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International Journal of Thermal Sciences 50 (2011) 2141e2153

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

Free convection in a triangle cavity filled with a porous medium saturated withnanofluids with flush mounted heater on the wall

Qiang Sun a,*, Ioan Pop b,*

aMechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, SingaporebUniversity of Cluj, Faculty of Mathematics, R-400082 Cluj-Napoca, CP 253, Romania

a r t i c l e i n f o

Article history:Received 15 January 2011Received in revised form31 May 2011Accepted 4 June 2011Available online 7 July 2011

Keywords:Porous mediaNanofluidFree convectionTriangular enclosure

* Corresponding authors.E-mail addresses: mpesq@nus.edu.sg (Q. Sun), pop

1290-0729/$ e see front matter � 2011 Elsevier Masdoi:10.1016/j.ijthermalsci.2011.06.005

a b s t r a c t

Steady-state free convection heat transfer behavior of nanofluids is investigated numerically insidea right-angle triangular enclosure filled with a porous medium. The flush mounted heater with finite sizeis placed on the left vertical wall. The temperature of the inclined wall is lower than the heater, and therest of walls are adiabatic. The governing equations are obtained based on the Darcy’s law and thenanofluid model proposed by Tiwari and Das [1]. The transformed dimensionless governing equationswere solved by finite difference method and solution for algebraic equations was obtained throughSuccessive Under Relaxation method. Investigations with three types of nanofluids were made fordifferent values of Rayleigh number Ra of a porous medium with the range of 10 � Ra � 1000, size ofheater Ht as 0.1 � Ht � 0.9, position of heater Yp when 0.25 � Yp � 0.75, enclosure aspect ratio AR as0.5 � AR � 1.5 and solid volume fraction parameter f of nanofluids with the range of 0.0 � f � 0.2. It isfound that the maximum value of average Nusselt number is obtained by decreasing the enclosure aspectratio and lowering the heater position with the highest value of Rayleigh number and the largest size ofheater. It is further observed that the heat transfer in the cavity is improved with the increasing of solidvolume fraction parameter of nanofluids at low Rayleigh number, but opposite effects appear when theRayleigh number is high.

� 2011 Elsevier Masson SAS. All rights reserved.

1. Introduction

Heat and fluid flow in cavities filled with porous media are well-known natural phenomenon and have attracted interest of manyresearchers due to its many practical situations. Among theseinsulation materials, geophysics applications, building heating andcooling operations, underground heat pump systems, solar engi-neering and material science can be listed. These are reviewed inseveral books: Pop and Ingham [2], Bejan et al. [3], Ingham and Pop[4], Nield and Bejan [5], Vafai [6,7], Vadasz [8] and in the papers byVarol et al. [9], Varol et al. [10] and Basak et al. [11,12].

A technique for improving heat transfer is using solid particlesin the base fluids, which has been used recently. The term nano-fluid, first introduced by Choi [13], refers to fluids in which nano-scale particles are suspended in the base fluid. He suggested thatintroducing nanoparticles with higher thermal conductivity intothe base fluid results in a higher thermal performance for theresultant nanofluid. It is expected that the presence of the

m.ioan@yahoo.co.uk (I. Pop).

son SAS. All rights reserved.

nanoparticles in the nanofluid increases its thermal conductivityand therefore, substantially enhances the heat transfer character-istics of the nanofluid [13]. Use of metallic nanoparticles with highthermal conductivity will increase the effective thermal conduc-tivity of these types of fluid remarkably. However, the increase inthe thermal conductivity depends on the shape, size and thermalproperties of the solid particles [14]. It must, however, be noted thatheat transfer enhancement by means of nanofluids is stilla controversial issue. Contradictory studies have also been reportedin the literature, which argue that the dispersion of nanoparticles inthe base fluid may results in a considerable decrease in the heattransfer [15,16]. It has been demonstrated that the augmentation ormitigation of the heat transfer found in the numerical studiesdepends on the existing models used to predict the properties ofthe nanofluids [17,18]. Buongiorno [19] noted that the nanoparticleabsolute velocity can be viewed as the sum of the base fluid velocityand a relative velocity (that he calls the slip velocity). He has shownthat in the absence of turbulent effects, it is the Brownian diffusionand the thermophoresis that are important and he has writtendown conservation equations based on these two effects. There areseveral numerical and experimental studies on the forced andnatural convection using nanofluids related with differentially

Nomenclature

AR enclosure aspect ration W/HCp specific heat at constant pressure (J kg�1 K�1)g gravitational acceleration (m s�2)h size of heater (m)H height of enclosure (m)Ht dimensionless size of heaterk thermal conductivity (W m�1 K�1)K permeability of porous medium (m2)NuY local Nusselt numberNu average Nusselt numberRa Rayleigh number for porous mediumT dimensional fluid temperature (K)u, v dimensional components of velocity (m s�1)U, V dimensionless components of velocityW width of enclosure (m)x, y dimensional coordinates

X, Y dimensionless coordinatesyp dimensional location of heater center (m)Yp dimensionless location of heater center

Greek symbolsa fluid thermal diffusivity (m2 s�1)b thermal expansion coefficient (K�1)f solid volume fraction of nanofluidQ dimensionless fluid temperatureJ dimensionless flow stream functionr density (kg m�3)m dynamic viscosity (kg m�1 s�1)

Subscriptsnf nanofluidf fluids solid

Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e21532142

heated enclosures and we mention here those by Khanafer et al.[20], Maïga et al. [21], Tannehill et al. [22], Oztop and Abu-Nada[23], Muthtamilselvan et al. [24], Ghasemi and Aminossadati[25,26], Popa et al. [27], Mahmoudi et al. [28,29], etc. The book byDas et al. [30] and the review papers by Daungthongsuk andWongwises [31], Ding et al. [32], Wang and Mujumdar [33,34], andKakaç and Pramuanjaroenkij [35] present excellent collection of upto now published papers on nanofluids.

It is obvious from the foregoing review that most of the studiesare performed considering the water-based nanofluids in cavities.Very little research is performed considering a porous mediumfilled with nanofluids. Recently, Nield and Kuznetsov [36] havestudied the Cheng and Minkowycz’s problem [37] for naturalconvective boundary layer flow over a vertical flat plate embeddedin a porous medium filled with nanofluid taking into account thecombined effects of heat and mass transfer in the presence ofBrownian motion and thermophoresis as proposed by Buongiorno[19]. In another paper, Kuznetsov and Nield [38] have providednumerical solution to the problem of natural convective heattransfer in the boundary layer flow of a nanofluid past a vertical flatplate embedded in a viscous (Newtonian) fluid using the same

Fig. 1. Sketch of the physical model.

Buongiorno’s model [19]. Also, Khan and Pop [39], and Bachok et al.[40] have studied the steady boundary layer flow of a nanofluidpast a stretching surface using Buongiorno’s nanofluid model [19],while Ahmad and Pop [41] have considered the steady mixedconvection boundary layer flow over a vertical flat plate embeddedin a porous medium saturated with a nanofluid using the nanofluidmodel proposed by Tiwari and Das [1]. However, Buongiorno [19]noted that the nanoparticle absolute velocity can be viewed asthe sum of the base fluid velocity and a relative velocity (that hecalls the slip velocity). He considered in turn seven slip mecha-nisms: inertia, Brownian diffusion, thermophoresis, diffusiopho-resis, Magnus effect, fluid drainage, and gravity settling.

In the present study, the problem of steady free convection heattransfer in a triangular enclosure filled with a nanofluid, where theenclosure, with a heater on its vertical wall and filled with a waterCu nanofluid considered by Ghasemi and Aminossadati [25] hasbeen extended to a triangular cavity filled with a porous mediumand saturated by nanofluids using the nanofluidmodel proposed byTiwari and Das [1]. Three different types of nanoparticles areconsidered, namely Cu, Al2O3 and TiO2. The present study has beenmotivated by the need to determine the detailed flow andtemperature characteristics as well al the local and average Nusseltnumbers. To the best knowledge of the authors, no study whichconsiders this problem has yet been reported in the literature. Assuch, the focus of this paper is to examine the effects of pertinentparameters such as Rayleigh number for a porous medium, solidvolume fraction parameter of nanofluids, heater size and position,and enclosure aspect ratio.

2. Physical model and basic governing equations

The physical domain for the free convection in a triangle cavityis sketched in Fig. 1 with dimensions and boundary conditions

Table 1Thermalephysical properties of fluid and nanoparticles [23].

Physical properties Fluid phase (water) Cu Al2O3 TiO2

Cp (J kg�1 K�1) 4179 385 765 686.2r (kg m�3) 997.1 8933 3970 4250k (W m�1 K�1) 0.613 400 40 8.9538a � 10�7 (m2 s�1) 1.47 1163.1 131.7 30.7b � 10�5 (K�1) 21 1.67 0.85 0.9

number of grid points

Nu Y

Y=

0.5

31 51 71 91 111 131 151 1718

8.5

9

9.5

10

10.5

|

Fig. 2. Dependence of the local Nusselt number NuY at Yp ¼ 0.5 on the number of gridpoints for Cuewater nanofluid when Ra¼ 1000, f ¼ 0.1, AR ¼ 1.0, Ht ¼ 0.8 and Yp ¼ 0.5.

Table 2Comparison of average Nusselt number Nu for f¼ 0.0 (purefluidewater) when AR ¼ 1.0 and Ra ¼ 1 � 103.

Literature Nu

Bejan [43] 15.800Goyeau et al. [44] 13.470Baytas and Pop [45] 14.060Varol et al. [9,10] 13.564Present study 13.575

Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e2153 2143

according to different positions of the heater. In this model, heatersize can be changed, which is denoted by h. The position of theheater is expressed by yp, which is measured from the middle pointof the heater to the bottom wall of the cavity. For the enclosure

Fig. 3. (a, c) Local NuY profile for Cuewater nanofluid; (b, d) Dependence of average NusseltYp ¼ 0.5.

(cavity), length of the bottom wall and height of the vertical wallare shown by W and H, respectively. Inclined wall of the cavity hasconstant cold temperature Tcold, while the heater has constant hottemperature Thot, and the remained walls are adiabatic.

The fluid within the cavity is a water-based nanofluid. Threedifferent types of nanoparticles are considered, namely Cu, Al2O3and TiO2, which thermalephysical properties are listed in Table 1. Inthis study, the nanofluid flow is set to be incompressible andlaminar. It is presumed that the base fluid (i.e. water) and thenanoparticles are in thermal equilibrium and no slip occursbetween them. Meanwhile, the Boussinesq approximation isemployed and homogeneity and local thermal equilibrium in theporous medium is assumed. It is also assumed that nanoparticlesare suspended in the nanofluid using either surfactant or surfacecharge technology. As a result, in keeping with the Darcy’s law andadopting the nanofluid model proposed by Tiwari and Das [1], thebasic continuity, momentum, and energy equations can bewritten as

numbers on Rayleigh number Ra for different nanoparticles when f ¼ 0.1, AR ¼ 1.0 and

Fig. 4. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when f ¼ 0.1, Ht ¼ 0.8, AR ¼ 1.0 and Yp ¼ 0.5.

Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e21532144

vuvx

þ vv

vy¼ 0; (1)

mnfK

u ¼ �vpvx

; (2)

mnfK

v ¼ �vpvy

þhfrsbs þ ð1� fÞrfbf

igðT � TcoldÞ; (3)

and

uvTvx

þ vvTvy

¼ anf

v2Tvx2

þ v2Tvy2

!: (4)

In Eqs. (1)e(4), x and y are Cartesian coordinates measuredalong the horizontal and vertical walls of the cavity respectively, uand v are the velocity components along the x- and y- axesrespectively, T is the fluid temperature, p is the fluid pressure, g isthe gravity acceleration, K is the permeability of the porousmedium, f is the solid volume fraction of the nanofluid, bf and bsare the coefficients of thermal expansion of the fluid and of thesolid fractions respectively, rf and rs are the densities of the fluidand of the solid fractions respectively, mf is the viscosity of the fluidfraction, mnf is the viscosity of the nanofluid, and anf is the thermaldiffusivity of the nanofluid. The flow is assumed to be slow so that

an advective term and a Forchheimer quadratic drag term do notappear in the Darcy’s Eqs. (2) and (3). The viscosity of the nanofluidmnf can be approximated as viscosity of a base fluid if containingdilute suspension of fine spherical particles, which is given byBrinkman [42] as

mnf ¼ mf

ð1� fÞ2:5: (5)

Meanwhile, the thermal diffusivity of the nanofluid is defined byOztop and Abu-Nada [23] as

anf ¼ knf�rCp�nf; (6)

where ðrCpÞnf is the heat capacity of the nanofluid, which is givenby Khanafer et al. [20] as�rCp�nf ¼ ð1� fÞ�rCp�fþf

�rCp�s; (7)

and knf stands for the effective thermal conductivity of the nano-fluid that can be obtained according to the Maxwell-Garnettsmodel

knfkf

¼�ks þ 2kf

�� 2f

�kf � ks

��ks þ 2kf

�þ f

�kf � ks

� : (8)

φ

Nu

0 0.05 0.1 0.15 0.22

2.3

2.6

2.9

3.2

3.5a

c d

b

Ra = 50

φ

Nu

0 0.05 0.1 0.15 0.22.6

2.8

3

3.2

3.4

3.6

3.8

Ra = 100

φ

Nu

0 0.05 0.1 0.15 0.25.5

5.9

6.3

6.7

7.1

7.5

7.9

Ra = 500

φ

Nu

0 0.05 0.1 0.15 0.28

8.5

9

9.5

10

10.5

11

11.5

12

Ra = 1000

Fig. 5. Dependence of average Nusselt numbers on solid volume fraction parameter f for different nanoparticles when Ht ¼ 0.4, AR ¼ 1.0 and Yp ¼ 0.5.

Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e2153 2145

In the above two equations, kf and ks are the thermal conduc-tivities of the fluid and the solid fractions respectively, ðrCpÞf standsthe heat capacity of the fluid fraction, and ðrCpÞs is the heat capacityof the solid fraction respectively.

The following dimensionless variables

X ¼ xH; Y ¼ y

H; U ¼ H

afu; V ¼ H

afv; Q¼ T�Tcold

Thot�Tcold; (9)

are introduced, and the dimensionless stream functionJ is definedin the usual way as

U ¼ vJ=vY ; V ¼ �vJ=vX: (10)

When Eqs. (9) and (10) are substituted into Eqs. (2)e(4), and thepressure terms are eliminated from Eqs. (2) and (3), it is found that

1

ð1�fÞ2:5 v2J

vX2 þv2J

vY2

!¼�Ra

hð1�fÞþf

�rs=rf

��bs=bf

�i vQvX

;

(11)

vJ

vYvQ

vX� vJ

vXvQ

vY¼ anf

af

v2Q

vX2 þ v2Q

vY2

!; (12)

where Ra is the Rayleigh number for a porous medium, which isdefined as Ra ¼ gKrfbf ðThot � TcoldÞH=ðmfaf Þ. Due to the viscosity

of the fluid and the impermeability of the cavity walls, non-slipcondition is composed along the whole boundary of the calcula-tion domain in this model. Consequently, the correspondingdimensionless boundary conditions are converted as8>><>>:Alongtheverticalwall : onheater;Q¼ 1;J¼ 0:Alongtheverticalwall : ontheunheatedpart;vQ=vX¼ 0;J¼ 0:OntheadiabaticðbottomÞwall;vQ=vY ¼ 0;J¼ 0:Ontheinclinedwall;Q¼ 0;J¼ 0:

(13)In the meantime, the length and position of the heater arerespectively nondimensionalized as

Ht ¼ hH

and Yp ¼ ypH: (14)

The definitions for local Nusselt number NuY on the left verticalwall and average Nusselt number Nu are

NuY ¼ �knfkf

vQ

vX

����X¼0

; (15)

and

Nu ¼Z10

NuYdY ; (16)

respectively.

φ

Nu

0 0.05 0.1 0.15 0.23.5

4

4.5

5

5.5

6

6.5a

c d

b

Ra = 50

φ

Nu

0 0.05 0.1 0.15 0.2

4.8

5.1

5.4

5.7

6

6.3

Ra = 100

φ

Nu

0 0.05 0.1 0.15 0.2

8.1

8.4

8.7

9

9.3

9.6

9.9

Ra = 500

φ

Nu

0 0.05 0.1 0.15 0.211

11.5

12

12.5

13

13.5

14

Ra = 1000

Fig. 6. Dependence of average Nusselt numbers on solid volume fraction parameter f for different nanoparticles when Ht ¼ 0.8, AR ¼ 1.0 and Yp ¼ 0.5.

Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e21532146

3. Numerical procedure

Finite difference method [22] was adopted to solve numericallythe governing Eqs. (11) and (12) together with the boundaryconditions in Eq. (13). The diffusion terms within Eqs. (11) and (12)were replaced by the second order central differencing schemes,while the second order upwind differencing scheme was chosen toapproximate the convective term in order to make the numericalprocedure stable. The solution for the corresponding linear alge-braic equations was obtained through the Successive UnderRelaxation (SUR) method. The temperature function Q and thestream function J were calculated through iteration when theinitial guess was made. The iteration process is terminated whenthe following criterion is satisfied

PMi¼1

PNj¼1

���cnþ1i;j � cni;j

���PMi¼1

PNj¼1

���cnþ1i;j

��� � 10�6; n � 1; (17)

in which c reflects either Q or J, M and N are the number of gridpoints in the X- and Y- directions respectively, and n is the iterationstep.

In this study, the grid points with uniform spaced mesh weregenerated along both X- and Y- directions. The number of the grid

points were examined from 31 � 31 to 171 �171. As can been seenin Fig. 2, the calculation results become mesh independent whenthe number of grid points are higher than 91 � 91. Meanwhile, inorder to verify the rationality and accuracy of the results, the abovenumerical simulation procedure was tested with the classicalnatural convection heat transfer problem in a differentially heatedsquare porous enclosure (AR ¼ 1.0) in context with the pure fluid(i.e. water), which is equivalent with f ¼ 0.0 in the presentedmodel. The obtained results for the average Nusselt number, asdefined in Eq. (16), were compared with those given by differentauthors, as listed in Table 2. It can be seen that the results obtainedhere are in good agreement with the results that have beenreported.

4. Results and analysis

Numerical simulation results on flow field, temperaturedistribution, and Nusselt number are presented in this section.The effects of different values of Rayleigh number Ra of a porousmedium, solid volume fraction parameter f of nanofluid, types ofthe nanoparticles, heater size Ht, heater position Yp and enclosureaspect ratio AR are analyzed. Enclosure aspect ratio AR changesfrom 0.5 to 1.5. Dimensionless size of the heater Ht varies from

Fig. 7. Dependence of average Nusselt numbers on aspect ratio AR for different nanoparticles when f ¼ 0.1 and Yp ¼ 0.5.

Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e2153 2147

0.1 to 0.9, and dimensionless heater position Yp varies from 0.25to 0.75. We assume, as in Oztop and Abu-Nada [23], that therange for solid volume fraction parameter f of nanoparticles istaken from 0.0 to 0.2. We assume also that Rayleigh number Rafor porous medium varies from 10 to 1000.

Rayleigh number Ra is a very important parameter that haseffects on heat transfer within a porous medium. The results inFig. 3a and c present the distribution of local Nusselt number NuYalong the heater for different values of Ra. From Fig. 3b and d, onecan see that average Nusselt number Nu is improved when theRayleigh number is increased. This conclusion is supported by Varolet al. [9] for pure fluid. Meanwhile, Fig. 3b and d also show that theaverage Nusselt number of Cuewater nanofluid is the highestamong the three kinds of nanofluids, which is probably due to thatthe thermal conductivity of nanoparticle Cu is higher than those ofthe other two nanoparticles (Al2O3 and TiO2). Fig. 4 presents thestreamlines and isotherms for Cuewater nanofluid under differentvalues of the Rayleigh number. Fig. 4a, b and c demonstrate thata single circulation flow cell is formed in the clockwise direction forall values of Ra that have been tested. When the Rayleigh numberincreases, the flow convection is strengthened, the width of flowcell is increased as egg-shaped cell is extended to triangle-shapedcell, and the boundary layers become more significant.

Solid volume fraction parameter f is a key factor to study hownanoparticles affect the heat transfer of nanofluids. Figs. 5 and 6present the average Nusselt number for different values of theRayleigh number and the solid volume fraction parameter. Onceagain, the average Nusselt number is elevated when the Rayleigh

number increases and nanoparticle Cu is used. However, the effectsof solid volume fraction parameter f on the heat transfer ofnanofluids are complicated. When Rayleigh number Ra is low, theaverage Nusselt number increases as solid volume fractionparameter f increases (Figs. 5a and 6a, b). However, elevating f hasadverse effects on the heat transfer of nanofluids when Ra is high,as shown in Figs. 5c, d and 6d. One possible explanation for theabove phenomena is as following. The density of nanofluid isdefined by Oztop and Abu-Nadaas [23] as

rnf ¼ frs þ ð1� fÞrf : (18)

Based on the definitions in Eqs. (5) and (18), and consid-ering the physical values listed in Table 1, the density andviscosity of nanofluids are elevated relative to the pure fluid(water) as f increases, which means that the inertial andviscous resistances for nanofluids are higher than those forthe pure fluid (water). When the Rayleigh number is low, theflow convection is insignificant. The heat transfer in the cavityis dominated by conduction. Along the increase of f, thethermal conductivity of nanofluids is increased relative to thatof pure fluid (water) due to the high thermal conductivity ofnanoparticles. Consequently, the heat transfer of nanofluids isimproved compared with pure fluid (water), as depicted inFigs. 5a and 6a, b. As the Rayleigh number increases, the flowconvection becomes stronger, and the heat transfer in thecavity is improved, as displayed in Fig. 3b and d. However, asf increases, the elevated inertial and viscous resistances of

Fig. 8. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when Ra ¼ 1000, f ¼ 0.1 and Yp ¼ 0.5.

Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e21532148

nanofluids would compromise the flow convection, which isable to lead to the adverse effects on the heat transfer in thenanofluids compared with the pure fluid (water). Neverthe-less, when f is upon some level, the positive effects ofincreased thermal conductivity of nanofluids can appear againto overtake the adverse effects of the elevated inertial andviscous resistances, which results in the value of averageNusselt number Nu to rise after falling. The curves in Fig. 5band solid line in Fig. 6c reflect the above analysis. WhenRayleigh number Ra is high (Ra ¼ 1000), convection domi-nates the fluid movement. Under such a circumstance, heattransfer, for all three types of nanofluids, is decreased alongthe increasing of the solid volume fraction parameter due tothe elevated inertial and viscous resistances of the nanofluids,as shown in Figs. 5c, d and 6d. It is possible that when solidvolume fraction parameter f is above 0.2, sedimentation setsin and the fluid loses the regular (Darcian) character as it wasconcluded by Muthtamilselvan et al. [24] for a copperewaternanofluids in a lid-driven enclosure.

Enclosure aspect ratio AR is another feature that has influenceon the heat transfer within the cavity. Normally, the value of

average Nusselt number Nu is reduced along the increase of AR, asshown in Fig. 7a, b and d. This is understandable because when ARincreases, the distance between the heater and cold wall of thecavity is enlarged, which can lead to the decrease in temperaturegradients. However, the effects of AR on Nu are also affected byother physical parameters. For example, when Ra ¼ 1000, the heattransfer within the cavity with small size of heater (Ht ¼ 0.4) fallsafter rising along the increasing of AR, as shown in Fig. 7c. Never-theless, when the heater size is enlarged as Ht ¼ 0.8, the heattransfer inside the enclosure again decreases consistently as ARincreases (Fig. 7d). The comparisons for streamlines and isothermsof Cuewater nanofluid with different values of AR are presented inFigs. 8 and 9.

The heat transfer can also be affected by the position of heater.When Ra ¼ 1000, convection leads the flow movement within thecavity. Considering the shape of the enclosure and the boundaryconditions, when the position of heater is high, for exampleYp¼0.75, theflowandheat transfer cannot be fully developed insidethe entire enclosure, as shown in Fig. 12. Under this circumstance,heat transfer within the cavity is decreased along the increase of Yp,as displayed in Figs. 10b and 11b. Nevertheless, when Ra is low as

Fig. 9. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when Ra ¼ 1000, f ¼ 0.1 and Yp ¼ 0.5.

a b

Fig. 10. Dependence of average Nusselt numbers on center position of heater Yp for different nanoparticles when f ¼ 0.1, Ht ¼ 0.4, and AR ¼ 1.0.

Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e2153 2149

Fig. 11. Local Nusselt number profile of Cuewater nanofluid for different center position of heater Yp when f ¼ 0.1, Ht ¼ 0.4, and AR ¼ 1.0.

Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e21532150

Ra¼ 1000, the flowconvection is insignificant, and the heat transferinside the cavity ismainly performed by conduction, as presented inFig. 13. In such a case, considering the shape of the enclosure, whenthe position of heater is high, the distance between the heater and

Fig. 12. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater

cold wall is shortened, which leads to the improvement of the heattransfer inside the cavity, as shown in Figs. 10a and 11a.

It is easily expected that enlarging the size of heater Ht is able toimprove the heat transfer inside the enclosure. Fig. 14 reflects that

nanofluids when Ra ¼ 1000, f ¼ 0.1, Ht ¼ 0.4, and AR ¼ 1.0.

Fig. 13. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when Ra ¼ 100, f ¼ 0.1, Ht ¼ 0.4, and AR ¼ 1.0.

Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e2153 2151

expectation for three types of nanofluids. It is also can be seen thatthe difference of the average Nusselt number becomes significantfor larger size of the heater since flow convection becomes stronger.These results are consistent with the findings reported by Oztop

Fig. 14. Dependence of average Nusselt numbers on size of heater Ht

and Abu-Nada [23] for the convection in a rectangular cavity ofhomogeneous nanofluids. In the meantime, the streamlines andisotherms of Cuewater nonafluid for different values of Ht areplotted in Fig. 15.

for different nanoparticles when f ¼ 0.1, AR ¼ 1.0, and Yp ¼ 0.5.

Fig. 15. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when Ra ¼ 1000, f ¼ 0.1, AR ¼ 1.0, and Yp ¼ 0.5.

Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e21532152

5. Conclusions

A numerical study has been performed in this paper to inves-tigate the free convection heat transfer problem in a partly heatedtriangle cavity filled with a porous medium saturated with nano-fluids. The governing equations were solved by finite differencemethod. It is found that the maximum value of the average Nusseltnumber can be achieved for the highest Rayleigh number, thelargest heater size. Meanwhile, under that circumstance, loweringthe heater position and decreasing the aspect ratio of the enclosureare beneficial to the heat transfer in the cavity. Among the threetypes of nanofluids, the highest value of the average Nusseltnumber is obtained when using Cu nanoparticles. The effects of thesolid volume fraction parameter on the heat transfer of nanofluidswithin a porous medium are complex. When the Rayleigh numberis low, increasing the value of the solid volume fraction parameterof nanofluids can improve the value of the average Nusselt number,while if Rayleigh number is high, elevating the solid volume frac-tion parameter of nanofluids reduces the value of the averageNusselt number. An optimization investigation might be requiredin future to search for the best value of the solid volume fractionparameter of nanofluids to achieve the most efficient way ofimproving heat transfer within a porous medium, which is never-theless beyond the scope of this study. It is worth mentioning tothis end that the study of nanofluids is still at its early stage and it

seems that it is very difficult to have a precise idea on the way theuse of nanoparticles acts in convective flow in porous media andcomplementary work is necessary to understand the heat transfercharacteristics of nanofluids and identify newapplications for thesefluids.

References

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