international journal of heat and fluid flow · 2009. 7. 22. · ostrach (1988), catton (1978),...

International Journal of Heat and Fluid Flow xxx (2009) xxx–xxx

ARTICLE IN PRESS

Contents lists available at ScienceDirect

International Journal of Heat and Fluid Flow

journal homepage: www.elsevier .com/locate / i jhf f

Effects of inclination angle on natural convection in enclosures filledwith Cu–water nanofluid

Eiyad Abu-Nada a,*, Hakan F. Oztop b

a Department of Mechanical Engineering, Hashemite University, Zarqa 13115, Jordanb Department of Mechanical Engineering, Fırat University, Elazig TR-23119, Turkey

a r t i c l e i n f o

Article history:Received 26 July 2008Received in revised form 20 December 2008Accepted 2 February 2009Available online xxxx

Keywords:NanofluidsEnclosureNatural convectionInclination angle

0142-727X/$ - see front matter � 2009 Elsevier Inc. Adoi:10.1016/j.ijheatfluidflow.2009.02.001

* Corresponding author. Tel.: +962 390 3333; fax: +E-mail addresses: [email protected], eiyad_abunada

Please cite this article in press as: Abu-NaHeat Fluid Flow (2009), doi:10.1016/j.ijhe

a b s t r a c t

Effects of inclination angle on natural convection heat transfer and fluid flow in a two-dimensional enclo-sure filled with Cu-nanofluid has been analyzed numerically. The performance of nanofluids is testedinside an enclosure by taking into account the solid particle dispersion. The angle of inclination is usedas a control parameter for flow and heat transfer. It was varied from = 0� to = 120�. The governing equa-tions are solved with finite-volume technique for the range of Rayleigh numbers as 103

6 Ra 6 105. It isfound that the effect of nanoparticles concentration on Nusselt number is more pronounced at low vol-ume fraction than at high volume fraction. Inclination angle can be a control parameter for nanofluidfilled enclosure. Percentage of heat transfer enhancement using nanoparticles decreases for higher Ray-leigh numbers.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

Analysis of natural convection heat transfer and fluid flow inenclosures has been extensively made using numerical techniquesexperiments because of its wide applications and interest in engi-neering such as nuclear energy, double pane windows, heating andcooling of buildings, solar collectors, electronic cooling, micro-elec-tromechanical systems (MEMS). Further applications are listed byOstrach (1988), Catton (1978), Bejan (1995), Khalifa Bdul-Jabbar(2001) and De Vahl Davis et al. (1983).

Elsherbiny (1996) made an experimental study to investigatethe natural convection heat transfer in inclined rectangular enclo-sures. They indicated that the lowest heat transfer is formed foru = 180� (the cavity is heated from above). Bairi et al. (2007) per-formed a study on natural convection for high Rayleigh numbersusing numerical and experimental techniques in rectangular in-clined enclosures. They obtained a correlation between Nusseltand Rayleigh numbers and a minimal value for Nusselt numbersat u = 270o. Varol et al. (2008) investigated the effects of inclinationangle on natural convection in an enclosure with corner heater. Loet al. (2007) used a novel numerical solution algorithm based on adifferential quadrature (DQ) method to simulate natural convectionin an inclined cubic cavity using velocity–vorticity form of the Na-vier–Stokes equations. Aounallah et al. (2007) made a study on tur-bulent natural convection of air flow in a confined cavity with twodifferentially heated side walls is investigated numerically up to

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962 382 [email protected] (E. Abu-Nada).

da, E., Oztop, H.F. Effects of iatfluidflow.2009.02.001

Rayleigh number of 1012. They analyzed the effect of the inclina-tion angle and the amplitude of the undulation on turbulent heattransfer. Cianfrini et al. (2005) performed a study on natural con-vection in air filled, inclined enclosures with two adjacent wallsheated and two opposite walls cooled. As indicated above thatmany researchers focused on effects of inclination angle on naturalconvection due to complexity of the flow. Aydin et al. (1999), Soonget al. (1996) and Lee and Lin (1995) made different applications forfluid filled inclined enclosures and found that inclination angle canbe use a control parameter for heat and fluid flow.

An innovative technique to enhance heat transfer is by usingnano-scale particles in the base fluid. Nanotechnology has beenwidely used in industry since materials with sizes of nanometerspossess unique physical and chemical properties. Nano-scale parti-cle added fluids are called as nanofluid which is firstly utilized byChoi (1995). Some numerical and experimental studies on nanofl-uids include thermal conductivity (Kang et al. (2006), convectiveheat transfer (Maiga et al., (2005), Abu-Nada, (2008) and Oztopand Abu-Nada (2008) boiling heat transfer and natural convectionXuan and Li (2000)) Polidori et al. (2007), Duangthongsuk andWongwises (2008) and Jang and Choi (2004). Detailed review stud-ies are published by Putra et al. (2003), Wang and Mujumdar(2006, 2007), and Trisaksri and Wongwises (2007). Recently,Daungthongsuk and Wongwises (2007) studied the influence ofthermophysical properties of nanofluids on the convective heattransfer and summarized various models used in literature for pre-dicting the thermophysical properties of nanofluids. They studiedconvective heat transfer coefficient in double-tube counter flowheat exchanger in the presence of TiO2–water nanofluid.

nclination angle on natural convection in enclosures filled ... Int. J.

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Nomenclature

Cp specific heat at constant pressure (kJ kg�1 K�1)g gravitational acceleration (m s�2)H height of the enclosure (m)h local heat transfer coefficient (W m�2 K�1)h dimensionless length of partial heater, h0/Hh0 length of heater (m)Gr Grash of numberk thermal conductivity (W m�1 K�1)Nu Nusselt number, Nu = hH/kNuavg average Nusselt numberPr Prandtl numberqw heat flux (W m�2)Ra Rayleigh numberT dimensional temperature (K)u, v dimensional x and v components of velocity (m s�1)U, V dimensionless x and v components of velocityW length of the enclosure (m)x, y dimensionless coordinatesyp dimensionless center of heatery0p center of location of heater (m)

Greek symbolsa fluid thermal diffusivity (m2 s�1)b thermal expansion coefficient (K�1)

e numerical toleranceu nanoparticle volume fraction/ transport quantitym kinematic viscosity (m2 s�1)h dimensionless temperatureW dimensionless stream functionw dimensional stream function (m2 s�1)X dimensionless vorticityx dimensional vorticity (s�1)q density (kg m�3)l dynamic viscosity(N s m�2)

Subscriptsavg averagenf nanofluidf fluidH hotL colds Solidw wallp particle

φ

xy

g

TH

TCWH

Fig. 1. Schematic of inclined enclosure.

Table 1Thermophysical properties of pure fluid and nanoparticles.

Physical properties Fluid phase (Water) Cu

Cp (J/kg K) 4179 385q (kg/m3) 997.1 8933K (W/mK) 0.613 400a � 107 (m2/s) 1.47 1163.1b � 10�5 (1/K) 21 1.67

2 E. Abu-Nada, H.F. Oztop / International Journal of Heat and Fluid Flow xxx (2009) xxx–xxx

ARTICLE IN PRESS

Studies on natural convection using nanofluids are very limitedand they are related with differentially heated enclosures. Hwanget al. (2007) investigated the buoyancy-driven heat transfer ofwater-based Al2O3 nanofluids in a rectangular cavity. They showedthat the ratio of heat transfer coefficient of nanofluids to that ofbase fluid is decreased as the size of nanoparticles increases, orthe average temperature of nanofluids is decreased. Khanaferet al. (2003) investigated the heat transfer enhancement in atwo-dimensional enclosure utilizing nanofluids for various perti-nent parameters. They tested different models for nanofluid den-sity, viscosity, and thermal expansion coefficients. It was foundthat the suspended nanoparticles substantially increase the heattransfer rate any given Grashof number. Jou and Tzeng (2006) usednanofluids to enhance natural convection heat transfer in a rectan-gular enclosure. They conducted a numerical study using Khana-fer’s model. They indicated that volume fraction of nanofluidscause an increase in the average heat transfer coefficient.

The main aim of this work is to present the effects of inclinationangle on flow field and temperature distribution in differentiallyheated and nanofluid filled enclosures. Based on above literaturesurvey and to the author’s knowledge, no previous study on effectsof inclination angle on natural convection in nanofluid filled enclo-sure has not been studied yet.

2. Definition of model

Plotting of considered model is shown in Fig. 1 with coordi-nates. It is a two-dimensional square enclosure with an inclinationangle. Its vertical wall has different isothermal temperature wheninclination angle is zero. Remaining walls are adiabatic. The enclo-sure is filled with nanofluid which is Newtonian, incompressibleand laminar.

3. Formulation

Fig. 1 shows a schematic diagram of the partially heated enclo-sure. The fluid in the enclosure is a water-based nanofluid contain-

Please cite this article in press as: Abu-Nada, E., Oztop, H.F. Effects of iHeat Fluid Flow (2009), doi:10.1016/j.ijheatfluidflow.2009.02.001

ing copper nano particles. It is assumed that the base fluid (i.e.water) and the nanoparticles are in thermal equilibrium and noslip occurs between them. The thermo-physical properties of thenanofluid are given in Table 1. The left wall is maintained at a con-stant temperature (TH) higher than the right wall (TL). The ther-mo-physical properties of the nanofluid are assumed to beconstant except for the density variation, which is approximatedby the Boussinesq model.

The governing equations for the laminar and steady state natu-ral convection in terms of the stream function-vorticity formula-tion are:


E. Abu-Nada, H.F. Oztop / International Journal of Heat and Fluid Flow xxx (2009) xxx–xxx 3

ARTICLE IN PRESS

Vorticity

@

@x0x@W@y0

� �� @

@y0x@W@x0

� �

¼ lnf

qnf

@x@x02þ @x@y02

� �

þ ðuqsbs þ ð1�uÞqfbfÞqnf

g@T@x0

cos /� @T@y0 sin /

� �ð1Þ

Energy

@

@x0T@W@y0

� �� @

@y0T@W@x0

� �¼ @

@x0anf

@T@x0

� �þ @

@y0anf

@T@y0

� �ð2Þ

Kinematics

@2W@x02þ @

2W@y02

¼ �x ð3Þ

anf ¼keff

ðqcpÞnfð4Þ

The effective density of the nanofluid is given as

qnf ¼ ð1�uÞqf þuqs ð5Þ

The heat capacitance of the nanofluid is expressed as Abu-Nada(2008) and Khanafer et al. (2003):

ðqcpÞnf ¼ ð1�uÞðqcpÞf þuðqcpÞs ð6Þ

The effective thermal conductivity of the nanofluid is approxi-mated by the Maxwell–Garnetts model:

knf

kf¼ ks þ 2kf � 2uðkf � ksÞ

ks þ 2kf þuðkf � ksÞð7Þ

The use of this equation is restricted to spherical nanoparticleswhere it does not account for other shapes of nanoparticles. Thismodel is found to be appropriate for studying heat transfer enhance-ment using nanofluids (Akbarinia and Behzadmehr, 2007; Abu-Nada, 2008; Maiga et al., 2005). The viscosity of the nanofluid canbe approximated as viscosity of a base fluid lf containing dilute sus-pension of fine spherical particles and is given by Brinkman (1952):

lnf ¼lf

ð1�uÞ2:5ð8Þ

The radial and tangential velocities are given by the followingrelations, respectively:

u ¼ @W@y0

ð9Þ

v ¼ � @W@x0

ð10Þ

The following dimensionless groups are introduced:

x ¼ x0

H; y ¼ y0

H; X ¼ xH2

af; W ¼ W

af; V ¼ vH

af;

U ¼ uHaf; h ¼ T � TL

TH � TLð11Þ

By using the dimensionless parameters the equations are writ-ten as

@

@xX@W@y

� �� @

@yX@W@x

� �

¼ Pr

ð1�uÞ0:25 ð1�uÞ þu qsqf

� �24

35 @2X

@x2 þ@2X@y2

!

þ GrPr2 1ð1�uÞ

uqfqsþ 1

bs

bfþ 1

uð1�uÞ

qfqsþ 1

" #@T@x

cos /� @T@y

sin /

� �ð12Þ


@

@xh@W@y

� �� @

@yh@W@x

� �¼ @

@xk@h@x

� �þ @

@yk@h@y

� �ð13Þ

@2W@x2 þ

@2W@y2 ¼ �X ð14Þ

k ¼knfkf

ð1�uÞ þuqcpð Þsqcpð Þf

ð15Þ

Grashof and Prandtl numbers are given in Eq. 16, respectively.

Gr ¼ gbH3ðTH � TLÞm2 ; Pr ¼ m

að16Þ

Rayleigh number is

Ra ¼ gbH3ðTH � TLÞma

ð17Þ

The dimensionless radial and tangential velocities are given as,respectively:

U ¼ @W@y

ð18Þ

V ¼ � @W@x

ð19Þ

The dimensionless boundary conditions are written as

1-On the left wall i:e:; x ¼ 0; W ¼ 0; X ¼ � @2W@x2 ; h ¼ 1

2-On the right wall i:e:; x ¼ 1; W ¼ 0; X ¼ � @2W@x2 ; h ¼ 0

3-On the top and bottom walls : W ¼ 0; X ¼ � @2W@y2 ;

@h@y ¼ 0

9>>=>>;ð20Þ

4. Numerical analysis

Equations (12)–(14) with corresponding boundary conditionsgiven in Eq. 20, are solved using the finite-volume approach (Pat-ankar, 1980; Versteeg and Malalasekera, 1995). The diffusion termin the vorticity and energy equations is approximated by a second-order central difference scheme which gives a stable solution. Fur-thermore, a second-order upwind differencing scheme is adoptedfor the convective terms. The algebraic finite-volume equationsfor the vorticity and energy equations are written into the follow-ing form:

aP/P ¼ aE/E þ aW/W þ aN/N þ aS/S þ b ð21Þ

where P, W, E, N, S denote cell location, west face of the control vol-ume, east face of the control volume, north face of the control vol-ume and south face of the control volume respectively. Similarexpression is also used for the kinematics equation where only cen-tral difference is used for the discritization at the cell P of the con-trol volume. The resulted algebraic equations are solved usingsuccessive over/under relaxation method. Successive under relaxa-tion was used due to the non-linear nature of the governing equa-tions especially for the vorticity equation at high Rayleighnumbers. The convergence criterion is defined by the followingexpression:

e ¼Pj¼M

j¼1

Pi¼Ni¼1 j/

nþ1 � /njPj¼Mj¼1

Pi¼Ni¼1 j/

nþ1j< 10�6 ð22Þ

where e is the tolerance; M and N are the number of grid points inthe x and y directions, respectively.

An accurate representation of vorticity at the surface is the mostcritical step in the stream function-vorticity formulation. A second-


01

2345

678

910

1.E+03 1.E+04 1.E+05 1.E+06Ra

Nu

Present Work

Khanafer et al. (2003)

Barakos and Mitsoulis (1994)

Markatos and Pericleous (1984)

De Vahl Davis (1962)

Fusegi et al. (1991)

a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1x

θ

Present Work

Krane and Jesse (1983)Khanafer et al. (2003)

b

Fig. 2. (a) Nusselt number versus Ra number and comparison with other publishedworks, and (b) comparison between present work and other published data for thetemperature distribution on the left wall (Ra = 105, Pr = 0.7, u = 0�).


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order accurate formula is used for the vorticity boundary condi-tion. For example, the vorticity at the bottom wall is expressed as

X ¼ �ð8W1;j �W2;jÞ2ðDyÞ2

ð23Þ

Algebraic equations are solved using successive under/over relaxa-tion method. After solving W, X, and T, further useful quantitiesare obtained. For example, the Nusselt number can be expressedas

Nu ¼ hHkf

ð24Þ

The heat transfer coefficient is expressed as

h ¼ qw

TH � TLð25Þ

The thermal conductivity is expressed as

knf ¼ �qw

@T=@xð26Þ

By substituting Eqs. (25) and (26) and Eq. (7) into Eq. (24), andusing the dimensionless quantities, the Nusselt number on the leftwall is written as

Nu ¼ � knf

kf

� �@T@x

ð27Þ

The average Nusselt number is defined as

Nuavg ¼Z 1

0NuðyÞdy ð28Þ

A 1/3rd Simpson’s rule of integration is used to evaluate Eq.(28).

5. Grid testing and code validation

An extensive mesh testing procedure was conducted to guaran-tee a grid independent solution. Seven different mesh combina-tions were used for the case of Ra = 105, Pr = 0.7, and zeroinclination angle. The present code was tested for grid indepen-dence by calculating the average Nusselt number on the left wall.It is found that a grid size of 51 � 51 ensures a grid independentsolution. The converged value (Nu = 4.644) was compared to otherknown values reported by other researchers as shown in Fig. 2a.Therefore, the converged value compares very well with other val-ues obtained in literature.

The present numerical solution is further validated by compar-ing the present code results for Ra = 105 and Pr = 0.70 and zeroinclination angle against the experiment of Krane and Jessee(1983) and numerical simulation of Khanafer et al. (2003). It isclear that the present code is in good agreement with other workreported in literature as shown in Fig. 2b.

6. Results and discussion

A numerical analysis has been conducted to investigate the ef-fects of inclination angle on a Cu–water nanofluid filling a two-dimensional inclined square enclosure. The enclosure is differen-tially heated and the other two opposite walls are adiabatic. Calcu-lations were made for various values of volume fraction ofnanoparticle (0 6 u 6 0.1), inclination angles (0� 6 6 120�) andRayleigh numbers (103

6 Ra 6 105). The thermophysical propertiesof pure fluid and copper nanoparticles are listed in Table 1.

Fig. 3a–c shows the effects of Rayleigh number on streamlines(on the left) and isotherms (on the right) for / = 0.1. The figure pre-sents a comparison for flow field and temperature distribution of


nanofluid (smooth lines) and pure fluid (dashed lines). Also, thisfigure is presented to compare the obtained result with literature(Khanafer et al., 2003). A singular cell is formed at low Rayleighnumber and the shape of the main cell is circular. As shown inthe figure, the shape of the cell becomes ellipsoidal with theincreasing of Rayleigh number. The addition of nanoparticle affectsthe diameter of the cell. In all cases, the flow rotates in clockwisedirection. Moreover, Fig. 3 illustrates how the thickness of thermalboundary layer next to the heated wall is influenced by the addi-tion of nanoparticles. This sensitivity of thermal boundary layerthickness to volume fraction of nanoparticles is related to the in-creased thermal conductivity of the nanofluid. In fact, higher val-ues of thermal conductivity are accompanied by higher values ofthermal diffusivity. The high value of thermal diffusivity cause adrop in the temperature gradients and accordingly increases theboundary thickness as demonstrated in Fig. 3b. This increase inthermal boundary layer thickness reduces the Nusselt number,however, according to Eq. (27), the Nusselt number is a multiplica-tion of temperature gradient and the thermal conductivity ratio(conductivity of the of the nanofluid to the conductivity of the basefluid). Since the reduction in temperature gradient due to the pres-ence of nanoparticles is much smaller than thermal conductivityratio therefore an enhancement in Nusselt is taking taken placeby increasing the volume fraction of nanoparticles. The enhance-ment in Nusselt number will be addressed later in detail in this dis-cussion section.

Fig. 4a–d illustrates the streamlines and isotherms at differentinclination angles for Ra = 105 and / = 0.1. Again, this figure makes


-0.18

-0.27

-0.45-0.63

-0.81

-0.90

0.94

0.81

0.69

0.56

0.44

0.31

0.19

0.06

a

-0.42-0.85

-1.70-2.55-3.39

-4.24-4.67-5.09

0.940.81

0.69

0.56

0.44

0.31

0.19 0.06

b

-10.95

-10.03-9.12-8.21-6.38-5.47-3.65-1.82-0.91

0.94

0.81

0.69

0.56

0.44

0.31

0.19

0.06

c

Fig. 3. Streamlines (left) and isotherms (right) for u = 0� and / = 0.1, (a) Ra = 103 (wmin = �1.156), (b) Ra = 104 (wmin = �5.228), and (c) Ra = 105 (wmin = �11.22), (pure fluid(dashed lines —), lines (nanofluids)).


ARTICLE IN PRESS

a comparison between Cu–water nanofluid and pure fluid. It is ob-served that the shape of the main cell is sensitive to the inclinationangle and addition of nanoparticles. Also, isotherms indicate theaddition of nanoparticle becomes more effective for = 90�. In thiscase, Rayleigh–Bénard type flow is formed inside the enclosure be-cause of the existence of a vertical thermal gradient in the cavity.The appearance of the plume is believed to be the reason for themore enhancement in heat transfer for = 90�. More discussion onthis behavior will be addressed later when studying Fig. 8. Rotationdirection of flow completely changes for = 120� from clockwise tocounterclockwise.

Effects of volume fraction on flow fields and temperature distri-bution are displayed in Fig. 5a and b for Ra = 105 and u = 30�. As


indicated in the literature (Khanafer et al., 2003), the velocity com-ponents of nanofluid increase as a result of an increase in the en-ergy transport in the fluid with the increasing of volume fraction.Thus, the dimension of main cell becomes smaller with increasingof volume fraction of nanofluid. The absolute values of streamfunc-tions indicate that the strength of flow increases with increasing ofvolume fraction of nanofluid.

Fig. 6a–e presents the variation of Nusselt number against Ray-leigh number for different inclination angles. Three different parti-cle concentrations are used as u = 0, 0.05, and 0.1. The figure showsthat the effect of nanoparticles concentration on Nusselt number ismore pronounced at low Rayleigh number than at high Rayleighnumber. This is related to the conduction dominated mechanism


-16.91-15.50-14.09

-12.68-11.27

-8.46-7.05-4.23-4.23

-21

0.81 0.690.56

0.44

0.31

0.190.06

0.94

a

-23.59

-21.62-17.69-15.72-13.76-9.83-7.86-3.93

-29.48

0.38

0.44

0.50

0.560.63

0.75

0.88

0.250.13

b

-11.50

13.94

13.94

11.63

9.327.004.69

-2.25

-6.88-9.19-9.19

16.26

4.69

0.53

0.47

0.470.35

0.820.71

0.65

0.59

0.47

c

23.59

21.62

17.69

13.76

7.86

3.93

0.310.440.50

0.56

0.63 0.69

0.75

0.81

0.25

0.19

d

Fig. 4. Streamlines (on the left) and isotherms (on the right) for Ra = 105 and / = 0.1, (a) u = 30�, (wmin = �17.42 (pure fluid)), wmin = � 22.55 (nanofluid)), (b) u = 60�,(wmin = �24.08 (pure fluid), wmin = � 31.44 (nanofluid)), (c) u = 90�, (wmin = �16.13 (nanofluid)), (wmax = 20.88 (nanofluid)), (d) u = 120�, (wmax = 26 (purefluid), wmax = 31.44(nanofluid)), (pure fluid (dashed lines —), lines (nanofluids)).


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Please cite this article in press as: Abu-Nada, E., Oztop, H.F. Effects of inclination angle on natural convection in enclosures filled ... Int. J.Heat Fluid Flow (2009), doi:10.1016/j.ijheatfluidflow.2009.02.001

-16.34-15.25

-13.07-10.89-8.71-6.54-4.36

-1.09

0.750.69

0.56

0.50

0.440.380.31

0.25 0.19 0.

13

0.88 0.81

a

-17.34-16.19-15.03-12.72-10.41-8.09-5.78-3.47-1.16

0.88

0.750.63 0.56

0.50

0.44

0.380.31 0.25

0.19 0.13

b

Fig. 5. Streamlines (on the left) and isotherms (on the right) for Ra = 105 and u = 30�, (a) = 0, (wmin = �17.42), b) = 0.05, (wmin = �18.49).


ARTICLE IN PRESS

for heat transfer at low Rayleigh number compared to convectionmechanism at higher Rayleigh number. Therefore, the effect ofhigh conductive nanoparticles on heat transfer becomes more sig-nificant at low Rayleigh number. However, the buoyancy forces in-crease and they overcome the viscous forces and the heat transferis dominated by convection at high Rayleigh number. Also, it isinteresting to note that at high nanoparticles concentration theBrinkman model of viscosity makes the fluid more viscous whichexplains the relatively smaller difference between the Nusseltnumber for the cases of Ra = 1 � 104 and Ra = 1 � 105 comparedto the difference between Ra = 1 � 104 and Ra = 1 � 103.

The inclination angle affects the heat transfer. Globally, heattransfer decreases with inclination angle and the lowest heattransfer is formed for = 90�. Fig. 7 shows the variation of meanNusselt number with Rayleigh number for different inclination an-gles. Heat transfer is increased with increasing of Rayleigh numbermonotonically. It is demonstrated that the case of inclination angleof 90� has the lowest Nusselt number due to the low velocityencountered at this angle; see Fig. 8, where the vertical velocityis plotted at the middle of the heated wall (y = 0.5) for = 0.1 andRa = 105. Also, Fig. 7 shows the maximum Nusselt number occursfor the case of inclination angle of 30� due to the increase of flowvelocity at this angle as shown in Fig. 8. Also, Fig. 7 shows that con-duction regime prevailed for Ra = 1000 and then the mean Nusseltnumber started to deviate from the value of Nu ffi 1.

Fig. 9 is presented to show the variation of the local Nusseltnumber along the heated wall for the case of inclination angle of90�. Fig. 9 shows that the minimum value of Nusselt number is lo-cated around the centerline of the plume. The temperature gradi-ent at the heated wall is a minimum in the plume region which


leads to minimum value of Nusselt number in the plume region.The temperature gradient is a minimum in the plume region be-cause the thermal boundary layer thickness in the plume regionbecomes very large. This is accompanied by a reduction in temper-ature gradients. Also, Fig. 9 shows that the addition of nanoparti-cles causes the Nusselt number to increase.

Fig. 10 illustrates the variation of local Nusselt number alongthe heated wall at different inclination angle for = 0.1. As seenfrom the figure, local Nusselt number decreases along the lowerhalf of the hot wall for the values of inclination angle from = 0�to = 60�. However, the Nusselt number enhances on the top por-tion of the wall. A wavy trend is observed for = 90� due to forma-tion of Rayleigh–Bénard type flow inside the enclosure. Thus, twomaximum points were formed at this angle but their values arelower than that of other values of inclination angles due to the re-duced value of velocity as shown in Fig. 8.

The previous discussions indicate that generally heat transferenhances with addition of nanoparticle. To estimate the enhance-ment of heat transfer between the case of u = 0.1 and the pure fluid(base fluid) case, the enhancement is defined as

E ¼ Nu ðu ¼ 0:1Þ � Nu ðbasefluidÞNu ðbasefluidÞ � 100% ð29Þ

Enhancement of heat transfer (u = 0.1) is plotted versus inclinationangle at different Rayleigh numbers in Fig. 11. For the whole rangeof Rayleigh numbers, the figure illustrates that enhancement of heattransfer is almost constant up to = 60�. Also, the percentage of heattransfer enhancement decreases with increasing of Rayleigh num-ber for all inclination angles. It is an interesting observation that


1

10N

u

ϕ=0ϕ=0.05ϕ=0.1

a b

c d

e

1

10

1,E+03 1,E+04 1,E+05Ra

1,E+03 1,E+04 1,E+05Ra

1,E+03 1,E+04 1,E+05

Ra

1,E+03 1,E+04 1,E+05

Ra

Nu

ϕ=0ϕ=0.05ϕ=0.1

1

10

Nu

ϕ=0ϕ=0.05ϕ=0.1

1

10

Nu

ϕ=0ϕ=0.05ϕ=0.1

1

10

50+E,140+E,130+E,1Ra

Nu

ϕ=0ϕ=0.05ϕ=0.1

Fig. 6. Variation of mean Nusselt numbers with Rayleigh numbers at different values of volume fraction for different inclination angles (a) = 0�, (b) = 30�, (c) = 60�, (d) = 90�,(e) = 120�.

0

1

2

3

4

5

6

7

0 20 40 60 80 100 120

φ

Nu

Ra = 1000Ra = 10000Ra = 100000

Fig. 7. Variation of mean Nusselt numbers with inclination angles for u = 0.1.

-150

-100

-50

0

50

100

150

0 0.2 0.4 0.6 0.8 1

X

V

φ=0

φ=30

φ=60

φ=90

φ=120

Fig. 8. Velocity profiles along the middle of hot wall for different inclination anglefor u = 0.1 and Ra = 105.


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Please cite this article in press as: Abu-Nada, E., Oztop, H.F. Effects of inclination angle on natural convection in enclosures filled ... Int. J.Heat Fluid Flow (2009), doi:10.1016/j.ijheatfluidflow.2009.02.001

0

2

4

6

8

10

12

Nu

φ=0φ=30φ=60φ=90φ=120

0 0.2 0.4 0.6 0.8 1Y

Fig. 10. Variation of local Nusselt number along the heated wall for = 0.1 atdifferent inclination angles.

Fig. 11. Ratio of enhancement of heat transfer due to addition of nanoparticles.

0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1Y

Nuy

ϕ=0.0ϕ=0.1

Fig. 9. Variation of Nusselt number along the heated wall for / ¼ 90� .


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the enhancement in heat transfer for Ra = 1 � 104 is the same for allinclination angles. Also, for Ra = 105, the enhancement in heattransfer is reduced by increasing the inclination angle.


7. Conclusions

In this paper the influence of inclination angle and volume frac-tion of nanoparticle have been studied for a square enclosure. Cop-per–water was used as nanofluid. Results have clearly indicatedthat the addition of copper nanoparticles has produced a remark-able enhancement on heat transfer with respect to that of the purefluid. Heat transfer enhances with increasing of Rayleigh numberalmost linearly but the effect of nanoparticles concentration onNusselt number is more pronounced at low Rayleigh number thanat high Rayleigh number. Inclination angle of the enclosure is pro-posed a control parameter for fluid flow and heat transfer. It wasfound that lower heat transfer is formed for = 90�. But higher val-ues of volume fraction become insignificant from the fluid flowpoint of view at this inclination angle. Effects of inclination angleon percentage of heat transfer enhancement become insignificantat low Rayleigh number but it decreases the enhancement of heattransfer with nanofluid. Finally, the inclination angle is a good con-trol parameter for both pure and nanofluid filled enclosures.

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