International Communications in Heat and Mass Transfer 50 (2014) 117–127
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ichmt
Augmentation of natural convection heat transfer in triangular shape solar collector by
utilizing water based nanofluids having a corrugated bottom wall☆
M.M. Rahman a,b,⁎, S. Mojumder c, S. Saha c, S. Mekhilef a, R. Saidur d
a
Department of Electrical Engineering, Faculty of Engineering, University of Malaya, 50603, Kuala Lumpur, Malaysia
Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh
Department of Mechanical Engineering, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh
d
Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia
b
c
a r t i c l e
i n f o
Available online 6 November 2013
Keywords:
Nanofluid
Solar thermal collector
Solid volume fraction
Corrugated wall
Finite element method
a b s t r a c t
Nanofluids have been introduced for the enhancement in the heat transfer phenomena in the last few years. In this
paper a corrugated bottom triangular solar collector has been studied introducing water based nanofluids inside
the enclosure. The corrugated bottom is kept at a constant high temperature whereas the side walls of the triangular enclosure are kept at a low temperature. Three types of nanoparticles are taken into consideration: Cu,
Al2O3, and TiO2. The effect of solid volume fraction (ϕ) of the nanoparticle of nanofluid has been studied numerically by Galerkin weighted residual method of finite element for a wide range of Grashof number (Gr) 104–106.
Calculations are carried out for ϕ = 0, 0.05, 0.08, and 0.1 and dimensionless time, τ = 0.1, 0.5, and 1. For the specified conditions streamlines and isotherm contours are obtained and detailed results of the interaction between
different parameters are studied using overall Nusselt number. It has been found that both Grashof number and
solid volume fraction have significant influence on streamlines and isotherms in the enclosure. It is also found
that heat transfer increased by 24.28% from the heated surface as volume fraction ϕ increases from 0% to 10% at
Gr = 106 and τ = 1 for copper water nanofluid.
© 2013 Elsevier Ltd. All rights reserved.
1. Introduction
A prodigious importance has been given to the natural convection
heat transfer phenomena as it has a very wide range of application in
heat exchangers, solar collector, electronics cooling, desalination process and so on [1–4]. However the conventional fluid used such as air,
water, ethylene glycol etc. for the natural convection has a very low
thermal conductivity and cannot fulfill the demand of high thermal conductivity of fluid which is a desired property of fluid nowadays. To enhance the thermal conductivity in addition to the heat transfer rate for
the last few years a new advanced technique has been deployed by
mixing nanosized (less than 100 nm) particles such as metal, metal
oxide and carbon materials etc. in the base fluid. And this fluid is called
nanofluid which has a very good characteristic of thermal conductivity
which can meet the challenge of higher heat transfer rate in modern engineering application. Basically nanosized particle suspension in the
base fluid enhances the heat transfer rate but still there are controversies over whether heat transfer is increased due to the nanoparticle or
not. Numerous research works have been carried out on this regard
which show both positive and negative results.
☆ Communicated by W.J. Minkowycz.
⁎ Corresponding author at: Department of Electrical Engineering, Faculty of Engineering,
University of Malaya, 50603 Kuala Lumpur, Malaysia.
E-mail addresses: [email protected], [email protected] (M.M. Rahman).
0735-1933/$ – see front matter © 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.10.008
Khanafer et al. [5] have investigated numerically the effect of
nanofluid assuming that the nanofluid is in single phase and reported
that heat transfer rate has been improved due to the increase of nanoparticle. Similar work has been carried out by Ozotop and Abu-Nada
[6], Tiwari and Das [7], Aminossadati and Ghasemi [8], Ghasemi and
Aminossadati [9] and they concluded that heat transfer rate increases
with an increase of the nanoparticle. Kim et al. [10] found that with an
increase of density and heat capacity of nanoparticle thermal conductivity and the shape factor decrease. Xuan and Li [11] experimentally
witnessed that copper water based nanofluid enhances the heat transfer
rate.
Hwang et al. [12] investigated Benard convection and found the adverse effect of using nanofluid. Santra et al. [13] also reported that nanoparticle decreases the heat transfer rate. Details review on the nanofluid
can be found in Wang and Mujumdar [14]. Shape of the enclosure plays
a vital role in convection, though the shape depends on practical
application. Different types of enclosure filled with nanofluid are studied in recent years. Most of them are rectangular, square, triangular
and trapezoidal enclosures. Square and rectangular shape enclosures
are mostly studied. Related studies are presented in these literatures
[15–28].
Triangular cavities are not studied in a great extent yet. Zi-Tao et al.
[29] inspected the effect of nanofluids in a bottom heated isosceles triangular enclosure in transient buoyancy driven condition. Rahman
et al. [30,31] studied the convection which was laminar mixed in an
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M.M. Rahman et al. / International Communications in Heat and Mass Transfer 50 (2014) 117–127
Nomenclature
cp
g
Gr
H
k
L
Nu
p
P
Pr
T
t
u
U
v
V
x
X
y
Y
specific heat (J kg−1 k−1)
gravitational acceleration (ms−2)
Grashof number
enclosure height (m)
thermal conductivity (Wm−1 k−1)
length of the enclosure (m)
Nusselt number
dimensional pressure (kg m−1 s−2)
dimensionless pressure
Prandtl number
fluid temperature (K)
dimensional time (s)
horizontal velocity component (ms−1)
dimensionless horizontal velocity component
vertical velocity component (ms−1)
dimensionless vertical velocity component
horizontal coordinate (m)
dimensionless horizontal coordinate
vertical coordinate (m)
dimensionless vertical coordinate
Greek symbols
α
thermal diffusivity (m2 s−1)
β
thermal expansion coefficient (K−1)
ϕ
solid volume fraction
μ
dynamic viscosity (kg m−1 s−1)
ν
kinematic viscosity (m2 s−1)
τ
dimensionless time
θ
non-dimensional temperature
ρ
density (kg m−3)
ψ
stream function
λ
wave length
Γ
general dependent variable
Subscripts
av
overall
h
hot
c
cold
f
fluid
nf
nanofluid
s
solid nanoparticle
max
maximum
min
minimum
inclined triangular enclosure filled with water based Cu nanofluid and
found that the angle of inclination plays a significant role along with
the nanofluid in the heat transfer. Billah et al. [32] studied unsteady
buoyancy-driven heat transfer enhancement of nanofluids in an inclined triangular enclosure. Corrugated bottom walls are also available
in different physical phenomena and play a significant role in the heat
transfer. Rahman et al. [33] show the effect of corrugated bottom on a
triangular cavity for a double diffusive buoyancy induced flow.
In the previous work triangular enclosure gets a very little attention
along with the corrugated bottom surface. In this paper a numerical
study has been performed for a triangular shape solar collector with a
corrugated bottom and the enclosure is filled with the copper–water
nanofluid. As in solar thermal collector higher heat transfer rate is required, introducing nanofluid can be a possible solution of this problem.
Different nanofluids such as TiO2–water, Al2O3–water, Cu–water etc. are
available nowadays. This paper also shows that Cu–water nanofluid is
the best nanofluid for the augmentation of heat transfer due to its
thermophysical properties. Enhancements of heat transfer with the increase of solid volume fraction are also shown.
2. Problem formulation
2.1. Physical model
The detail of the physical model is presented in Fig. 1(a) along
with its specified co-ordinate system and boundary conditions. In
this model a triangular type solar collector with a corrugated bottom
is shown in Fig. 1(b). The enclosure is filled with the nanofluids
which are Cu–water, Al2O3–water and TiO2–water. The height of
the cavity is H and the length of the cavity is L. The effect of the gravity is shown in the negative Y axis. Here the cavity is formed by the
two inclined glass covers where the horizontal base plate is corrugated and acts as an absorber plate. The absorber plate has a constant
higher temperature than the inclined glass cover.
2.2. Thermophysical property of nanofluid
For this numerical study Cu, Al2O3 and TiO2 are taken as the nanoparticle and water is taken as the base fluid. Different experiments
have been carried out by different researchers. Different nanofluids
such as TiO2–water, Al2O3–water, and Cu–water are recently available.
The data used for the numerical simulation [23] is given in Table 1.
2.3. Mathematical modeling
The governing equations which define the system behavior are conservation of mass, energy and momentum. The thermophysical properties of the nanofluid are presumed to be constant excluding the density
variation in the buoyancy force, which is established on the Boussinesq
approximation. Cu–water, Al2O3–water and TiO2–water nanofluids filled
the free space of the enclosure and which are modeled as a Newtonian
fluid. The flow is assumed to be unsteady, laminar and incompressible.
Thermal equilibrium between the base fluid and nanoparticles is considered, and no slip arises between the two media. In the light of these suppositions stated above, the continuity, momentum and energy equations
in two-dimensional form can be written as [15]:
∂u ∂v
þ
¼0
∂x ∂y
ð1Þ
∂u
∂u
∂u
1 ∂p μ nf
þu þv
¼−
þ
ρnf ∂x ρnf
∂t
∂x
∂y
!
∂2 u ∂2 u
þ
∂x2 ∂y2
∂v
∂v
∂v
1 ∂p μ nf
þu þv
¼−
þ
ρnf ∂y ρnf
∂t
∂x
∂y
∂2 v ∂2 v
þ
∂x2 ∂y2
∂T
∂T
∂T
∂2 T ∂2 T
þ
þu
þv
¼ α nf
∂t
∂x
∂y
∂x2 ∂y2
!
þ
ð2Þ
ðρβÞnf
ρnf
g ðT−T c Þ
ð3Þ
!
ð4Þ
where, the effective density ρnf of the nanofluid is described by
ρnf ¼ ð1−ϕÞρ f þ ϕρs
ð5Þ
and ϕ is the solid volume fraction of nanoparticles. Furthermore, the thermal diffusivity αnf of the nanofluid is specified by:
α nf ¼ knf
:
ρcp
nf
ð6Þ
M.M. Rahman et al. / International Communications in Heat and Mass Transfer 50 (2014) 117–127
119
a
u = v = 0 T = Tc
u = v = 0 T = Tc
g
H
Nano fluids
y
λ = 0.1
x
u = v = 0, T = Th
L
Glass cover
b
Fluid
Nano particle
Solar thermal collector
Fig. 1. (a) Schematic of the problem with the domain and boundary conditions and (b) 3D view of a solar thermal collector filled with nanofluid.
The effective thermal conductivity of nanofluid was given by
Khanafer et al. [5] and Wasp [35] as follows:
The heat capacitance of nanofluids can be defined as
ρcp
nf
¼ ð1−ϕÞ ρcp þ ϕ ρcp :
f
ð7Þ
s
Moreover, (ρβ)nf is the thermal expansion coefficient of the nanofluid
and it can be found by
ðρβÞnf ¼ ð1−ϕÞðρβÞ f þ ϕðρβÞs :
ð8Þ
Furthermore, μnf is the dynamic viscosity of the nanofluid introduced
by Brinkman [34] as
μ nf ¼
μf
ð1−ϕÞ
:
2:5
ð9Þ
knf ks þ 2k f −2ϕ k f −ks
¼
kf
ks þ 2k f þ ϕ k f −ks
ð10Þ
where, ks is the thermal conductivity of the nanoparticles and kf is the
thermal conductivity of base fluid.
The suitable initial and boundary conditions in dimensional form
are:
t¼0
Entire domain : u ¼ v ¼ 0;
T ¼ Tc
ð11aÞ
t N0
Table 1
Thermophysical properties of water and nanoparticles [23].
Properties
Water
Cu
Al2O3
TiO2
cp
ρ
k
β
4179
997.1
0.613
2.1 × 10−4
385
8933
400
1.67 × 10−5
765
3970
40
0.85 × 10−5
686.2
4250
8.9538
0.9 × 10−5
on the bottom wall : u ¼ v ¼ 0;
on the inclined walls : u ¼ v ¼ 0;
T ¼ Th
T ¼ T c:
ð11bÞ
ð11cÞ
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M.M. Rahman et al. / International Communications in Heat and Mass Transfer 50 (2014) 117–127
Eqs. (1)–(4) are nondimensionalized using the following dimensionless variables:
p þ ρ f gy L2
αft
x
y
uL
vL
;V ¼
;P ¼
; θ
X ¼ ;Y ¼ ;τ ¼ 2 ;U ¼
L
L
αf
αf
ρnf α f 2
L
ðT−T c Þ
:
¼
ðT h −T c Þ
ð12Þ
By employing Eq. (12) the resulting dimensionless equations are reduced to:
∂U ∂V
þ
¼0
∂X ∂Y
ð13Þ
∂U
∂U
∂U
∂P
Pr
1
2
i
∇ U
þU
þV
¼−
þh
2:5
∂τ
∂X
∂Y
∂X
ð
Þ
1−ϕ
ð1−ϕÞ þ ϕ ρs =ρ f
ð14Þ
∂V
∂τ
þU
∂V
∂V
∂P
Pr
1
2
i
∇ V
þV
¼−
þh
2:5
∂X
∂Y
∂Y
ð1−ϕÞ þ ϕ ρs =ρ f ð1−ϕÞ
þ
ð15Þ
ϕρs βs þ ð1−ϕÞρ f β f
2
GrPr θ
ð1−ϕÞρ f β f þ ϕρs β f
∂θ
∂θ
∂θ ð1 þ 2ϕÞks þ 2ð1−ϕÞk f
þU
þV
¼
ð1−ϕÞks þ ð2 þ ϕÞk f
∂τ
∂X
∂Y
1
∇2 θ: ð16Þ
ð1−ϕÞ þ ϕ ρcp = ρcp
s
f
Here the Prandtl number and Grashof number are defined as
3
Pr ¼ ν f =α f
and Gr ¼
gβ f L ðT h −T c Þ
νf 2
initial and boundary conditions in the dimensionless form for the present problems become
τ¼0
Entire domain : U ¼ V ¼ 0; θ ¼ 0
ð17aÞ
τN0
on the bottom wall : U ¼ V ¼; θ ¼ 1
ð17bÞ
on the inclined walls : U ¼ V ¼ 0; θ ¼ 0:
ð17cÞ
The overall Nusselt number at the heated surface of the enclosure
can be expressed by
Nuav ¼ −
knf
kf
Z1
0
∂θ
dX:
∂Y
ð18Þ
discretize the equations. A non-uniform triangular element is located
over the enclosure. For each element the dependent variables are estimated using interpolation function. A set of algebraic equation has
been formed by the governing equation to lessen the continuum area
into the discrete triangular regions. Then the algebraic equation is
solved using the iteration technique. The iteration has been done till
the solution becomes convergent. |Γm + 1 − Γm| ≤ 10−6 where m is
the number of iteration and Γ is the general dependent variable.
3.2. Grid independency test
A grid independency test has been performed for this model and the
result is shown in Fig. 2. The grid independency test has been performed
for the dimensionless time τ = 0.1 and ϕ = 10 and thermal Grashof
number Gr = 105 for Cu–water nanofluid. Element number is varied
for the five specified conditions. The element numbers are 1214, 2042,
3456, 4544 and 5928. It has been found that for the element number
1214 overall Nusselt numbers are increasing. For the element number
2042 the Nusselt number also increases. When the element number is
3456 the Nusselt number becomes more or less constant. For the higher
element number of 4544 and 5928 the Nusselt number decreases and
increases slightly. As the Nusselt number is just about constant for the
grid of 3456 element the study has been performed for this grid of element number 3456.
3.3. Code validation
The present work is validated with the published literature. Code
validation was done with a view to checking the accuracy of the numerical simulation and the solution procedure of the problem. The present
study was compared with Khanafer et al. [5] on the basis of overall
Nusselt number and the deviation with the study is reported in
Table 2. From the table it is evident that the present code and the
numerical method are completely reliable as it shows good agreement
with the previous published literature. The present study varies not
more than 2% with the previous one.
4. Results and discussion
In this section detail analysis of the results has been presented in
light of the figures and graphs obtained from the numerical analysis.
In this paper time dependent solution has been obtained for the
resulting differential equations using finite element analysis. Firstly the
effect of changing the solid volume fraction (Cu–water nano fluid)
from ϕ = 0, 0.05, 0.08 to 0.1 on the streamlines and isotherms has
been shown for different values of Gr (104 to 106) at different instants
(for τ = 0.1, 0.5 and 1). Later the variation of overall value of Nusselt
number at the heated surface is shown with the change in solid volume
fraction for different values of Gr at different instants. Finally the effect of
The fluid motion is displayed by means of the stream function ψ acquired from velocity components U and V. The relationships among
stream function and velocity components from Batchelor [36] for two
dimensional flows are
U¼
∂ψ
∂ψ
; V ¼−
:
∂Y
∂X
ð19Þ
3. Numerical methods
3.1. Numerical scheme
Finite element method was employed to solve non-dimensional
form of the governing Eqs. (14)–(16) subject to the initial and boundary
conditions (17). The Galerkin weighted residual scheme is applied to
Fig. 2. Grid independency study with τ = 0.1, ϕ = 0.05 and Gr = 105 (Cu–water).
M.M. Rahman et al. / International Communications in Heat and Mass Transfer 50 (2014) 117–127
Table 2
Comparison of Nuav with those of Khanafer et al. [5].
Gr
Nuav
3
10
104
105
Ref [5]
Present study
Error (%)
1.9806
4.0653
8.3444
1.9541
4.0727
8.5216
1.34
0.18
2.12
different commonly used nanofluids such as TiO2–water, Al2O3–water
and Cu–water is discussed on the light of overall Nusselt number.
4.1. Effect of solid volume fraction on streamlines varying the dimensionless
time
In Fig. 3 effect of changing the percentage of solid Cu in the base fluid
water on the streamlines has been presented at τ = 0.1 for different
values of Gr. For Gr = 104 and 105 the effect of change in percentage
of nanoparticle is not that pronounced. At ϕ = 0 and Gr = 104 two opposite rotating vortices are formed with very low value of stream function showing poor convective heat transfer. Here ψmax = 0.25 and
ψmin = −0.27 have been observed. At Gr = 105 there are four cells instead of two with dominating cells in the middle having opposite sense
of rotation. Two smaller vortices are formed at the two corners of bottom corrugated wall. Here the maximum value of stream function is observed to be ψmax = 5.72 and the minimum value ψmin = −5.87. In
the two cases discussed cells are formed in symmetric manner showing
very normal nature of convection. Here from the bottom heated wall
fluid takes up heat energy and becomes lighter to induce a convective
current. Lighter fluid goes up and due to the symmetric nature of colder
wall; the colder portion of fluid follows the line traced by the colder
walls. As a consequence symmetric cells are obtained. But in the case
of Gr = 106 this symmetry is completely broken and a very strong convective current pattern is formed. Here six cells are formed; three at
each three corner and the cells have low to moderate strength. The
121
dominating cell is at the middle of the triangular enclosure having a
stream function value of 39.57 and rotating in clockwise direction.
There is another cell adjacent to it having a lower strength and opposite
sense of rotation. Overall a significant change in convection is observed
just by increasing the value of Gr having no nanoparticle in the base
fluid. For Gr = 105 and ϕ = 0.1 strongest flow pattern is achieved
with four cells formed in a symmetric manner, having ψmax = 12 and
ψmin = −12. But there is an interesting pattern of change in the value
of stream function in the case of Gr = 106 for varying percentage of
Cu in water. At ϕ = 0.05 a very strong convective current is observed
at the middle having ψ = 45.79 and a counter clockwise direction of rotation unlike the case of ϕ = 0. Here three cells are formed at each corner of the enclosure with quite strong flow strength (ψmax = 5.49 and
ψmin = −5.56). The convective currents are more adjacent to the bottom heated wall for obvious reason and that is from this wall heat energy
is entering into the fluid. In general a very good convective characteristic
is shown by the flow at ϕ = 0.05. At ϕ = 0.08 it is seen that the strongest vortex is formed in the middle region but has a considerable portion
of its flow along the left inclined wall. Its stream function value is 40.12
and it has a positive sense of rotation. There are smaller cells formed
near the bottom wall and at the corners. These cells have lower strength
than the cells which were formed in case of ϕ = 0.05. At ϕ = 0.1 the
cells are formed in a totally random manner having irregular cloudy
shapes. The stronger cell has ψmax = 21.83 and ψmin = −24.79. Two
small cells form at two corners and have higher strength than previous
cases (ψmax = 9.09 and ψmin = −5.94). However, the overall convective current pattern suggests that the convective heat transfer is weaker
in cases of ϕ = 0.08 and ϕ = 0.1 than ϕ = 0.05 at τ = 0.1.
Fig. 4 shows the influence of changing proportion of Cu nanoparticle
in base fluid at different Gr values at τ = 0.5. So it is the continuation of
Fig. 3 with respect to time. For Gr = 104 the cells obtained are just like
the cells obtained at τ = 0.5. But there is a difference in the nature of
change in strength. In this case not only stronger cells are obtained
but also the strength clearly decreases as the value of ϕ increases. So
with the addition of nanoparticle the heat transfer characteristics are
Fig. 3. Effect of solid volume fraction on streamlines for the selected values of Gr with τ = 0.1 (Cu–water).
122
M.M. Rahman et al. / International Communications in Heat and Mass Transfer 50 (2014) 117–127
becoming poorer. For Gr = 105 there is no such change and flow pattern is just the same as it was in the case of τ = 0.1 but the strength
of the flow has increased. Now in the case of Gr = 106 the vortices
again break the symmetric nature.
Fig. 5 shows the effect of changing solid volume fraction on the
streamlines at different Gr values at τ = 1. Just like the cases discussed
in Figs. 3 and 4, at Gr = 104 and 105 the effect of changing the percentage of Cu nanoparticle is not that pronounced. Although the flow
strength in both cases decreases with the increment of values of ϕ. So
in general it can be said that for lower value of Gr it is not fruitful to induce Cu nanoparticle in water based fluid to increase the convective
heat transfer. However, for Gr = 106 at ϕ = 0 there is a very strong
convective current in the middle of the cell having ψ = 40.57 and rotating in clockwise direction. There are other smaller cells formed at the
corners and along the right wall having low to moderate strength. So
it is obvious that without inducing nanoparticle convective heat transfer
characteristics can be improved just by increasing the value Gr. For
ϕ = 0.05, 0.08 and 0.1 multiple cells are obtained in this case.
4.2. Effect of solid volume fraction on isotherms varying dimensionless time
In Fig. 6 the effect of changing solid volume fraction on isotherms is
shown for specific values of Gr at τ = 0.1. At Gr = 104 convection is not
significant at all even after the percentage of nanoparticles being increased. The isotherms are parallel to each other and are densely distributed near the bottom heated wall. The parallel nature of isotherms
shows the domination of conductive heat transfer. At Gr = 105 the isotherms are seen to be densely packed along the boundaries and very
densely packed at the corners for ϕ = 0. It shows that convection is
very strong in those regions. The isotherms assume a distorted shape
near the middle and the top showing introduction of convection. At
Gr = 105 and ϕ = 0.05 the isotherms' distribution is almost the same
as it was in the case of ϕ = 0. But here the parallel lines are more separated along the walls, indicating the expansion of thermal boundary
layer. Also in the middle the isotherms assume an inverted mushroom
shape showing that convection is dominant in those regions. At the corners isotherms are very densely packed showing the presence of a
strong conduction heat transfer. This pattern remains the same for
ϕ = 0.08 and 0.1. At Gr = 106 and ϕ = 0 there is a clear formation of
thermal boundary layer along the walls and at the corners where isotherms are parallel to each other. In these regions convection is the
dominant mode of heat transfer. At the middle the isotherms assume
mushroom shape showing the presence of convective heat transfer. As
the value of ϕ increases the thickness of thermal boundary layer at the
walls also increases indicating a strong convective heat transfer in
those regions. Fig. 7 shows the same effect as Fig. 6 but at a later moment or at τ = 0.5. For Gr = 104 conduction is the dominant heat
transfer mode for all values of ϕ. For Gr = 105 at ϕ = 0 thermal boundary layer is formed along the walls of the enclosure which thickens as
the value of ϕ increases from 0 to 0.1. At the middle of the enclosure
the isotherms are distorted and the pattern is almost same for different
values of ϕ. It shows that with the increase of Cu nanoparticle in the
base fluid water, the convection heat transfer doesn't change that
much but convection becomes stronger. It supports our observations
made during the discussion on the streamlines' distribution. At
Gr = 106 again thermal boundary layers are formed along the walls.
These walls are thinner compared to Gr = 105 and isotherms are closely packed at the corners. The shape of the isotherms at the middle is
distorted for different values of ϕ. As the percentage of Cu nanoparticle
in the base fluid increases, the thickness of boundary layer increases too.
At ϕ = 0.1 and Gr = 106 both conduction and convection heat transfer
modes are dominant at τ = 0.5.
Fig. 8 shows the outcome of change in the solid volume fraction of Cu
nanoparticle in water based fluid on the isotherms for different Gr
values at τ = 1. This is a continuation of the process described in the
previous passages with respect to time variable. There is no noticeable
change in the distribution of isotherms in this case. Just like the previous
cases, at lower values of Gr, there is strong presence of conduction inside the enclosure. As the value of Gr and ϕ is increased both heat transfer modes become effective.
Fig. 4. Effect of solid volume fraction on streamlines for the selected values of Gr with τ = 0.5 (Cu–water).
M.M. Rahman et al. / International Communications in Heat and Mass Transfer 50 (2014) 117–127
123
Fig. 5. Effect of solid volume fraction on streamlines for the selected values of Gr with τ = 1 (Cu–water).
4.3. Effect of solid volume fraction on overall Nusselt number
Fig. 9 shows the variation in overall Nusselt number with varying
solid volume fractions with a view to comparing the situations created at different values of Gr. (a), (b) and (c) show these effects for
three different instants (τ = 0.1, τ = 0.5 and τ = 1). For all cases
it is observed that for the value of Gr = 106, the magnitude of overall Nusselt number is maximum. The value of overall Nusselt number increases for all values of Gr with the increase in the value of
solid volume fraction. These results indicate that convection heat
transfer becomes stronger at a higher value of Gr and solid volume
fraction.
Fig. 6. Effect of solid volume fraction on isotherms for the selected values of Gr with τ = 0.1 (Cu–water).
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M.M. Rahman et al. / International Communications in Heat and Mass Transfer 50 (2014) 117–127
Fig. 7. Effect of solid volume fraction on isotherms for the selected values of Gr with τ = 0.5 (Cu–water).
4.4. Comparison of overall Nusselt number for the different nanofluids
In Fig. 10 comparison of overall Nusselt number at the heated
surface among different nanofluids for τ = 0.1, 0.5 and 1 for
Gr = 105 has been depicted. From the figure it is seen that there is
a general pattern in the change of overall Nusselt number value
with the change of solid volume fraction for the nanofluids. The
overall Nusselt number value at the heated wall increases with the
value of solid volume fraction but the rate of change is not the
same for all values of solid volume fraction. At first the rate of
change in the value of Nusselt number is high up to a certain value
of solid volume faction and then the rate of change in overall
Fig. 8. Effect of solid volume fraction on isotherms for the selected values of Gr with τ = 1 (Cu–water).
M.M. Rahman et al. / International Communications in Heat and Mass Transfer 50 (2014) 117–127
125
Fig. 10. Comparison of overall Nusselt number at the heated surface among different
nanofluids for (a) τ = 0.1, (b) τ = 0.5 and (c) τ = 1, while Gr = 105.
Fig. 9. Overall Nusselt number at the heated surface versus solid volume fraction
for different values of Gr at (a) τ = 0.1, (b) τ = 0.5 and (c) τ = 1 (Cu–water).
5. Conclusions
In this paper the problem involving the addition of Cu, Al2O3 and TiO2
nanoparticles in water based fluid inside a triangular enclosure with corrugated bottom wall is solved thoroughly. All the numerical results are
discussed and from the analysis the following points have come out.
Nusselt number retards for higher values of solid volume fraction.
Cu–water nanofluid shows higher value of overall Nusselt number
at the heated surface for all values of solid volume fraction and τ.
On the other hand, water–TiO2 shows the lowest value of overall
Nusselt number for all values of solid volume fraction and τ.
Water–Al2O3 nanofluid shows the moderate Nusselt number
value for all the cases. Another point to note that, overall Nusselt
number value is highest at τ = 0.1 for all the nanofluids indicating
maximum heat transfer at that instant from the heated wall. So
from the comparison it can be said that from heat transfer point
of view Cu–water nanofluid gives better performance.
Comparison of overall Nusselt number at the heated surface among
different nanofluids for τ = 0.1, 0.5 and 1 while Gr = 106 has been
shown in Fig. 11. Unlike the case of Gr = 105, here there is no general
trend in the change of overall Nusselt number with the change of solid
volume fraction.
▪ Solid suspended particles are added to the base fluid with a view to
increasing the heat transfer by the conduction of heat through solid
particles' medium. So it is logical to assume that additional solid particles mean better heat transfer. But the analysis made here suggests
otherwise. It has been found that, addition of nanoparticle does increase the convective heat transfer but only up to a certain limit.
The above results evidently indicate that the convective heat transfer performance is better when the solid volume fraction is kept at
0.05 or 0.08.
▪ Increasing the Gr number is sufficient to increase the convective
heat transfer effectively.
▪ At higher values of Gr, convection is very strong for solid volume
fraction values from 0.05 to 0.08
▪ For lower values of Gr conduction is the primary mode of heat transfer for any value of solid volume fraction.
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M.M. Rahman et al. / International Communications in Heat and Mass Transfer 50 (2014) 117–127
References
Fig. 11. Comparison of overall Nusselt number at the heated surface among different
nanofluids for (a) τ = 0.1, (b) τ = 0.5 and (c) τ = 1, while Gr = 106.
▪ High value of both Gr and solid volume fraction confirms better heat
transfer through convection and conduction.
▪ Overall Nusselt number at the heated surface increases with increasing value of solid volume fraction
▪ Cu–water nanofluid performs better from heat transfer point of view
than other commonly used nanofluids.
▪ Heat transfer can be increased up to 24.28% from the heated surface
as volume fraction ϕ increases from 0% to 10%.
The use of solar energy has been increasing by leaps and bounds and
many countries are making specific policies in order to use the renewable energy in a more efficient manner. In this circumstance, this
paper is written to enlighten the scientific world about the effect of
nanoparticle in common fluid on heat transfer characteristics and it is
hoped that the results and discussion will be helpful for the betterment
of existing solar energy technologies.
Acknowledgment
The authors would like to thank the Ministry of Higher Education of
Malaysia and the University of Malaya for providing financial support
under the research grant No. UM.C/HIR/MOHE/ENG/16001-00-D000024.
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