin et al. [11] for hydromagnetic transport of nanofluids over a stretching sheet, and
Malvandi et al. [12–15] for considering the effects of magnetic field
on nanoparticle migration. A good review on this subject is given
by Bahiraei and Hangi [16].
Among different passive techniques, particles as additives in
the working fluids have burst onto the scene of engineering research which emerged in 1873 [17] and is developing rapidly. The
motivation was to improve the thermal conductivity of the most
common fluids such as water, oil, and ethylene–glycol mixture,
with the solid particles which have intentionally higher thermal
conductivity. Then, many researchers studied the influence of solid–liquid mixtures on potential heat transfer enhancement. But,
they were confronted with problems such as abrasion, clogging,
fouling and additional pressure loss of the system, which makes
these unsuitable for heat transfer systems. In 1995, the word
“nanofluid” was proposed by Choi [18] to indicate dilute suspensions formed by functionalized nanoparticles smaller than 100 nm
in diameter which had already been created by Masuda et al. [19]
as Al2O3–water. These nanoparticles are fairly close in size to the
A. Malvandi et al. / Journal of Magnetism and Magnetic Materials 382 (2015) 296–306
Nomenclature
B
d
cp
DB
Dh
DT
g
h
Ha
HTC
k
k BO
Ng
Nr
NBT
Np
p
qw
R
T
u
x, r
uniform magnetic field strength
nanoparticle diameter (m)
specific heat (m2/s2 K)
Brownian diffusion coefficient
hydraulic diameter (m)
thermophoresis diffusion coefficient
gravity (m/s2)
heat transfer coefficient (W/m2 K)
Hartmann number
dimensionless heat transfer coefficient
thermal conductivity (W/m K)
Boltzmann constant (¼ 1.3806488 10 23 m2 kg/s2 K)
mixed convective parameter due to temperature
gradient
mixed convective parameter due to nanoparticle
distribution
ratio of the Brownian to thermophoretic diffusivities
non-dimensional pressure drop
pressure (Pa)
surface heat flux (W/m2)
radius (m)
temperature (K)
axial velocity (m/s)
coordinate system
molecules of the base fluid and, thus, can enable extremely stable
[20] suspensions with only slight gravitational settling over long
periods.
Along with the same proposition, theoretical studies emerged
to model the nanofluid behaviors. In the beginning, the models
were twofold: homogeneous models and dispersion models. In
2006, Buongiorno [21] stated that the homogeneous models are in
disagreement with the experimental observations and tend to
underpredict the nanofluid heat transfer coefficient. In addition,
the dispersion effect is completely negligible due to the nanoparticle size. Thus, Buongiorno developed an alternative model to
explain the abnormal convective heat transfer in nanofluids and so
eliminate the shortcomings of the homogeneous and dispersion
models. He asserted that nanoparticle migration is responsible for
the abnormal heat transfer rate in nanofluids. Taking this finding
as a basis, he proposed a two-component four-equation nonhomogeneous equilibrium model for convective transport in nanofluids. Then, a comprehensive survey of convective transport of
nanofluids were conducted by Kuznetsov and Nield [22] to study
influence of nanoparticles on natural convection boundary-layer
flow past a vertical plate, Goodarzi et al. [23] for two-phase simulation of nanofluids in a shallow cavity, Soleimani et al. [24] for
CFD simulation of free convection of nanofluids in a semi-annulus
enclosure, Safaei et al. [25] for heat transfer enhancement in a
forward-facing contradicting channel, Malvandi et al. [26–30], and
Garoosi et al. [31,32]. Recently, Buongiorno's model has been
modified by Yang et al. [33,34] to fully account for effects of the
nanoparticle volume fraction. Next, Malvandi et al. [35] considered
the modified model for fully developed mixed convection of nanofluids in a vertical annulus. They indicate that the modified
model is suitable for considering effects of nanoparticle migration
in nanofluids. Then, the modified Buongiorno's model has been
applied to different heat transfer concepts, including forced [36–
39], mixed [40–42], and natural convections [43,44].
In the current research, the effects of nanoparticle migration on
hydromagnetic (MHD) mixed convection of alumina/water
297
Greek symbols
θ
ϕ
γ
η
μ
ρ
σ
ζ
ε
non-dimensional temperature
nanoparticle volume fraction
ratio of wall and fluid temperature difference to absolute temperature
transverse direction
dynamic viscosity (kg/m s)
density (kg/m3)
electric conductivity
radius ratio
heat flux ratio
Subscripts
B
bf
i
o
p
bulk mean
base fluid
condition at the inner wall
condition at the outer wall
nanoparticle
Superscripts
n
dimensionless variable
nanofluid inside a vertical annular pipe is theoretically investigated. Since the thermophoresis is the main mechanism of
the nanoparticle migration, different temperature gradients have
been imposed using the assymetric heating. The effects of a uniform magnetic field, asymmetric heating effects, the migration of
nanoparticles, and how these affect the hydrodynamic and thermal characteristics of the system are of particular interest.
2. Migration of nanoparticles
It is well known that nanoparticles do not passively follow the
fluid streamlines. Migration of nanoparticles has considerable effects on rheological and thermophysical properties of the nanofluids. For considering the nanoparticle migration, it can be assumed that the suspended nanoparticles can homogeneously be in
motion with the fluid, considering a slip velocity relative to the
fluid. Because of the very small dimension of the nanoparticles
(o100 nm), Brownian and thermophoretic diffusivities are the
only significant slip mechanisms which are responsible for nanoparticle migration in nanofluids, as Buongiorno [21] stated.
Brownian diffusion can be observed due to random drifting of
suspended nanoparticles within the base fluid which comes from
continuous collisions between nanoparticles and liquid molecules.
It is proportional to the concentration gradient and described by
the Brownian diffusion coefficient, DB, which is given by the Einstein–Stokes's equation
DB =
kB T
3πμ bf d p
(1)
where kB is the Boltzmann's constant, μbf is the dynamic viscosity
of the base fluid, T is the local temperature and dp is the nanoparticle diameter. The nanoparticle flux due to Brownian diffusion
( Jp, B ) can be given as
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J p, B = − ρ pDB ∇ϕ
(2)
On the other hand, the thermophoresis (“particle” equivalent of
the Soret effect), tries to induce the nanoparticles migration in the
opposite direction of the temperature gradient (warmer to colder
region), causing a non-uniform nanoparticle distribution. The
thermophoresis is described by the thermal diffusion, DT, which is
given by
DT = β
μ bf
ρ bf
ϕ
(3)
where β = 0.26(kbf /(2k bf + k p )). The nanoparticle flux due to
thermophoresis ( Jp, T ) can be calculated as
Jp, T = − ρp DT
∇T
T
(4)
Therefore, the total nanoparticle flux consists of two parts as described above
⎛
∇T ⎞
⎟
Jp = Jp, B + Jp, T = − ρp ⎜DB ∇ϕ + DT
⎝
T ⎠
(5)
Since D B ∼ T and DT ∼ ϕ (depend on the flow field), it is advantageous to re-write Eq. (5) as follows [45]:
⎛
∇T ⎞
⎟
Jp = − ρp ⎜CBT ∇ϕ + CT ϕ
⎝
T ⎠
(6)
where CB = D B /T and CT = DT /ϕ do not depend on the flow field.
As a result, distribution of nanoparticle can be obtained via
∂t (ϕ) + ∇⋅ (uϕ) = −
⎛
∇T ⎞
1
⎟
∇ J p = ∇⋅ ⎜CB T ∇ϕ + CT ϕ
⎝
ρp
T ⎠
( )
(7)
3. Problem formulation and governing equations
Consider an MHD, laminar and two-dimensional flow of the
alumina/water nanofluid inside a vertical annular pipe, which is
subjected to different heat fluxes at the inner (qi″) and outer (qo″)
qi″/qo″,
which characterizes
walls. The ratio of the heat fluxes is ε =
the degree of the thermal asymmetry. The geometry of the problem is shown in Fig. 1. A two-dimensional coordinate frame has
been selected where the x-axis is aligned vertically and the r-axis
is normal to the walls. A modified two-component heterogeneous
model is employed for the nanofluid in the hypothesis that the
Brownian motion and the thermophoresis are the only significant
bases of nanoparticle migration. This model involves the following
assumptions: incompressible flow, no chemical reactions, dilute
mixture, negligible viscous dissipation, negligible radiation, and
local thermal equilibrium between the nanoparticles and base
fluid. Consequently, the basic incompressible conservation equations of the mass, momentum, thermal energy, and nanoparticle
fraction can be expressed in the following manner [21,35]:
Continuity equation
∂t (ρ) + ∇⋅ (ρu) = 0
(8)
Fig. 1. The geometry of physical model and coordinate system.
τ = μ (∇u + (∇u)t ) is the shear stress, β is the nanofluid thermal
expansion, g is the gravity, and q is the energy flux relative to the
nanofluid velocity, which can be expressed as the sum of the
conduction and diffusion heat flux as below
q=
) (ϕ − ϕB ) ⎤⎥⎦ g
is
(11)
= ϕρp + (1 − ϕ) ρ bf
cp
ϕρp c p p + (1 − ϕ) ρ bf c pbf
ρ
,
k
= k bf (1 + 7.47ϕ),
β
∂t (ρcT ) + ∇⋅ (ρc uT ) = − ∇⋅q + h p ∇⋅J p
hp
⏟
ρ
(9)
Energy equation
where
h p Jp
nanoparticle diffusion heat flux
μ = μ bf (1 + 39.11ϕ + 533.9ϕ2),
=
(
+
Further, ρ , μ, k , c are the density, dynamic viscosity, thermal
conductivity, and specific heat capacity of alumina/water nanofluid respectively, depending on the nanoparticle volume fraction
as follows:
Momentum equation
∂ t (ρu) + ∇. (ρuu) = − ∇p + ∇⋅τ − σB 2u
⎡
+ ⎢⎣ (1 − ϕ B ) ρbf β (T − TB ) − ρp − ρbf
k∇
T
−

conduction heat flux
the
specific
(10)
enthalpy
of
nanoparticles,
=
ϕρp βp + (1 − ϕ) ρ bf βbf
ρ
(12)
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A. Malvandi et al. / Journal of Magnetism and Magnetic Materials 382 (2015) 296–306
Table 1
Validation of the results with the ones reported by Kays and Crawford [46] when
Ha = Nr = Ng = ϕ B = 0 .
ζ
ε
Kays and Crawford [46]
Present study
HTCo
HTCi
HTCi
HTCo
0.2
0
0.5
1
1.5
0
10.493
89.463
21.426
4.883
5.151
5.450
5.787
0
10.513
88.712
21.397
4.883
5.150
5.449
5.784
0.6
0
0.5
1
1.5
0
109.481
11.218
8.635
5.099
5.812
6.758
8.071
0
115.490
11.248
8.646
5.099
5.813
6.759
8.072
1a
0
0.5
1
1.5
0
17.484
8.234
7.000
5.385
6.511
8.234
11.195
0
17.500
8.235
7.000
5.385
6.512
8.235
11.200
a
In order to avoid singularity at ζ ¼1, the results are obtained at ζ ¼0.99999.
Table 2
Comparison of NuB with the reported data of Yang et al. [33] when
Nr = Ha = γ = Ng = 0, ϕ B = 0.02.
NBT
Yang et al.
Present work
Error (%)
0.2
0.4
0.6
0.8
1
2
4
6
8
10
10.359
10.238
10.189
10.163
10.146
10.11
10.09
10.086
10.082
10.08
10.3591
10.239
10.19
10.1635
10.1467
10.111
10.092
10.0861
10.0828
10.081
0.0010
0.0098
0.0098
0.0049
0.0069
0.0099
0.0198
0.0010
0.0079
0.0099
Table 3
Grid independence test for different values of dη when Ha ¼0, NBT ¼ 1, ϕ B = 0.02,
ε¼ 0.5, ζ¼ 0.4, and Nr ¼ Ng ¼50.
where bf stands for base fluid and p for particle. In addition, the
thermophysical properties of Al2O3 nanoparticle and base fluid
(water) are also provided as follows:
⎧
⎪
⎪ c p p = 773 J /kg K ,
= 998.2 kg/m3 ,
⎪
⎪ ρp = 3880 kg/m3 ,
Al 2O 3⎨
= 0.597 W /m K,
⎪ k p = 36 W /m K ,
⎪
= 9.93 × 10−4 kg/m s,
⎪ β = 8.4 × 10−6 1/K
⎪ p
4
−
= 2.066 × 10 1/K
⎩
k bf
μ bf
β bf
10−5
25.8497
8.1136
148.0673
5 × 10−6
25.8520
8.1124
148.0666
10−6
25.8573
8.1109
148.0641
HTCo
Np
dp
1 d ⎛ du ⎞
− σB 2u + ⎡⎣ (1 − ϕ B ) ρβ (T − TB ) − (ρ − ρ bf )(ϕ − ϕ B ) ⎤⎦ g = 0
⎜ rμ ⎟ −
r dr ⎝ dr ⎠
dx
(14)
ρc p u
c pbf = 4182 J /kg K,
ρbf
HTCi
equating ∇h p = c p ∇T , one may simply obtain governing equations
for steady, incompressible, hydrodynamically and thermally fully
developed flow as follows:
Fig. 2. Algorithm of the numerical method.
⎧
⎪
⎪
⎪
⎪
Water⎨
⎪
⎪
⎪
⎪
⎩
dη
⎛ ∂ϕ
DT ∂T ⎞ ∂T
dT
1 d ⎛ dT ⎞
⎟
⎜rk ⎟ + ρp c p p ⎜DB
+
=
⎝ ∂r
dx
r dr ⎝ dr ⎠
T ∂r ⎠ ∂r
CT ϕ ∂T ⎞
1 ∂ ⎛
∂ϕ
+
⎟=0
⎜CB T
r ∂r⎝
T ∂ r⎠
∂r
(13)
Since the rheological and thermophysical properties
( ρ , μ, k and c ) are dependent to the nanoparticle concentration,
the nanoparticle distribution equation, Eq. (7), should be coupled
with Eqs. (8)–(10). Thus, substituting Eq. (11) into Eq. (10) and
(15)
(16)
According to Buongiorno [21], heat transfer associated with
nanoparticle diffusion (second RHS term of Eq. (15)) can be neglected in comparison with the other terms. In addition, By averaging Eq. (15) from r = Ri to Ro and according to the thermally fully
developed condition for the uniform wall heat flux
(dT /dx = dTB/dx ) and introducing the following non-dimensional
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A. Malvandi et al. / Journal of Magnetism and Magnetic Materials 382 (2015) 296–306
parameters:
η=
2r
,
Dh
θ=
T − Ti
,
(qo + qi ) D h
u⁎ =
Eqs. (14)–(16) can be reduced as
u
,
D h2/ μbf ( − dp/ dx)
γ=
⎛1
d2u⁎
1 dμ dϕ ⎞ du⁎
=−⎜ +
⎟
2
μ dϕ dη ⎠ dη
⎝η
dη
μ bf
⎡1 − Nr ϕ − ϕ + Ngρ⁎ β ⁎ (θ − θ ) − Ha2u⁎⎤
−
B
(
⎣
⎦
B)
4μ
(qo + qi ) D h
TB kbf
kbf
⎤
⎡
βp
1
1
⎥,
β⁎ = ⎢
+
⎢⎣ 1 + ((1 − ϕ) ρ bf / ϕρ p ) β bf
1 + (ϕρ p /(1 − ϕ) ρ bf ) ⎥⎦
q″
ε= i,
qo″
k bf
d2θ
=−
2
k
dη
ρ⁎ = (1 − ϕ) + ϕρ p / ρ bf
(D h (qo + qi )/ kbf ) ρ bf β bf g
,
Ng =
−dp/ dx
Nr =
(ρ − ρ bf ) g
Ha2 =
− (dp/ dx)
σB 2D h2
μbf
,
⎡ ρcu⁎
⎛
k/k bf ⎞ dθ ⎤
(1 + ζε)
dϕ
⎢−
⎟ ⎥
+ ⎜7.47
+
⎢⎣ ρcu⁎ (1 + ζ)(1 + ε) ⎝
η ⎠ dη ⎥⎦
dη
∂θ
∂ϕ
ϕ
=−
∂η
NBT [1 + γθ]2 ∂η
Ri
,
ζ=
Ro
(17)
(18)
(19)
(20)
where the average value of parameters can be calculated over the
cross-section by
Fig. 3. The effects of NBT (ϕ B = 0.06 ) and ϕ B (NBT ¼0.5) on nanoparticle distribution (ϕ/ϕ B ), velocity (u/uB) and temperature (θ/θB) profiles when ε ¼0.5, Nr ¼ 50, Ng¼ 50,
Ha¼ 5, and ζ ¼0.6.
A. Malvandi et al. / Journal of Magnetism and Magnetic Materials 382 (2015) 296–306
301
Fig. 4. The effects of Ng and Nr (Ha¼ 5) and Ha (Nr ¼Ng ¼ 50) on nanoparticle distribution (ϕ/ϕ B ), velocity (u/uB) and temperature (θ/θB) profiles when ε¼ 0.5, ϕ B = 0.06 ,
NBT ¼ 0.5, and ζ¼ 0.6.
Γ ≡
1
A
∫A Γ dA=
1
π(
R o2
−
Ri2
)
∫R
Ro
Γ (2πr) dr
i
(21)
T B⁎ ,
Hence, the bulk mean dimensionless temperature
and the bulk
mean nanoparticle volume fraction ϕ B can be obtained by
θB ≡
ρc puθ
ρc pu
,
ϕB =
u⁎ϕ
u⁎
the solid boundary (no-slip condition). Different heat fluxes are
taken into account at the walls, qi″ for the inner wall and the heat
flux at outer wall is considered to be qo″. As a result, appropriate
boundary conditions for this problem can be expressed as
r = Ri : u = 0,
(22)
3.1. Boundary conditions
The fluid velocity at all fluid–solid boundaries is equal to that of
− ki
∂T
= qi″,
∂r
CT ϕ ∂T
∂ϕ
+
= 0. r = R o: u = 0,
T ∂r
∂r
CT ϕ ∂T
∂T
∂ϕ
ko
= qo″, CB T
+
= 0.
T ∂r
∂r
∂r
CB T
(23)
Substuting Eq. (20) into Eq. (26), the boundary conditions can
be expressed as
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Fig. 5. The effects of ζ (ε¼ 0.5) and ε (ζ ¼ 0.6) on nanoparticle distribution (ϕ/ϕ B ), velocity (u/uB) and temperature (θ/θB) profiles when ϕ B = 0.06 , Nr ¼50, Ng ¼50, Ha ¼5, and
NBT ¼0.5.
η=
ζ
: u⁎ = 0,
1−ζ
ϕ = ϕw = ϕi η =
ε
∂θ
,
=−
∂η
2 (1 + 7.47ϕi ) (1 + ε)
1
: u⁎ = 0,
1−ζ
1
∂θ
=
∂η
2 (1 + 7.47ϕo ) (1 + ε)
(24)
4. Numerical method and accuracy
Eqs. (18)–(20) are solved in conjunction with the boundary
conditions of Eq. (24) by means of the Runge–Kutta–Fehlberg
scheme. Convergence criterion is considered to be 10 6 for relative errors of the velocity, temperature, and nanoparticle volume
fraction. The numerical procedure involves a reciprocal algorithm
in which ϕw , ρcu⁎, and θ B are used to solve the governing equations.
The process is repeated until a prescribed value of ϕ B reached, and
the relative errors between the assumed values of ρcu⁎ and θ B with
the calculated ones after solving Eqs. (18)–(20) are lower than
10 6. In view of helping others to regenerate their own results and
provide possible future references, the numerical algorithm is
shown graphically in Fig. 2.
To check the accuracy of the numerical code, the results obtained for a horizontal annulus with Nr ¼ ϕB ¼Ha¼ Ng¼ 0 and
different values of ε and ζ are compared to the reported results of
Kays and Crawford [46] in Table 1. Further, a comparison for
HTCt (kbf /k B ) with the reported results of Yang et al. [33] is presented in Table 2. Obviously, the maximum percentage difference
is less than 0.02%; so, the results are in a desirable accuracy. In
A. Malvandi et al. / Journal of Magnetism and Magnetic Materials 382 (2015) 296–306
Fig. 6. The effects of NBT and ϕ B on the total heat transfer coefficient (a), and the
pressure drop (b) when ε¼ 0.5, Nr ¼50, Ng¼ 50, Ha¼ 5, and ζ ¼0.6.
addition, the numerical code developed here is run on three different integration steps (dη ) of 10 5, 5 10 6, and 10 6 to verify
the results are independent of the grid size. The obtained numerical results are presented in Table 3. The results clearly indicate
the grid independent of the code. Accordingly, all the numeric
results obtained here are carried out using the integration step
dη = 10−6 .
5. Results and discussions
Migration of nanoparticles, the viscosity and thermal conductivity distributions are determined by the mutual effects of the
Brownian diffusion and the thermophoresis. Here, these effects are
considered by means of NBT, which is the ratio of the Brownian
diffusion to the thermoporesis. With d p ≅ 20 nm and ϕ B ≅ 0.1, the
ratio of Brownian motion to thermophoretic forces NBT ∝ 1/d p can
be changed over a wide range of 0.2–10 for alumina/water nanofluid. In addition, the results have been carried out for
γ ≅ (Tw − TB )/Tw = 0.1, since its effects on the solution is
303
Fig. 7. The effects of Ha and Nr on the total heat transfer coefficient (a), and the
pressure drop (b) when ε ¼0.5, ϕ B = 0.06 , NBT ¼ 0.5, Ng ¼50, and ζ¼ 0.6.
insignificant, see Refs. [33,34].
5.1. Velocity, temperature and concentration profiles
The effects of NBT on the nanoparticle volume fraction (ϕ/ϕ B ),
velocity (u/uB), and temperature (θ/θB) profiles are shown in
Fig. 3a. For the lower values of NBT, the nanoparticle volume
fraction ejects themselves at the heated walls and accumulate at
the core region. In contrast, at the higher values of NBT, a more
uniform nanoparticle distribution can be obtained. Migration of
the nanoparticles from the walls toward the core region at the
lower values of NBT constructs nanoparticle-depleted regions near
the walls. This reduces the viscosity, and so reduces the shear
stress of the nanofluid on the walls. However, the viscosity and
shear stress increase in the core region. Hence, the velocity near
the walls increases, especially on the inner wall, which has a lower
nanoparticle volume fraction. Furthermore, it can be easily seen
that the nanoparticle concentration takes its lowest value at the
inner wall (the higher wall heat flux), a slight increase to the
maximum in the region far from the wall, but decreased rapidly
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A. Malvandi et al. / Journal of Magnetism and Magnetic Materials 382 (2015) 296–306
Fig. 8. The effects of ζ and ε on the total heat transfer coefficient (a), and the
pressure drop (b) when ϕ B = 0.06 , Nr ¼50, Ng¼ 50, Ha¼ 5, and NBT ¼ 0.5.
towards the outer wall (the lower wall heat flux). This is purely
because the heat flux at the inner wall is higher than at the outer
wall; so the temperature gradient and the thermophoresis are
greater at the inner wall. It is not surprising that the peak of the
nanoparticle concentration is closer to the outer wall, having a
lower heat flux. The variations in the nanoparticle volume fraction
(ϕ/ϕ B ), velocity (u/uB), and temperature (θ/θB) profiles for different
values of ϕ B are shown in Fig. 3b. There is a downward trend for
the volume fraction of nanoparticles in the core region as well as a
reversed behavior for that at the walls, as increasing ϕ B . In other
words, an increase in ϕ B leads to a more uniform nanoparticle
volume fraction distribution. Also, it can be observed that for the
lower values of ϕ B , the nanoparticle depletion effects become
significant and the velocities move further to the outer walls.
Fig. 4a shows the effects of mixed convective parameters due to
temperature gradient (Ng) and nanoparticle distribution (Nr) on
the profiles. When Ng and Nr increase, peak of the velocity no
longer remains at the core region, but moves toward the heated
walls, particularly near the outer wall where the nanoparticle
concentration is the lowest. This is because increasing Nr and Ng,
intensify the buoyancy forces, which accelerate the momentum
near the walls and due to a constant mass flow rate inside the
annulus, the velocities in the core region decrease. Accordingly,
the nanoparticle concentration and temperature gradients reduce;
so, their profiles become more uniform, as mixed convective
parameters increase. An examination of Fig. 4b reveals a continuing increase in the velocities of the fluid close to the outer wall
(lowest nanoparticle concentration), followed by a decrease in the
core region, as Ha increases. In essence, the velocities are forced to
move slowly close to the outer wall (low viscosity region). This is
because the transverse magnetic field induces a resistive type
force (Lorentz force), which is a retarding force on the velocity
field. Thus, the velocities reduce at the core region and due to a
constant mass flow rate, the velocities near the walls should increase. The increment of the velocities is more likely to take place
near the outer wall which has a lower viscosity region.
The effects of radii (ζ) and heat fluxes (ε) ratios on the profiles
have been demonstrated in Fig. 5a and b, respectively. As ζ increases, the effects of inner wall is increased, which strength the
effects of viscous forces; so the velocities shift toward the inner
wall. As a result, the temperature gradients are reduced. Regarding
Fig. 5b, for ε < 1 the nanoparticle rich region is constructed near
the inner wall, having a lower heat flux. Increasing ε shifts the
peak of the nanoparticle volume fraction toward the outer wall.
This is due to the fact that the thermophoresis, which is related to
the temperature gradient, is the mechanism of the nanoparticle
migration. Any change in ε leads the temperature gradient at the
walls to change, so changes the thermophoresis. For ε < 1, the
temperature gradient at the outer wall is more than that at the
inner wall; so the nanoparticle migration at the outer wall is
greater, leading the nanoparticle accumulated region to move toward the inner wall. This phenomena continues until ε = 0, in
which there is no temperature gradient at the inner wall; so a
nanoparticle accumulation region formed at the inner wall. The
effects of ε on nanoparticles distribution have considerable influence on the velocity and temperature profiles. Evidently, a regular
symmetry in the velocity profile disappears and the peak of the
velocity profile moves toward the outer wall (the lower viscosity
region), as ε decreases. However, the dip point of the temperature
profile increases and moves toward the inner wall in which the
nanoparticles accumulated (the higher thermal conductivity region). In fact, the velocity profile has a tendency to shift toward the
nanoparticle depleted region, however, for the temperature profile
it is the opposite.
5.2. Heat transfer rate and pressure drop
The dimensionless heat transfer coefficient (HTC) at the inner
and the outer walls can be defined respectively as
HTCi =
qi″ Dh
hi Dh
ε
=
=−
k bf
(Ti − TB ) k bf
(1 + ε) θ B
(25)
HTCo =
qo″ Dh
ho D h
1
=
=
k bf
(To − TB ) k o
(1 + ε)(θo − θ B )
(26)
The total heat transfer ratio can be expressed as
HTCt =
HTCi Ri + HTCo R o
HTCi ζ + HTCo
=
Ri + R o
ζ+1
(27)
and the non-dimensional pressure drop can be defined as
⎛ dp ⎞ ⎛ μ bf uB ⎞
ρ
⎟= B
Np = ⎜− ⎟/⎜⎜
⎝ dx ⎠ ⎝ Dh2 ⎟⎠
ρu⁎
(28)
Figs. 6–8 show the effects of the parameters ϕ B , NBT, Nr, Ha, ζ,
A. Malvandi et al. / Journal of Magnetism and Magnetic Materials 382 (2015) 296–306
and ε on the total heat transfer rate and the pressure drop, respectively. In addition, the solid line represents the corresponded
value for the base fluid. It is shown in Fig. 6a and b that increasing
NBT intensifies the heat transfer rate, however, for the pressure
drop it is the opposite. Furthermore, increasing ϕ B leads to a rise in
both the heat transfer rate and the pressure drop. In fact, the total
heat transfer coefficient and the pressure drop for the nanofluid
are always greater than that of the base fluid. Fig. 7a and b shows
the effects of Nr and Ha on the heat transfer rate and the pressure
drop. For nanofluids, increasing Nr reduces the heat transfer rate,
whereas the pressure drop is increased. Accordingly, Nr has a
negative effects on the performance. When Ha increases, the heat
transfer rate of the base fluid is increased, while for the nanofluid
it is vice versa. Therefore, it can be concluded that there is no merit
in coupling of the magnetic field and nanparticles inclusion. In
other words, inclusion of nanoparticles would decrease the heat
transfer rate of the fluids in the presence of the magnetic field. The
effects of radii ratio ζ and heat flux ratio ε on heat transfer rate and
pressure drop are shown in Fig. 8a and b. It is obvious that the heat
transfer rate is very sensitive to ζ for ε o1. In this range, inclusion
of nanoparticles enhanced the heat transfer rate, which is more
apparent for the lower values of ζ. It can be observed that for the
lower values of ε and ζ, where the heat flux and surface at the
outer wall is relatively greater than that at the inner wall, the heat
transfer rate becomes negative (direction of the heat transfer rate
has been changed at the inner wall). Increasing ζ at a constant
value of ε, enhances the heat transfer rate. Further rise in ζ, increases the effects of viscous forces and decreases the velocities;
so, the heat transfer rate is reduced. In conclusion, there is an
optimum value for ζ at ε o1, which results in a greatest heat
transfer coefficient. In contrast, for ε 41, the heat transfer rate has
an increasing trend with ε and ζ. In addition, the pressure drop is
almost reduced with increasing ζ, except for the higher values of ε,
as it can be observed in Fig. 8b.
6. Summary and conclusions
An MHD mixed convection of Al2O3–water nanofluid inside a
vertical annulus is theoretically investigated. Walls are subjected
to different heat fluxes; qi″ for the inner wall and qo″ for the right
wall, and nanoparticles are assumed to have a slip velocity relative
to the base fluid. Assuming a fully developed flow and heat
transfer, the basic partial differential equations including continuity, momentum, and energy equations have been reduced to
two-point ordinary boundary value differential equations before
they are solved numerically. The effects of different parameters
including the ratio of Brownian motion to thermophoretic diffusivities NBT, the ratio of heat fluxes at the walls ε, Hartmann
number Ha, and bulk mean nanoparticle volume fraction ϕ B on the
heat transfer rate and the pressure drop were investigated in detail. The major findings of this paper are as follows:
The imposed thermal asymmetry would change the direction
of nanoparticle migration and distorts the velocity, temperature and nanoparticle concentration profiles. Accordingly, the
heat transfer rate and the pressure drop for the nanofluids are
significantly sensitive to the imposed thermal boundary
condition.
The nanoparticles are more likely to accumulate toward the
wall with a lower heat flux. As a result, an in-homogeneous
distribution of nanoparticles developed which leads to a nonuniform distribution of the viscosity and thermal conductivity
of nanofluids. Thus, the velocities shift toward the nanoparticle
depleted region, however, the dip point of the temperature
305
profiles move toward the nanoparticle accumulated region.
Inclusion of nanoparticles in the presence of a magnetic field
has a negative effect on the performance.
There is an optimum value for ζ at ε o1, which results in a
greatest heat transfer rate. In contrast, for ε 4 1, the heat
transfer rate has an increasing trend with ε and ζ.
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