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Augmentation of the Heat Transfer Performance of a Sinusoidal Corrugated Enclosure by Employing Hybrid Nanofluid

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This paper numerically examines laminar natural convection in a sinusoidal corrugated enclosure with a discrete heat source on the bottom wall, filled by pure water, Al2O3/water nanofluid, and Al2O3-Cu/water hybrid nanofluid which is a new advanced nanofluid with two kinds of nanoparticle materials. The effects of Rayleigh number (103≤Ra≤106) and water, nanofluid, and hybrid nanofluid (in volume concentration of 0% ≤ ϕ ≤ 2%) as the working fluid on temperature fields and heat transfer performance of the enclosure are investigated. The finite volume discretization method is employed to solve the set of governing equations. The results indicate that for all Rayleigh numbers been studied, employing hybrid nanofluid improves the heat transfer rate compared to nanofluid and water, which results in a better cooling performance of the enclosure and lower temperature of the heated surface. The rate of this enhancement is considerably more at higher values of Ra and volume concentrations. Furthermore, by applying the modeling results, two correlations are developed to estimate the average Nusselt number. The results reveal that the modeling data are in very good agreement with the predicted data. The maximum error for nanofluid and hybrid nanofluid was around 11% and 12%, respectively.
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Research Article
Augmentation of the Heat Transfer Performance of a Sinusoidal
Corrugated Enclosure by Employing Hybrid Nanofluid
Behrouz Takabi1and Saeed Salehi2
1Young Researchers Club, Central Tehran Branch, Islamic Azad University, Tehran, Iran
2School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran 1439957131, Iran
Correspondence should be addressed to Behrouz Takabi; btakabi@iust.ac.ir
Received  September ; Revised  January ; Accepted  January ; Published  March 
Academic Editor: Marco Ceccarelli
Copyright ©  B. Takabi and S. Salehi. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
is paper numerically examines laminar natural convection in a sinusoidal corrugated enclosure with a discrete heat source on the
bottom wall, lled by pure water, Al2O3/water nanouid, and Al2O3-Cu/water hybrid nanouid which is a new advanced nanouid
with two kinds of nanoparticle materials. e eects of Rayleigh number (103Ra ≤10
6)and water, nanouid, and hybrid
nanouid (in volume concentration of 0%≤≤2%) as the working uid on temperature elds and heat transfer performance
of the enclosure are investigated. e nite volume discretization method is employed to solve the set of governing equations. e
results indicate that for all Rayleigh numbers been studied, employing hybrid nanouid improves the heat transfer rate compared
to nanouid and water, which results in a better cooling performance of the enclosure and lower temperature of the heated surface.
e rate of this enhancement is considerably more at higher values of Ra and volume concentrations. Furthermore, by applying the
modeling results, two correlations are developed to estimate the average Nusselt number. e results reveal that the modeling data
are in very good agreement with the predicted data. e maximum error for nanouid and hybrid nanouid was around % and
%, respectively.
1. Introduction
In the past years, many dierent techniques were utilized to
improve the heat transfer rate for reaching a satisfactory level
of thermal eciency. A way for this purpose is enhancement
in thermal conductivity. So many eorts for dispersing solid
particles with high thermal conductivity in the liquid coolant
have been conducted to enhance thermal properties of the
conventional heat transfer uids. Maxwell [,]wasthe
rst to show the possibility of augmentation of thermal
conductivity of a solid-liquid mixture by increasing the
volume fraction of solid particles. However, large particles
cause many troublesome problems such as sedimentation of
large sized particles in base uid. us, a new class of uids
for improving both thermal conductivity and suspension
stability was developed that is known as nanouid.
Choi [] presented the benet of using the nanoparticles
dispersed in a base uid in dierent thermal systems to
enhance the heat transfer rate. Eastman et al. []presented
that with .% volume concentration of Cu nanoparticles dis-
persed in ethylene glycol, its thermal conductivity increased
by %.
Almost all published papers in nanouid eld until
now dened “nanouid” as a base uid with suspended
nanosized particles from just one type of material. ey
have studied the eect of size, shape, concentration, and
material of nanoparticles on thermophysical properties of
nanouid and its inuence on heat transfer and pressure drop
characteristics. Nevertheless, very recently, an experimental
study has been carried out on nanouid with two types of
nanoparticles dispersed simultaneously in a base uid that is
called “hybrid nanouid” [].
e most important exclusivity of hybrid nanouid refers
to composition of two variant types of dispersed nanopar-
ticles in a base uid. us, when materials of particles are
chosen properly, they could enhance the positive features of
each other and cover the disadvantages of just one material.
For example, alumina (i.e., a ceramic material) has many
Hindawi Publishing Corporation
Advances in Mechanical Engineering
Volume 2014, Article ID 147059, 16 pages
http://dx.doi.org/10.1155/2014/147059
Advances in Mechanical Engineering
benecial properties such as chemical inertness and a great
deal of stability, while Al2O3exhibits lower thermal conduc-
tivity with respect to the metallic nanoparticles. Aluminum,
zinc, copper, and the other metallic nanoparticles encompass
great thermal conductivities. However, the use of metallic
nanoparticles for nanouid applications is limited due to
the stability and reactivity. According to these features of
metallic and nonmetallic nanoparticles, it can be expected
that the addition of metallic nanoparticles (such as Cu) into
a nanouid composed based on Al2O3nanoparticles can
enhance the thermophysical properties of this mixture.
Suresh et al. carried out an experimental study to
synthesize Al2O3-Cu/water hybrid nanouid []. To reach
the stable hybrid nanouid, they used a thermomechanical
method (two-step method). ey added Cu nanoparticles
to Al2O3/water nanouid and synthesized Al2O3-Cu/water
hybrid nanouid by dierent volume concentrations as .,
., ., , and %.
According to the above mentioned benets of hybrid
nanouids, it is clearly expected that this advanced nanouid
plays a vital role in the future of science of nanouid and
researchers show a greater tendency toward investigation of
hybrid nanouids and eect of such uids on heat transfer
and pressure drop characteristics.
Natural convection heat transfer is an important phe-
nomenon in engineering systems due to its wide applications
in nuclear energy, double pane windows, heating and cooling
of buildings, solar collectors, electronic cooling, microelec-
tromechanical systems (MEMS) []. erefore, the inves-
tigation of thermal and hydrodynamic behaviors for dierent
shapes of the heat transfer surfaces is necessary to ensure the
ecient performance of the various heat transfer equipment.
e problem of laminar natural convection in two-
dimensional enclosures has been widely studied in sev-
eral literatures. Many of these investigations (e.g., [])
focused on natural convection heat transfer in enclosures
with corrugated surfaces. Ali and Husain [] examined the
eect of corrugation frequencies on natural convection heat
transfer and ow characteristics in a square enclosure of
vee-corrugated vertical walls. In addition to regular geome-
tries like square or rectangle, many studies on wavy-walled
enclosures were performed in the literatures due to their
application in many engineering problems related to geomet-
rical design requirements []. Natural convection in a wavy
enclosure like double-wall thermal insulation, underground
cable systems, and cooling of micro-electronic devices has
several applications in industrial and engineering purposes
[]. erefore, because of the practical importance of ow
and heat transfer in corrugated geometry many researchers
have reported results on this geometry theoretically as well as
experimentally (e.g., []).
Saha [] performed a numerical investigation of the
steady sate magneto convection in a sinusoidal corrugated
enclosure with a heat source on the bottom wall. He showed
that as the heat source surface area increased, the average
Nusselt number decreased, indicating that the heat source
size has signicant eect on the heat transfer rate. In addition,
Hussain et al. [] numerically analyzed the eect of inclina-
tion angles on natural convection in the geometry introduced
W
H
0.1H
X
Yg
Adiabatic
Adiabatic Adiabatic
Cold wall, Tc
Cold wall, Tc
Heat flux q󳰀󳰀
F : Schematic of conguration under investigation.
by Saha []. ey reported that Nusselt number rstly
increased by rising inclination angles and then decreased for
all values of Hartmann number.
In this paper, laminar natural convection ow of
Al2O3/water nanouid and Al2O3-Cu/water hybrid nano-
uid in a sinusoidal corrugated enclosure with a discrete heat
source on the bottom wall is investigated. In fact this paper
concerns the ow and heat transfer characteristics in the same
geometry which was studied by Saha []andHussainet
al. [] and develops their works by considering the eect
of existence of nanoparticles from one and two types of
materials in working uid on heat transfer rate.
2. Mathematical Modeling
Figure  shows the considered geometrical conguration,
with the important geometric parameters. is enclosure
consists of two vertical sinusoidal corrugated and two at
horizontal walls with dimensions of and ,respectively.A
constant low temperature (𝑐)issubjectedtotwosinusoidal
sidewalls and a xed ux heat source (󸀠󸀠),isdiscretely
imposed at the bottom wall, while the remaining parts of the
bottom wall and the upper wall are considered to be thermally
insulated.eenclosurehasthesameheightandwidth,=
. e sidewalls prole is assumed sinusoidal with single
corrugation period and the amplitude of % of the enclosure
length. e ratio of the size of the heating element to the
enclosure width is taken at =0.4. Rayleigh number is
dened as
Ra =3Pr
]2.()
Ra has been varied from 3to 6, where Pr is Prandtl
number and is specied in
Pr =]
.()
Advances in Mechanical Engineering
2.1. ermophysical Properties of Nanouid and Hybrid
Nanouid. As previously mentioned, although some litera-
tures studied the determination of thermophysical proper-
ties, the classical models are not certain for nanouids. Of
course, experimental results allow us to select an appropriate
model for a specied property.
e eective properties of the Al2O3/water nanouid and
Al2O3-Cu/water hybrid nanouid are dened as follows:
nf =𝑝𝑝+1−𝑝bf.()
Equation () was originally introduced in [] for deter-
miningdensityandthenwidelyemployedin[]. So, the
density of hybrid nanouid is specied by
hnf =Al2O3Al2O3+Cu Cu +1−bf.()
is the overall volume concentration of two dierent
types of nanoparticles dispersed in hybrid nanouid and is
calculated as
=Al2O3+Cu.()
Equation ()thatisutilizedforspecifyingheatcapacity
wasrstemployedin[] and then used in many articles [,
,]:
nf =𝑝𝑝𝑝+1−𝑝bf bf
nf .()
According to (), heat capacity of hybrid nanouid can be
determined as follows:
hnf =Al2O3Al2O3Al2O3+CuCuCu +1−bf bf
hnf .
()
e thermal expansion coecient of nanouid can be
determined by ()employedinsomeliteratures[,]:
nf =𝑝𝑝𝑝+1−𝑝bfbf
nf .()
Hence, for hybrid nanouid, thermal expansion can be
dened as follows:
hnf =1
hnf Al2O3Al2O3Al2O3+CuCuCu
+1−bfbf. ()
In addition, for calculating conductivity of nanouids
equation () was proposed by Hamilton and Crosser [].
nf
bf =𝑝+(−1)bf (−1)𝑝bf −𝑝
𝑝+(−1)bf +𝑝bf −𝑝.()
Here is the empirical shape factor in order to account
the eect of particles shape and can be varied from . to ..
e shape factor is given by 3/,whereis the particle
0.7
0.6
0.5 0 0.5 1 1.5 2 2.5
Experimental (Suresh et al., 2011) [5]
Maxwell model [1, 2]
Bruggeman model
khnf (W/m·K)
Volume concentration, 𝜙(%)
F : ermal conductivity of Al2O3-Cu/water hybrid
nanouid.
sphericity, dened as surface area of a sphere to the surface
area of the particle. erefore for spherical nanoparticles
equals . is case of Hamilton and Crosser model (=3)is
thesameasMaxwellmodel[]; see the following:
nf
bf =𝑝+2bf −2𝑝bf −𝑝
𝑝+2bf +𝑝bf −𝑝.()
If the thermal conductivity of hybrid nanouid is dened
according to Maxwell model, ()mustbeemployedforthis
purpose:
hnf
bf =Al2O3Al2O3+Cu Cu
+2bf +2Al2O3Al2O3+CuCu −2bf
×Al2O3Al2O3+Cu Cu
+2bf −Al2O3Al2O3+Cu Cu+bf−1.
()
To thermal conductivities of hybrid nanouid in dierent
volume concentrations, as shown in Figure , a comparison is
made between the predicted values by Maxwell and Brugge-
man models and experimental results by Suresh et al. []. It
showsthattheseclassicalmodelscouldnotpreciselycalculate
the thermal conductivity, especially for higher volume con-
centrations. Even though Bruggeman model predicts better
Advances in Mechanical Engineering
T : ermophysical properties of nanouid and hybrid nanouid.
(%) Cu (%) Al2O3(%) nf (W/mK) nf (kg/ms) hnf (W/mK) hnf (kg/ms)
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
results compared to Maxwell model, the classical models,
generally, could not be as useful and ecient as experimental
data. erefore, to accomplish the most accurate numerical
results in this study, thermal conductivity values for both
nanouid and hybrid nanouid in dierent concentrations
have been extracted from experimental data reported by
Suresh et al. [].
Topredicttheviscosityofnanouid,threemodels
frequently were employed theoretically. ese models are
presented as follows.
Einstein model []:
nf =bf 1+𝜇, ()
where 𝜇=2.5.
Brinkman model []:
nf =bf
12.5 .()
Batchelor model []:
nf =bf 1+1+22, ()
where 1is . and 2describes the deviation from the
very dilute limit of suspension; by allowing a superimposed
Brownian motion, the value of the coecient 2is calculated
as  [].
In ()–(), for calculating the viscosity of nanouid,
is the volume concentration of Al2O3nanoparticles in
nanouid; whereas to dene this property of hybrid nanouid
(hnf), must be the overall volume concentration of
nanoparticles indicated in ().
Considering Figure , it can be understood that these
classical models signicantly underestimate the hybrid
nanouid viscosity, particularly in high volume concentra-
tion. erefore, to have a high accuracy in numerical results,
in the present study, viscosity of hybrid nanouid is obtained
from experimental data [].isresultthattheclassical
models underpredict hnf and hnf for hybrid nanouid has
been also reported in the earlier literature [].
e thermophysical properties of both Al2O3/water
nanouid and Al2O3-Cu/water hybrid nanouid for all
volume concentrations are available in Tab l e  . In addition
to this, the volume concentrations of Al2O3and Cu are
separately presented.
2.2. Governing Equations. e governing equations for lam-
inar natural convection in an enclosure are continuity,
momentum, and energy equations. Flow assumed to be
0.003
0.002
0.001
00 0.5 1 1.5 2 2.5
Experimental, Suresh et al. [5]
Einstein [29]
Brinkman [30]
Batchelor [31]
𝜇hnf (N·s/m2)
Volume concentration, 𝜙(%)
F : Dynamic viscosity of Al2O3-Cu/water hybrid nanouid.
steady state and incompressible. e density of uid is
assumedtobeconstantexceptinthebodyforceterminthe
momentum equation, which varies linearly with temperature
(Boussinesq’s hypothesis). Also the behavior of uid ow
is supposed to be Newtonian. Assumption of Newtonian
behavior for Al2O3-Cu/water hybrid nanouid with volume
concentration of lower than % seems acceptable according
to Suresh et al. []. ey have reported viscosity of hybrid
nanouid as a function of shear rate and showed an inde-
pendency between viscosity and shear rate of this hybrid
nanouid with volume concentration of lower than %. In
addition, other physical properties of uid (thermal conduc-
tivity, thermal expansion coecient, and specic heat) are
taken constant in a specic volume concentration. Hence, the
governing equations are transformed into a dimensionless
form under the following nondimensional variables []:
=−𝑐
 ,=
,=
,=

,
=V
,=
2
2,=
󸀠󸀠
,()
where and represent the nondimensional coordinate
and velocity along the horizontal axes, respectively; and
Advances in Mechanical Engineering
also dene the nondimensional vertical component and
velocity. According to (), the Rayleigh number is dened
as Ra =
3Pr(󸀠󸀠/)/]2. e nondimensional forms of
the governing equations are expressed in the following forms.
Continuity:

 +
 =0. ()
Momentum:

 +
 =
 +Pr 2
2+2
2,

 +
 =
+Pr 2
2+2
2+RaPr. ()
Energy:

 +
 =2
2+2
2,()
where Ra and Pr are Rayleigh and Prandtl number which are
specied in ()and(), respectively.
e average Nusselt number (Nu)at the heated surface
canbecalculatedas[,,]:
Nu =1
𝜀
01
𝑠(), ()
where 𝑠()is the local dimensionless temperature distribu-
tion of the heated surface; has been dened in ().
e corresponding boundary conditions for the above
problem are given by
All walls: =0,=0
Top wall : (/)=0
Right and le sidewalls: =0
Bottom wall: 

=0 0<<0.5(1−), 0.5 (1+)<<1,
−1 0.5 (1−)≤≤0.5(1+).()
3. Numerical Method and Validation
e set of coupled nonlinear governing equations have been
discretized using nite volume approach. e second-order
upwind and linear methods are employed to approximate the
convection and diusion terms in the momentum and energy
equation, respectively. Also pressure eld is corrected utiliz-
ing the well-known pressure correction algorithm SIMPLE.
To achieve inconsistency of the numerical results to the
grid resolution, ten dierent grids (from very coarse grid
of 10 × 10 cells to very ne grid of 200 × 200 cells) have
beenproducedandthemostcriticalcaseofthepresent
study (highest Rayleigh number lled with hybrid nanouid
16
15.8
15.6
15.4
15.2
15
14.8
14.6
14.4
14.2 0 5 10 15 20 25 30 35 40
×103
Number of cells
Nu
F : Variation of average Nusselt number along the heated
bottom wall with grid renement for hybrid nanouid of =2%
at Ra =106.
F : Grid used in this study.
by =2%) is solved using these grids. e variation of
average Nusselt number along the heated bottom wall with
grid renement is illustrated in Figure .isgraphshows
that an acceptable grid independency is achieved by rening
the grid and the results obtained from grids ner than 
cells can be assumed to be grid independent. e dierence
of average Nusselt number of each grid from that of the nest
grid is less than .% for grids 120×120and ner. Hence, the
grid 120×120is selected for the rest of calculations. As can
be seen in Figure , this nonuniform grid is ner near walls
and also the cell faces are aligned with enclosure walls.
For the validation of the employed code, two dierent
problems of natural convection in a square and a corrugated
enclosure are studied. Firstly, the problem of natural convec-
tion in a square enclosure with two vertical isothermal walls
Advances in Mechanical Engineering
1
0.8
0.6
0.4
0.2
00 0.2 0.4 0.6 0.8 1
𝜃
X
Present study
Numerical, Khanafer et al. [34]
Experimental, Krane and Jessee [33]
(a)
0 0.2 0.4 0.6 0.8 1
X
100
50
0
−50
−100
V
Present study
Numerical, Khanafer et al. [34]
Experimental, Krane and Jessee [33]
(b)
F : Comparison of results for natural convection in square enclosure along horizontal midline with those in the previous literatures.
(a) Dimensionless temperature (b) dimensionless vertical velocity.
(one hot and one cold) and two horizontal adiabatic walls,
which is a common problem and there are many published
results on, in Ra =10
6and Pr = 0.71,isinvestigated.
e dimensionless temperature and vertical velocity along
horizontal midline inside the enclosure obtained from the
present numerical code, experimental results of Krane and
Jessee [], and numerical results of Khanafer et al. []are
shown in Figure . is comparison reveals a very precise
match between results of present study and numerical results
of Khanafer et al. [] and also a good agreement with
experimental results of Krane and Jessee []. e dierences
between the numerical and experimental results can be
explained to not considering the radiation heat transfer due
to the high and low temperatures exposed on the walls
of the enclosure, in numerical procedure. In addition, the
assumptions employed for the numerical method (such as
Boussinesq’s hypothesis) can be another source of the error.
Asthesecondcaseofvalidation,theproblemofnatural
convection of air in sinusoidal corrugated enclosure is solved.
A comparison is made for isotherms based on dimensionless
temperature ()in various Grashof numbers (Gr =Ra/Pr)
between the current study and Saha results []inFigure .
A good accordance is found between the present results and
Saha results [].
4. Results and Discussion
e laminar natural convection in a sinusoidal corrugated
enclosure, the eect of adding the nanoparticles in one and
twokindsofmaterialstoabaseuidonheattransfercharac-
teristics and hydrodynamic behavior have been investigated
in this paper. is enclosure is partially heated via a discrete
heatsourcewithaconstantheatuxlocatedinthemiddleof
the bottom wall. Also, the eect of Rayleigh number (103
Ra ≤106)and dierent working uids including pure water,
nanouid, and hybrid nanouid (in volume concentration
of 0%≤≤2%) on heat transfer and hydrodynamic
performance of this enclosure is investigated.
To get a deep understanding of the thermal and ow
behavior, the temperature and velocity proles along the
midsection of the enclosure are assessed in the present study.
Accordingly, Figure (a) shows the prole of nondimensional
temperature along the horizontal midline inside the enclo-
sure, in a xed Rayleigh number (Ra =10
6)for pure water
and hybrid nanouid in dierent volume concentrations
as the working uid. In Figure (a) abumpcanbeseen
near the center of temperature proles which is clearly due
tothepresenceofpartiallyheatedsurfaceonthebottom
wall. It is revealed from this gure that by increasing the
volume concentration of hybrid nanouid, the temperature
decreases. is fact signies that employing hybrid nanouid,
especially in higher volume concentrations, can enhance the
cooling performance of the enclosure. is favorable eect on
temperature is also observed in Figure (b) along the vertical
midline inside the enclosure. As is expected, the temperature
decreases by increasing Y. In addition, the temperature prole
gradients on bottom and top walls certify the accuracy of
imposing temperature boundary condition on these walls
(niteandzerotemperaturegradientsonheatedsurfaceand
top wall, resp.).
Figure  hasmadeacomparisonbetweenpurewater,
nanouid %, and hybrid nanouid with the same volume
Advances in Mechanical Engineering
0.012
Saha results Present results
Gr =10
3
Gr =10
4
Gr =10
5
Gr =10
6
0.049
0.0023
0.0023
0.0023
0.002
0.007
0.007 0.007
0.007
0.007
0.012
0.012
0.012 0.012
0.0164
0.0164
0.0211
0.0211
0.0211
0.025
0.025
0.03 0.035
0.035
0.039
0.045
0.045
0.049
0.053
0.155
0.105
0.059 0.068
0.068
0.0855
0.0755 0.0955
0.14
0.0955 0.135
0.135
0.125
0.0023 0.002
0.007
0.045
0.007
0.007
0.012
0.012
0.012
0.012 0.012
0.012
0.0164
0.0164
0.0164
0.0164
0.0211
0.025
0.03
0.10
0.035
0.035
0.039
0.049
0.049
0.053
0.059
0.068
0.068
0.0755
0.0955
0.0955
0.08550.0755
0.125
0.145
0.155
0.135
0.0023
0.0025
0.007
0.007
0.007
0.007
0.012
0.0211
0.0211
0.0211
0.03
0.03
0.03
0.09555
0.068
0.035
0.135
0.039 0.039
0.045
0.049
0.059
0.0855
0.755
0.139
0.053
0.0590.0955
0.0023
0.0023
0.0023
0.0023
0.0023
0.0023
0.03
0.007
0.007
0.049
0.039
0.007
0.007
0.007
0.012
0.0120.012
0.012
0.0164
0.0164
0.0164
0.0164
0.0164
0.0211
0.0211
0.0211
0.0211
0.025
0.025
0.035
0.045 0.059
0.0023
0.0023
0.0023
0.0023
0.007
0.007 0.007
0.007
0.0211
0.0211
0.0211
0.025
0.0164
0.039
0.053
0.059
0.035
0.0164
0.025
0.025
0.0023
0.035
0.012
0.0755
0.025
0.03
0.025
0.0164 0.03
0.053
0.059
0.0211
0.035
0.039
0.045
0.012
0.0164
0.045
0.053
0.1
0.035
0.039
0.049
0.03
0.045
0.068
0.035
0.0755
F : Comparison of the isotherms in dierent Grashof numbers between the present study and results by Saha [].
concentration as the working uids. Figures (a) and (b)
clearly illustrate that hybrid nanouid has a better cooling
performance compared to that of nanouid and also a
conventional one.
Figure  shows the nondimensional temperature distri-
bution along the horizontal and vertical midlines in enclosure
for dierent Rayleigh numbers. It is observed in Figure (a)
that by increasing Ra up to 5,themaximumofdimen-
sionless temperature on horizontal midline increases and
also slightly tended towards the right of enclosure due to
itsgeometry.Nevertheless,thisbehaviorisnotobserved
for higher Rayleigh number, in a way that in Ra =10
6
the temperature prole is totally changed. is fact can be
explained that for lower Ra (Ra ≤105), the natural convection
ow is weak and, thus, conduction dominates the ow
and heat transfer regimes, although when Rayleigh number
increases (Ra >10
5),thebouncyforcesaregraduallymore
pronounced compared to viscous forces. As a consequence,
convection becomes dominant which results in a better
cooling performance. Hence, the maximum temperature for
Advances in Mechanical Engineering
0.035
0.03
0.025
0.02
0.015
0.01
0.005
00 0.2 0.4 0.6 0.8 1
𝜃
X
Wat e r
Al2O3-Cu/water, 𝜙 = 0.1
Al2O3-Cu/water, 𝜙 = 1.0
Al2O3-Cu/water, 𝜙 = 2.0
(a)
0 0.2 0.4 0.6 0.8 1
𝜃
Y
Wat e r
Al2O3-Cu/water, 𝜙 = 0.1
Al2O3-Cu/water, 𝜙 = 1.0
Al2O3-Cu/water, 𝜙 = 2.0
0.12
0.1
0.08
0.06
0.04
0.02
(b)
F : Prole of nondimensional temperature for pure water and hybrid nanouid in dierent volume concentrations in Ra =106along
the (a) horizontal midline (=0.5), (b) vertical midline (=0.5)in sinusoidal corrugated enclosure.
0.035
0.03
0.025
0.02
0.015
0.01
0.005
00 0.2 0.4 0.6 0.8 1
𝜃
X
Wat e r
Al2O3/water, 𝜙 = 2.0
Al2O3-Cu/water, 𝜙 = 2.0
(a)
0 0.2 0.4 0.6 0.8 1
0.12
0.1
0.08
0.06
0.04
0.02
𝜃
Y
Wat e r
Al2O3/water, 𝜙 = 2.0
Al2O3-Cu/water, 𝜙 = 2.0
Al2O3/
(b)
F : Prole of nondimensional temperature for pure water and nanouid and hybrid nanouid % in Ra =106along the (a) horizontal
midline (=0.5), (b) vertical midline (=0.5)in sinusoidal corrugated enclosure.
Ra =106is decreased. e vertical temperature distribution
along line  = 0.5canbeseeninFigure (b).Itisfound
outthatthetemperaturenearthepartiallyheatedsurface
is reduced by increasing Ra that indicates enhancing the
cooling performance of the enclosure. According to the afore-
mentionedexplanation,thesimilarbehavioroftemperature
prole is seen near the top wall that is far from the heat source
which means that temperature distribution far from the heat
Advances in Mechanical Engineering
Ra =10
6
Ra =10
5
Ra =10
4
Ra =10
3
0.06
0.05
0.04
0.03
0.02
0.01
00 0.2 0.4 0.6 0.8 1
𝜃
X
(a)
Ra =10
6
Ra =10
5
Ra =10
4
Ra =10
3
0 0.2 0.4 0.6 0.8 1
0.3
0.25
0.2
0.15
0.1
0.05
0
𝜃
Y
(b)
F : Prole of nondimensional temperature for hybrid nanouid % in range of Ra along the (a) horizontal midline ( = 0.5),
(b) vertical midline (=0.5)in sinusoidal corrugated enclosure.
Ra =10
6
Ra =10
5
Ra =10
4
Ra =10
3
100
80
60
40
20
0
−20
−40
−60
−80 0 0.2 0.4 0.6 0.8 1
X
V
(a)
Ra =10
6
Ra =10
5
Ra =10
4
Ra =10
3
00.2 0.4 0.6 0.8 1
10
5
0
−5
−10
Y
U
(b)
F : Velocity distribution for hybrid nanouid % in range of Ra (a) 𝑦along the horizontal midline (=0.5),(b)𝑥along the vertical
midline (=0.5)in sinusoidal corrugated enclosure.
source increases when conduction is dominant (Ra ≤10
5)
and decreased when convection dominates the ow regime
(Ra >105).
Figure  illustrates the velocity distribution on the hor-
izontal and vertical midline of the sinusoidal corrugated
enclosure or hybrid nanouid % in range of Ra. It is
 Advances in Mechanical Engineering
understood from Figure (a) that the absolute magnitude
of vertical velocity along the horizontal midline increases
by Ra. is fact is because of stronger buoyant ow due to
higher Rayleigh numbers. Moreover, the maximum of this
parameter is seen at the midsection of the enclosure that is
due to the place of heat source on the midline of the bottom
wall. See Figure (b) where the nonsymmetrical horizontal
velocity prole is presented which indicates the direction of
the ow rotation within the enclosure due to its geometry and
boundarycondition.Itcanbeseenthathorizontalvelocity
along the vertical midline also increases by Ra and then the
natural convection ow becomes stronger.
e evolution of thermal elds of nanouid and pure
water and also hybrid nanouid and pure water in range of
Rayleigh number and volume concentration for a sinusoidal
corrugated enclosure with  = 0.4is presented in Figure .
It is realized that for lower Ra (3and 4)theconvection
intensity inside the enclosure is very weak. us, viscous
forces are more dominant than the buoyancy forces and
diusion is the principal mode of heat transfer; such phe-
nomena have been already reported by Saha []andHussain
et al. []. Hence, the isotherm proles remain similar to
conduction heat transfer pattern and are almost invariant
up to Ra =10
4. At higher Rayleigh numbers, when the
intensity of convection increases, the isotherm pattern is
signicantly changed which indicates that the convection is
the dominating heat transfer mechanism in the enclosure.
In Ra =10
5and more signicantly for Ra =10
6the
isotherm proles start getting shied towards the side walls
and they break into two symmetric contour lines, as shown
inthisgure.ItcanbeseenthatwiththeincreaseofRayleigh
number, the isotherms are squeezed toward the heated part of
the bottom wall. erefore, the developing thermal boundary
layer thickness at the bottom wall becomes thinner and
thus indicates higher heat transfer rate and results in higher
average Nusselt number. Moreover, in this gure, it is clear
that the isotherm patterns are aected by the presence of
nanoparticles. In fact, the existence of nanoparticles results
in compression of isotherms near the heat section of bottom
wall which means improving in heat transfer performance.
is eect is more obvious for hybrid nanouid rather than
nanouid.
Intheelectroniccomponentswithaconstantheatux,
the temperature on the heated section is not uniform. is
uncontrolled surface temperature has an adverse eect on the
life and functionality of these components. Accordingly, in
this part of the current study, the variation of local Nusselt
number along the heated section on the bottom wall (0.5(1
) ≤  ≤ 0.5(1+)) is investigated in Figure .Itisseen
in Figure (a) that for each working uid there is a point
on which the Nusselt number is minimum. It can be noted
that the maximum of the temperature prole of the partially
heated surface is located at this point, where the temperature
dierence with the adjacent ow is minimal. It is understood
from this gure that employing hybrid nanouid % is more
eective compared to the similar nanouid and base uid on
decreasing the maximum temperature. Moreover, it is found
out from Figure (b) that by increasing the volume concen-
tration of nanoparticles in hybrid nanouid, the maximum
surface temperature of the heat source decreases. is fact
can be clearly observed in Figure (b) and (b). Also, the
similarbehaviorisreportedinthepreviousworks(i.e.,[]).
is reduction is less evident as the heat transfer mechanism
within the enclosure shis from conduction (low Rayleigh
numbers) to convection (high Rayleigh numbers) dominated
ow.
As previously mentioned, the decrease in the maximum
temperature of the heated section is a result of the augmented
thermal energy transfer from the wall to uid. However, since
the inclusion of nanoparticles enhances the eective thermal
conductivity of the nanouid and hybrid nanouid, the
decrease in the maximum temperature of the heated section
is more remarkable where the conduction regime prevails.
is phenomenon can be described in two microscopic and
macroscopic perspectives. From microscopic standpoint, the
nanoparticles hit the wall, absorb thermal energy, reduce
thewalltemperature,andmixbackwiththebulkofthe
uid. In macroscopic viewpoint, by adding nanoparticles in
one kind of material (nanouid) and two types of it (hybrid
nanouid), the thermal properties of the resulting mixture
have improved. erefore, hybrid nanouid possesses a
higher thermal conductivity than that of nanouid and also a
conventional one. us, this higher thermal conductivity has
the positive eect on the heat transfer performance.
Figure  presents the eect of Rayleigh number on the
local Nusselt number of heated section for hybrid nanouid
%. According to the above mentioned explanation about the
maximum temperature of the heated surface, it is expected
that the local Nusselt number increases by Rayleigh number
and this trend is seen in this gure.
Figures (a)(c) show the eect of Rayleigh number
ontheaverageNusseltnumberofdiscreteheatedbottom
wall for pure water, Al2O3/water nanouid, and Al2O3-
Cu/water hybrid nanouid in a xed volume concentration.
As is expected, the average Nusselt number increases by
Rayleigh number. Also, it is obvious that adding nanopar-
ticles enhances heat transfer characteristics. ese results
are in agreement with previous observations [,,].
In addition, comparison between Figures (a),(b),and
(c) revealsthataugmentationofheattransferofhybrid
nanouid and nanouid compared to pure water increases
by volume concentration in a constant Rayleigh number.
More importantly, it is found out that employing hybrid
nanouid ameliorates the average Nusselt number more than
that of nanouid in the same volume concentration. is
behavior could be explained in enhancing thermophysical
properties of the mixture compared to pure water, thanks to
particles inclusion. Consequently, hybrid nanouid possesses
a higher thermal conductivity that results in augmentation
of the heat transfer rate. For example, in Rayleigh number
of 6one can nd .% enhancement of average Nusselt
number for nanouid with =2%comparedtopure
water; however hybrid nanouid with the same volume
concentration provides an increase of .%. e average
Advances in Mechanical Engineering 
Ra =10
3
Ra =10
4
Nano
Nano
Hybrid
Hybrid
𝜙 = 0.1% 𝜙 = 1.0% 𝜙 = 2.0%
(a)
Ra =10
5
Ra =10
6
Nano
Nano
Hybrid
Hybrid
𝜙 = 0.1% 𝜙 = 1.0% 𝜙 = 2.0%
(b)
F : Isotherms for dierent Rayleigh number and volume concentration of nanouid and hybrid nanouid (red line) and pure water
(blue line).
 Advances in Mechanical Engineering
Wat e r
Al2O3-Cu/water, 𝜙 = 0.1
Al2O3-Cu/water, 𝜙 = 1.0
Al2O3-Cu/water, 𝜙 = 2.0
30
25
20
15
10
5
0.3 0.4 0.5 0.6 0.7
X
Nu
(a)
Wat e r
Al2O3/water, 𝜙 = 2.0
Al2O3-Cu/water, 𝜙 = 2.0
30
25
20
15
10
5
0.3 0.4 0.5 0.6 0.7
X
Nu
(b)
F : Variation of local Nusselt number along the partially heated surface in Ra =106for (a) pure water and hybrid nanouid in dierent
volume concentrations (b) water, nanouid, and hybrid nanouid %.
35
30
25
20
15
10
5
0
0.3 0.4 0.5 0.6 0.7
X
Ra =10
6
Ra =10
5Ra =10
4
Ra =10
3
Nu
F : Variation of local Nusselt number along the partially
heated surface in dierent Rayleigh numbers.
Nusselt numbers for all the considered cases are reported in
Table . ese scenarios also can be observed for other cases.
Two correlations based on the numerical results have
been developed to predict the average Nusselt number for
T : Average Nusselt numbers for all studied cases.
Ra =103Ra =104Ra =105Ra =106
Wat e r
=0.0% . . . .
Nanouid
=0.1% . . . .
=1.0% . . . .
=2.0% . . . .
Hybrid nanouid
=0.1% . . . .
=1.0% . . . .
=2.0% . . . .
nanouid (see ())andhybridnanouid(see()) as
follows:
Nu =3.852+0.0104Ra0.4891+5.9,()
Nu =3.935+0.0106Ra0.4881+8.59.()
ese equations coecients were assessed with the help
of classical least square method and the correlations are
valid for laminar regime (103Ra ≤10
6),Al
2O3/water
nanouid, and Al2O3-Cu/water hybrid nanouid with the
volume concentrations less than %.
A parity plot for the above correlations is shown in
Figure . It shows that the correlated Nusselt data were in
good agreement with the simulated ones. e maximum
Advances in Mechanical Engineering 
Wat e r
Al2O3/water, 𝜙 = 0.1
Al2O3-Cu/water, 𝜙 = 0.1
14
12
10
8
6
4
103104105106
Ra
Nu
(a)
Wat e r
Al2O3/water, 𝜙 = 1.0
Al2O3-Cu/water, 𝜙 = 1.0
14
12
10
8
6
4
103104105106
Ra
Nu
(b)
Wat e r
Al2O3/water, 𝜙 = 2.0
Al2O3-Cu/water, 𝜙 = 2.0
14
12
10
8
6
4
103104105106
Ra
Nu
(c)
F : e average Nusselt number of sinusoidal corrugated enclosure versus Rayleigh number for pure water, Al2O3/water nanouid,
and Al2O3-Cu/water hybrid nanouid by (a) =0.1%, (b) =1%, and (c) =2%.
error observed in Figures (a) and (b) was around % and
%, for nanouid and hybrid nanouid, respectively.
5. Conclusions
e eect of Rayleigh number and the inclusion of Al2O3
and Al2O3-Cu nanoparticles in the base uid, for a laminar
natural convection in a sinusoidal corrugated enclosure with
adiscreteheatsourceonthebottomwall,onheattransferand
ow characteristics have been numerically investigated in this
study.
Some of important conclusions drawn from the present
analysis are as follows.
(i) Classical models for specifying nanouids thermo-
physical properties signicantly underestimate the
hybrid nanouid viscosity and thermal conductivity,
 Advances in Mechanical Engineering
+11%
−8%
15
12
9
6
3
003691215
Nu (correlation)
Nu (simulation)
(a)
15
12
9
6
3
00 3 6 9 12 15
+12%
−7%
Nu (correlation)
Nu (simulation)
(b)
F : Parity plot comparing the prediction data and simulation results, for (a) nanouid, (b) hybrid nanouid.
particularly in the higher volume concentrations. is
result has been also observed by Suresh et al. [].
(ii) e average Nusselt number increases by Rayleigh
number. Moreover, nanouid clearly enhances the
heat transfer rate, thanks to the presence of nanopar-
ticles. More importantly, hybrid nanouid improves
theaverageNusseltnumbermorethanthatof
nanouid. In the present study, the highest value of
the average Nusselt number (Nu =14.402)is related
to Rayleigh number of 6andhybridnanouid%,
whilethisvalueforthesimilarnanouidhasbeen
calculated . (in a xed Ra). erefore, employing
hybrid nanouid is more ecient in heat transfer
performance with respect to the similar nanouid.
(iii) e increase of Rayleigh number strengthens the nat-
ural convection ows which results in increasing the
local Nusselt number of the heated section and reduc-
tion of the corresponding temperature. e increase
of volume concentration of nanoparticles causes the
maximum surface temperature (i.e., located in point
in maximum Nusselt number) to decrease, particu-
larly at low Rayleigh numbers, because conduction
regime prevails.
(iv) e velocity distribution along the vertical midline
in sinusoidal corrugated enclosure is clearly nonsym-
metric due to the geometry and boundary condi-
tions. e velocity increases by Rayleigh number, as
expected.
(v)TopredicttheaverageNusseltnumberofnanouid
and hybrid nanouid, two correlations have been
developed. ese equations are based on the model-
ing results and calculated by employing the classical
least square method.
Nomenclature
: Heat capacity, J/kg K
: gravitational acceleration, m/s2
Gr: Grashof number
: Height of the sinusoidal corrugated enclosure, m
:ermalconductivity,W/mK
: Length of the heat source, m
Nu: Local Nusselt number
Nu: Average Nusselt number
: Dimensionless pressure
: Pressure, Pa
Pr: Prandtl number
󸀠󸀠:Heatux,W/m
2
Ra: Rayleigh number
:Temperature,K
,V:Velocitycomponents,m/s
,: Dimensionless velocity components
: Width of the sinusoidal corrugated enclosure, m
,: Cartesian coordinates, m
,: Dimensionless Cartesian coordinates.
Greek
:ermaldiusivity,m
2/s
:Coecient of volumetric expansion, /K
: Pressure drop, Pa
: Discrete heat source size ratio, L/W
: Dimensionless temperature
: Dynamic viscosity, kg/ms
: Density, kg/m3
]: Kinematic viscosity, m2/s
: Volume concentration
:Particlesphericity.
Advances in Mechanical Engineering 
Subscripts
Al2O3:RelatedtoAl
2O3nanoparticle
bf: Base uid
:ecoldsurface
Cu: RrelatedtoCunanoparticle
:Fluid
hnf: Hybrid nanouid
nf: Nanouid
:Particle.
Conflict of Interests
e authors declare that there is no conict of interests
regarding the publication of this paper.
Acknowledgments
e authors would like to thank the Delta Oshore Technol-
ogy Co. for nancial support and the R&D department of this
company for allocating computing facilities.
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Chapter
This chapter discusses the complex nature of the natural convection phenomena in enclosures. It discusses the two basic configurations of natural convection— that is, a rectangular cavity and a horizontal circular cylinder. In rectangular cavities, consideration is given to the two-dimensional convective motion generated by the buoyancy force on the fluid in a rectangle and to the associated heat transfer. The two long sides are vertical boundaries held at different temperatures and the short sides can either be heat conducting or insulated. Particular attention is given to the different flow regimes that can occur and the heat transfer across the fluid space between the two plane parallel vertical boundaries. Although heat transfer by radiation may not be negligible it is independent of the other types of heat transfer and can be fairly accurately calculated separately. To formulate the boundary value problem that describes this phenomena it is assumed that: (a) the motion is two-dimensional and steady, (b) the fluid is incompressible and frictional heating is negligible, and (c) the difference between the hot wall and cold wall temperatures is small relative to the absolute temperatures of the cold wall. In horizontal circular cylinder, consideration is given to the large Rayleigh number flow with the Prandtl number large and the Grashof number of unit order of the magnitude.
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In the present work, the effects of the longitudinal magnetic field and the heat source size on natural convection heat transfer through a tilted sinusoidal corrugated enclosure for different values of enclosure inclination angles are analyzed and solved numerically by using the finite volume technique based on body fitted control volumes with a collected variable arrangement. A constant heat flux source is discretely embedded at the central part of the bottom wall whereas the remaining parts of the bottom wall and the upper wall are assumed adiabatic, and two vertical sinusoidal corrugated walls are maintained at a constant low temperature. The range of the variable parameters considered in the present analysis is as follows: the enclosure inclination angle is varied from 0° to 135°, the ratio of the size of the heating element to enclosure width varied from 20 to 80% of enclosure reference length, Hartmann number is varied from 0 to 100, and Rayleigh number varied from 103 to 106. Liquid gallium with constant Prandtl number (0.02) is used as a working fluid with constant properties except the density. The obtained results indicated that streamlines are affected strongly by the magnetic field especially for small values of inclination angle (Φ=0°Φ=0°) and Rayleigh number (Ra=103–106). The magnetic field effect decreases with an increase in the enclosure inclination angle (Φ>0°Φ>0°) especially for large values of Rayleigh number. The increase in Hartmann number will cause the temperature lines to become symmetrical in shape for large values of Rayleigh number (Ra=105–106). The results also explain that the temperature lines are very little affected by the inclination angle especially for small values of (ε=0.4ε=0.4) and (Ra=104), but this effect will increase especially for (ε=0.8ε=0.8) and (Ra=106). The Nusselt number increases first with an increase in inclination angle (0°≤Φ≤45°)(0°≤Φ≤45°), then is slightly affected for (45°<Φ≤90°)(45°<Φ≤90°), and finally decreases for (90°<Φ≤135°)(90°<Φ≤135°). An empirical correlation is developed by using Nusselt number versus Hartmann and Rayleigh numbers, and enclosure inclination angle. The increase in Hartmann number and the ratio of heating element to enclosure width will decrease the Nusselt number. Furthermore, four mathematical correlations are extracted from the results and presented, which can be used to accurately predict the average Nusselt number in terms of enclosure inclination angle, Hartmann, and Rayleigh numbers.
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In the present study, mathematical modeling is performed to simulate natural convection of Al2O3/water nanofluids in a vertical square enclosure using the lattice Boltzmann method (LBM). Results indicate that the average Nusselt number increases with the increase of Rayleigh number and particle volume concentration. The average Nusselt number with the use of nanofluid is higher than the use of water under the same Rayleigh number. However, the heat transfer rate of the nanofluid takes on a lower value than water at a fixed temperature difference across the enclosure mainly due to the significant enhancement of dynamic viscosity. Furthermore, great deviations of computed Nusselt numbers using different models associated with the physical properties of a nanofluid are revealed. The present results are well validated with the works available in the literature and consequently LBM is robust and promising for practical applications.