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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 nanouid, and Al2O3-Cu/water hybrid nanouid which is a new advanced nanouid

with two kinds of nanoparticle materials. e eects of Rayleigh number (103≤Ra ≤10

6)and water, nanouid, and hybrid

nanouid (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 nanouid improves the heat transfer rate compared

to nanouid 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 nanouid and hybrid nanouid was around % and

%, respectively.

1. Introduction

In the past years, many dierent techniques were utilized to

improve the heat transfer rate for reaching a satisfactory level

of thermal eciency. A way for this purpose is enhancement

in thermal conductivity. So many eorts 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 nanouid.

Choi [] presented the benet of using the nanoparticles

dispersed in a base uid in dierent 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 nanouid eld until

now dened “nanouid” as a base uid with suspended

nanosized particles from just one type of material. ey

have studied the eect of size, shape, concentration, and

material of nanoparticles on thermophysical properties of

nanouid and its inuence on heat transfer and pressure drop

characteristics. Nevertheless, very recently, an experimental

study has been carried out on nanouid with two types of

nanoparticles dispersed simultaneously in a base uid that is

called “hybrid nanouid” [].

e most important exclusivity of hybrid nanouid 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

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Advances in Mechanical Engineering

Volume 2014, Article ID 147059, 16 pages

http://dx.doi.org/10.1155/2014/147059

Advances in Mechanical Engineering

benecial 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 nanouid 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 nanouid 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 nanouid []. To reach

the stable hybrid nanouid, they used a thermomechanical

method (two-step method). ey added Cu nanoparticles

to Al2O3/water nanouid and synthesized Al2O3-Cu/water

hybrid nanouid by dierent volume concentrations as .,

., ., , and %.

According to the above mentioned benets of hybrid

nanouids, it is clearly expected that this advanced nanouid

plays a vital role in the future of science of nanouid and

researchers show a greater tendency toward investigation of

hybrid nanouids and eect 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 dierent

shapes of the heat transfer surfaces is necessary to ensure the

ecient 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

eect 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 signicant eect on the heat transfer rate. In addition,

Hussain et al. [] numerically analyzed the eect 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 conguration 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 nanouid 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 eect

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 conguration,

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 prole 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

dened as

Ra =3Pr

]2.()

Ra has been varied from 3to 6, where Pr is Prandtl

number and is specied in

Pr =]

.()

Advances in Mechanical Engineering

2.1. ermophysical Properties of Nanouid and Hybrid

Nanouid. As previously mentioned, although some litera-

tures studied the determination of thermophysical proper-

ties, the classical models are not certain for nanouids. Of

course, experimental results allow us to select an appropriate

model for a specied property.

e eective properties of the Al2O3/water nanouid and

Al2O3-Cu/water hybrid nanouid are dened as follows:

nf =𝑝𝑝+1−𝑝bf.()

Equation () was originally introduced in [] for deter-

miningdensityandthenwidelyemployedin[–]. So, the

density of hybrid nanouid is specied by

hnf =Al2O3Al2O3+Cu Cu +1−bf.()

is the overall volume concentration of two dierent

types of nanoparticles dispersed in hybrid nanouid and is

calculated as

=Al2O3+Cu.()

Equation ()thatisutilizedforspecifyingheatcapacity

wasrstemployedin[] and then used in many articles [,

,]:

nf =𝑝𝑝𝑝+1−𝑝bf bf

nf .()

According to (), heat capacity of hybrid nanouid can be

determined as follows:

hnf =Al2O3Al2O3Al2O3+CuCuCu +1−bf bf

hnf .

()

e thermal expansion coecient of nanouid can be

determined by ()employedinsomeliteratures[,]:

nf =𝑝𝑝𝑝+1−𝑝bfbf

nf .()

Hence, for hybrid nanouid, thermal expansion can be

dened as follows:

hnf =1

hnf Al2O3Al2O3Al2O3+CuCuCu

+1−bfbf. ()

In addition, for calculating conductivity of nanouids

equation () was proposed by Hamilton and Crosser [].

nf

bf =𝑝+(−1)bf −(−1)𝑝bf −𝑝

𝑝+(−1)bf +𝑝bf −𝑝.()

Here is the empirical shape factor in order to account

the eect 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

nanouid.

sphericity, dened 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 −2𝑝bf −𝑝

𝑝+2bf +𝑝bf −𝑝.()

If the thermal conductivity of hybrid nanouid is dened

according to Maxwell model, ()mustbeemployedforthis

purpose:

hnf

bf =Al2O3Al2O3+Cu Cu

+2bf +2Al2O3Al2O3+CuCu −2bf

×Al2O3Al2O3+Cu Cu

+2bf −Al2O3Al2O3+Cu Cu+bf−1.

()

To thermal conductivities of hybrid nanouid in dierent

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 nanouid and hybrid nanouid.

(%) Cu (%) Al2O3(%) nf (W/m⋅K) nf (kg/m⋅s) hnf (W/m⋅K) hnf (kg/m⋅s)

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

results compared to Maxwell model, the classical models,

generally, could not be as useful and ecient as experimental

data. erefore, to accomplish the most accurate numerical

results in this study, thermal conductivity values for both

nanouid and hybrid nanouid in dierent 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 coecient 2is calculated

as [].

In ()–(), for calculating the viscosity of nanouid,

is the volume concentration of Al2O3nanoparticles in

nanouid; whereas to dene this property of hybrid nanouid

(hnf), must be the overall volume concentration of

nanoparticles indicated in ().

Considering Figure , it can be understood that these

classical models signicantly underestimate the hybrid

nanouid viscosity, particularly in high volume concentra-

tion. erefore, to have a high accuracy in numerical results,

in the present study, viscosity of hybrid nanouid is obtained

from experimental data [].isresultthattheclassical

models underpredict hnf and hnf for hybrid nanouid has

been also reported in the earlier literature [].

e thermophysical properties of both Al2O3/water

nanouid and Al2O3-Cu/water hybrid nanouid 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 nanouid.

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 nanouid with volume

concentration of lower than % seems acceptable according

to Suresh et al. []. ey have reported viscosity of hybrid

nanouid as a function of shear rate and showed an inde-

pendency between viscosity and shear rate of this hybrid

nanouid with volume concentration of lower than %. In

addition, other physical properties of uid (thermal conduc-

tivity, thermal expansion coecient, and specic heat) are

taken constant in a specic 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 dene the nondimensional vertical component and

velocity. According to (), the Rayleigh number is dened

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

specied 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 dened 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 diusion 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 dierent 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 nanouid

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 renement for hybrid nanouid 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 renement is illustrated in Figure .isgraphshows

that an acceptable grid independency is achieved by rening

the grid and the results obtained from grids ner than

cells can be assumed to be grid independent. e dierence

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 dierent

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 dierences

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 eect of adding the nanoparticles in one and

twokindsofmaterialstoabaseuidonheattransfercharac-

teristics and hydrodynamic behavior have been investigated

in this paper. is enclosure is partially heated via a discrete

heatsourcewithaconstantheatuxlocatedinthemiddleof

the bottom wall. Also, the eect of Rayleigh number (103≤

Ra ≤106)and dierent working uids including pure water,

nanouid, and hybrid nanouid (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 proles along the

midsection of the enclosure are assessed in the present study.

Accordingly, Figure (a) shows the prole of nondimensional

temperature along the horizontal midline inside the enclo-

sure, in a xed Rayleigh number (Ra =10

6)for pure water

and hybrid nanouid in dierent volume concentrations

as the working uid. In Figure (a) abumpcanbeseen

near the center of temperature proles which is clearly due

tothepresenceofpartiallyheatedsurfaceonthebottom

wall. It is revealed from this gure that by increasing the

volume concentration of hybrid nanouid, the temperature

decreases. is fact signies that employing hybrid nanouid,

especially in higher volume concentrations, can enhance the

cooling performance of the enclosure. is favorable eect 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 prole

gradients on bottom and top walls certify the accuracy of

imposing temperature boundary condition on these walls

(niteandzerotemperaturegradientsonheatedsurfaceand

top wall, resp.).

Figure hasmadeacomparisonbetweenpurewater,

nanouid %, and hybrid nanouid 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 dierent Grashof numbers between the present study and results by Saha [].

concentration as the working uids. Figures (a) and (b)

clearly illustrate that hybrid nanouid has a better cooling

performance compared to that of nanouid and also a

conventional one.

Figure shows the nondimensional temperature distri-

bution along the horizontal and vertical midlines in enclosure

for dierent 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 prole 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 : Prole of nondimensional temperature for pure water and hybrid nanouid in dierent 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 : Prole of nondimensional temperature for pure water and nanouid and hybrid nanouid % 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

prole 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 : Prole of nondimensional temperature for hybrid nanouid % 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 nanouid % 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 nanouid % 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 prole 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 nanouid and pure

water and also hybrid nanouid 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

diusion is the principal mode of heat transfer; such phe-

nomena have been already reported by Saha []andHussain

et al. []. Hence, the isotherm proles 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

signicantly changed which indicates that the convection is

the dominating heat transfer mechanism in the enclosure.

In Ra =10

5and more signicantly for Ra =10

6the

isotherm proles start getting shied towards the side walls

and they break into two symmetric contour lines, as shown

inthisgure.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 aected 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 eect is more obvious for hybrid nanouid rather than

nanouid.

Intheelectroniccomponentswithaconstantheatux,

the temperature on the heated section is not uniform. is

uncontrolled surface temperature has an adverse eect 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 prole of the partially

heated surface is located at this point, where the temperature

dierence with the adjacent ow is minimal. It is understood

from this gure that employing hybrid nanouid % is more

eective compared to the similar nanouid 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 nanouid, 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 shis 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 eective thermal

conductivity of the nanouid and hybrid nanouid, 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 (nanouid) and two types of it (hybrid

nanouid), the thermal properties of the resulting mixture

have improved. erefore, hybrid nanouid possesses a

higher thermal conductivity than that of nanouid and also a

conventional one. us, this higher thermal conductivity has

the positive eect on the heat transfer performance.

Figure presents the eect of Rayleigh number on the

local Nusselt number of heated section for hybrid nanouid

%. 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 eect of Rayleigh number

ontheaverageNusseltnumberofdiscreteheatedbottom

wall for pure water, Al2O3/water nanouid, and Al2O3-

Cu/water hybrid nanouid 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

nanouid and nanouid compared to pure water increases

by volume concentration in a constant Rayleigh number.

More importantly, it is found out that employing hybrid

nanouid ameliorates the average Nusselt number more than

that of nanouid 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 nanouid 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 nanouid with =2%comparedtopure

water; however hybrid nanouid 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 dierent Rayleigh number and volume concentration of nanouid and hybrid nanouid (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 nanouid in dierent

volume concentrations (b) water, nanouid, and hybrid nanouid %.

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 dierent 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% . . . .

Nanouid

=0.1% . . . .

=1.0% . . . .

=2.0% . . . .

Hybrid nanouid

=0.1% . . . .

=1.0% . . . .

=2.0% . . . .

nanouid (see ())andhybridnanouid(see()) as

follows:

Nu =3.852+0.0104Ra0.4891+5.9,()

Nu =3.935+0.0106Ra0.4881+8.59.()

ese equations coecients were assessed with the help

of classical least square method and the correlations are

valid for laminar regime (103≤Ra ≤10

6),Al

2O3/water

nanouid, and Al2O3-Cu/water hybrid nanouid 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 nanouid,

and Al2O3-Cu/water hybrid nanouid by (a) =0.1%, (b) =1%, and (c) =2%.

error observed in Figures (a) and (b) was around % and

%, for nanouid and hybrid nanouid, respectively.

5. Conclusions

e eect 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 nanouids thermo-

physical properties signicantly underestimate the

hybrid nanouid 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) nanouid, (b) hybrid nanouid.

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, nanouid clearly enhances the

heat transfer rate, thanks to the presence of nanopar-

ticles. More importantly, hybrid nanouid improves

theaverageNusseltnumbermorethanthatof

nanouid. 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 nanouid is more ecient in heat transfer

performance with respect to the similar nanouid.

(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 nanouid, 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

:Coecient of volumetric expansion, /K

: Pressure drop, Pa

: Discrete heat source size ratio, L/W

: Dimensionless temperature

: Dynamic viscosity, kg/m⋅s

: 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 nanouid

nf: Nanouid

:Particle.

Conflict of Interests

e authors declare that there is no conict of interests

regarding the publication of this paper.

Acknowledgments

e authors would like to thank the Delta Oshore Technol-

ogy Co. for nancial support and the R&D department of this

company for allocating computing facilities.

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