Latin American Applied Research 45:165-172 (2015)

FLUID FLOW AND HEAT TRANSFER CHARACTERISTICS OF NATURAL CONVECTION IN SQUARE NANOFLUID-FILLED

ENCLOSURES SATURATED POROUS MEDIA DUE TO DISCRETE SOURCE-SINKS PAIR

M.A. MANSOUR†, S.E. AHMED‡, A.M. ALY‡ and S.-W. LEE§

† Department of Mathematics, Assuit University, Faculty of Science, Assuit, Egypt. ‡ Department of Mathematics, Faculty of Sciences, South Valley University, Qena, Egypt,

§ School of Mechanical Engineering, University of Ulsan, Ulsan, South Korea E-mail: leesw@ulsan.ac.kr

Abstract−− Steady natural convection heat trans-

fer behavior of nanofluids is investigated numerical-ly inside a square enclosure filled with a porous me-dium. The flush mounted heater with finite size is placed on the bottom wall as well as two sinks with finite size are placed on the vertical walls, whereas the rest walls are adiabatic. The governing equations are obtained based on the Darcy's law with the nanofluid model. The transformed dimensionless governing equations were solved by finite difference approach. The influence of governing parameters, namely, Rayleigh number, location and geometry of the heat source, the type of nanofluid and solid vol-ume fraction of nanoparticles on the cooling perfor-mance is studied. The present results are validated by favorable comparisons with previously published results. The results of the problem are presented in graphical and tabular forms and discussed.

Keywords−− Nanofluid, natural convection, heat source, finite difference method.

I. INTRODUCTION Heat and fluid flow in cavities filled with porous media are well known natural phenomenon and have attracted interest of many researchers to its many practical situa-tions. Among these insulation materials, geophysics ap-plications, building heating and cooling operations, un-derground heat pump systems, solar engineering and and many other applications. Indeed, the books of Pop and Ingham (2001), Bejan et al. (2004), Ingham and Pop (2005), Nield and Bejan (2006), Vafai (2005, 2010), Vadasz (2008) and the papers made by Varol et al. (2006), Mansour et al. (2011), Misirlioglu et al. (2005), Moya et al. (1987) and Baytas and Pop (1999), contributed some important theoretical results in free convection in a porous cavity. In most natural convection studies, the base fluid in the enclosure has a low thermal conductivity, which limits the heat transfer enhancement. However, the con-tinuing miniaturization of electronic devices requires further heat transfer improvements from an energy sav-ing viewpoint. A technique for improving heat transfer is using particles in the base fluids, which has been used recently. The dispersed particles in the base fluid called nanofluid are first used by Choi (1995). Nanofluids have attracted attention as a new generation of heat

transfer fluids in building heating, in heat exchangers, in plants and in automotive cooling applications, because of their excellent thermal performance. Various benefits of the application of nanofluids include: improved heat transfer, heat transfer system size reduction, minimal clogging, micro channel cooling and miniaturization of systems (Choi, 1995). Therefore, research is underway to apply nanofluids in environments where higher heat flux is en-countered and the conventional fluid is not capable of achieving the desired heat transfer. Xuan and Li (2003) have examined the transport properties of nanofluid and have expressed that thermal dispersion, which takes place due to the random movement of par-ticles, takes a major role in increasing the heat transfer rate between the fluid and the wall. This requires a thermal dispersion coefficient, which is still unknown. Brownian motion of the particles, ballistic phonon transport through the particles and nanoparticles cluster-ing can also be the possible reason for this enhancement Xuan et ai. (2005). Das et al. (2003) has observed that the thermal conductivity for nanofluid increases with increasing temperature. They have also observed the stability of Al2O3–water and Cu–water nanofluid. Ex-periments on heat transfer due to natural convection with nanofluid have been studied by Putra et al. (2003) and Wen and Ding (2006). They have observed that heat transfer decreases with increase in concentration of nanoparticles. The viscosity of this nanofluid increases rapidly with inclusion of nanoparticles as shear rate de-creases. Recently, Kumar et al. (2010) used a single phase thermal dispersion model to study the flow and thermal field in nanofluid. Talebi et al. (2010) investigated nu-merically the problem of mixed convection flows through a Cu-water nanofluid in a square lid-driven cav-ity. The problem of natural convection cooling of a lo-calized heat source at the bottom of a nanofluid-filled enclosure was discussed by Aminossadati and Ghasemi (2009). Oztop and Abu-Nada (2008) presented a nu-merical study for heat transfer and fluid flow due to buoyancy forces in a partially heated enclosure using nanofluids made with different types of nanoparticles. Putra et al. (2003) and Wen and Ding (2005) found a systematic and definite deterioration in the heat transfer for a particular range of Rayleigh numbers and density

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Latin American Applied Research 45:165-172 (2015)

and concentration of nanoparticles. Similar results were also obtained by Santra et al. (2008) who modeled the nanofluid as a non-Newtonian fluid. Ho et al. (2008) argued that the heat transfer in a square enclosure filled nanofluids can be enhanced or mitigated depending on the formulas used for the estimated dynamic viscosity of the nanofluid. Hwang et al. (2007) theoretically investi-gated thermal characteristics of natural convection in a rectangular cavity filled with a water-based nanofluid containing alumina. Abu-Nada et al. (2010) investigat-ed the heat transfer enhancement in a differentially heated enclosure using variable thermal conductivity and variable viscosity of Al2O3–water and Cu–water nanofluids. In the present study, the problem of steady natural convection heat transfer in a square enclosure filled with nanofluidhas been studied numerically using finite dif-ference method. Here, the enclosure has a heater on its horizontal wall and sink on the vertical walls and filled with a porous medium. Four different types of nanopar-ticles are considered, namely Cu, Ag, Al2O3 and TiO2.The focus of the present study is on the analysis of several pertinent parameters such as Rayleigh number for a porous medium, length and location of the heat source and solid volume fraction parameters of nanoflu-ids.

II. PROBLEM DESCRIPTION Figure 1 displays the schematic diagram of the two-dimensional enclosure considered in this study. In the current problem, the following assumptions have been made: • The heater and the sinks can be changed size, which is denoted by B. • The positions of the heater and sinks are expressed by D, which are measured from the middle point of the heater and the sinks to the bottom and vertical walls of the cavity. • The sink has constant cold temperature Tc, while the heater has constant hot temperature Th, and the re-mained walls are adiabatic. • The nanofluids used in the analysis are assumed to be Newtonian, incompressible and laminar. • The base fluid (water) and the solid spherical nano-particles (Cu, Ag, Al2O3 and TiO2) are in thermal equi-librium. • The thermo-physical properties of the base fluid and the nanoparticles are given in Table 1. • The thermo-physical properties of the nanofluid are assumed constant except for the density variation, which is determined based on the Boussinesq approximation.

III. MATHEMATICAL FORMUALTION The governing equations are obtained based on the Dar-cy's law and the nanofluid model proposed by Tiwari and Das (2007): 0=

∂∂

+∂∂

yv

xu (1)

Adiabatic

Fig. 1.Physical model of the problem.

xpu

Knf

∂∂

−=µ (2)

[ ] )()1( cffppnf TTg

xpv

K−−++

∂∂

−= βρϕβϕρµ (3)

∂∂

+∂∂

=∂∂

+∂∂

2

2

2

2

yT

xT

yTv

xTu nfα (4)

where the effective density of the nanofluid is given as pfnf ϕρρϕρ +−= )1( , (5)

and ϕ is the solid volume fraction of nanoparticles. Thermal diffusivity of the nanofluid is: nfpnfnf Ck )/(ρα = , (6) where the heat capacitance of the nanofluid given is, ppfpnfp CCC )())(1()( ρϕρϕρ +−= , (7) The thermal expansion coefficient of the nanofluid can be determined by pfnf )())(1()( ρβϕρβϕρβ +−= . (8) The effective dynamic viscosity of the nanofluid given by Brinkman (1952) is:

5.2)1( ϕ

µµ

−= f

nf. (9)

In Eq. (6), knf is the thermal conductivity of the nanoflu-id which for spherical nanoparticles, according to Max-well (1904), is:

−++

−−+=

)()2()(2)2(

pffp

pffpfnf kkkk

kkkkkk

ϕϕ (10)

where kp is the thermal conductivity of dispersed nano-particles and kf is the thermal conductivity of pure fluid. The following dimensionless variables ,,,,, ch

c

f

TTTTTT

LyY

LxX −=∆

∆−

==Y== θαy (11)

are introduced, and the dimensionless stream function y is defined in the usual way as

,,X

VY

U∂Y∂

−=∂Y∂

= (12)

When Eqs. (12) and (11) are substituted into Eqs. (2)-(4), and the pressure terms are eliminated it is found that

166

M.A. MANSOUR, S.E. AHMED, A.M. ALY, S.-W. LEE

[ ] ,)/)(/()1(

)1(1

2

2

2

2

5.2

XRa

YX

fpfp ∂∂

+−−

=

∂Y∂

+∂Y∂

−

θββρρϕϕ

ϕ (13)

,

)()(

1

12

2

2

2

∂∂

+∂∂

+−

=∂∂

∂Y∂

−∂∂

∂Y∂

YXCCk

kYXXY

fp

ppf

nf θθ

ρρϕ

ϕ

θθ

(14)

where Ra=gKρfβf∆TL/(µ fαf) is the Rayleigh number for a porous medium. The dimensionless forms of the boundary conditions are:

0,0:1)5.0(:0

1,0:)5.0()5.0(:0

0,0:)5.0(0:0

0,0:1)5.0(:1,0

0,0:)5.0()5.0(:1,0

0,0:)5.0(0:1,0

=∂∂

=Y≤<+=

==Y+≤≤−=

=∂∂

=Y−<≤=

=∂∂

=Y≤<+=

==Y+≤≤−=

=∂∂

=Y−<≤=

YXBDY

BDXBDYY

BDXY

XYBDX

BDYBDXX

BDYX

θθ

θ

θθ

θ

(15)

The definitions for local Nusselt number Nus(X) on the bottom wall and average Nusselt number Num are

0

)(=

∂∂

−=Yf

nfs Yk

kXNu θ (16)

∫+

−=

2/

2/)(1 BD

BD sm dXXNuNuB

(17)

IV. NUMERICAL METHOD In this investigation, the finite difference method was employed to solve the governing equations with the boundary conditions. Central difference equations were used to approximate the second derivatives in both the-XandY-directions. Then, the obtained discretized equa-tions are solved using a Gauss-Seidel iteration tech-nique. The solution procedure is iterated until the fol-lowing convergence criterion is satisfied: ∑ −≤−

ji

oldji

newji

,

7,, 10χχ (18)

where χ is the general dependent variable. The numeri-cal methodology was coded in FORTRAN. Accuracy tests were made for grid independence using the finite difference method using six sets of grids: 31x31, 41x41, 61x61, 81x81, 101x101, 121x121. As it can be seen from table 2 a good agreement was found between (61x61) and (81x81) grids, so the numerical computa-tions were carried out for (81x81) grid nodal points. In order to check the method validity, the obtained results in special cases are compared with the results obtained by Moya et al. (1987), Baytas and Pop (1999), Misirlioglu et al. (2005) and Mansour et al. (2011). Ta-ble 3 shows a very good agreement. Also, another vali-

dation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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0.3

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0.8

0.9

1

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0.1

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0.9

1

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0.9

1

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0.1

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0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 2. Streamlines (left) and isotherms (right) for the enclo-sures filled with Cu-water nanofluid for different heat source lengths, B=0.2, 0.4, 0.6, 0.8. The referenced case is Ra=500, ϕ=0.1 and D=0.5.

was performed with Aminossadatia and Ghasemi (2009) results. Table 4, also, shows an excellent agreement. As a result, the confidence in the present numerical solution is enhanced.

V. RESULTS AND DISCUSSION Two–dimensional natural convection is studied for nanofluid in a square cavity for 10 1000Ra = − , solid volume fraction (0≤ϕ≤0.2), heat source lengths (0.2≤B≤0.8), heat source locations (0.2≤D≤0.5) and a choice of nanoparticles (Cu, Ag, Al2O3 and TiO2) for all simulations , pure water is considered as the base fluid with Pr=6.2. Figure 2 shows effects of the heat source length on the stream lines and isotherms contours for cavity filled with Cu-water nanofluid. It is found that, increasing the heat source length, leads to increasing activity of the fluid motion and there are two clockwise and anti-clockwise circular cellswere formed inside the cavity. This increasing relates to the association between the buoyancy forces generated due to the fluid temperature differences and activity of the fluid motion. On the other hand, as length of the heat source increases, the genera-tion rates of temperature increase. So the higher temper-ature patterns corresponds the taller heat source and the

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Latin American Applied Research 45:165-172 (2015)

opposite is valid. Thus, an increase in heat source length leads to greater heat generation rates and stronger buoy-ant forces, which intensify the circulating vortices and increase the value of the temperature field. It is also noted that since the heater remains in the middle of the bottom wall, symmetrical circulating cells are generated regardless of the length of the heat source. In order to understand the flow behavior in this situ-ation, the vertical velocity profiles at the enclosure mid-section are presented in Fig. 3. It is clear that, velocity components increase with increasing the heat source length (0.2≤B≤0.8). This is because of stronger buoyant flow for higher heat generation rates. Symmetrical ve-locity profiles are observed which indicate the direction of the flow rotation within the enclosure. It is clear that heat flow rate increased due to a greater heat source length; consequently, an increase in the buoyancy forces is observed, resulting in a high vertical velocity. Figure 4 shows the profiles of the local Nusselt number along the heater for different heat source lengths. As stated earlier, the highest and the lowest lo-cal Nusselt number are obtained at both ends of the heater, respectively. In general, increasing the heat source length leads to increased heat generation rate and causes the local Nusselt number profile to decrease. Figure 5 presents the streamlines and isotherms for the enclosure filled with Cu-water nanofluidfor different

0.0 0.2 0.4 0.6 0.8 1.0

-160-140

-120-100

-80-60-40

-200

2040

6080

V(Y=

0.5)

X

B=0.2 B=0.4 B=0.6 B=0.8

Fig. 3. Vertical velocity Profiles along the mid-section of the enclosure for various heat source length (Cu-water, φ=0.1, D=0.5 and Ra=500).

0.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.800.850.900.95

0

10

20

30

40

50

60

70

80

Nus

X

B=0.2 B=0.4 B=0.6 B=0.8

Fig. 4. Profile of local Nusselt number along the heat source for various heat source length (Cu-water, ϕ=0.1, D=0.5, and Ra=500).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

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1

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1

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1

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1

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1

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1

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1

Fig. 5. Streamlines (left) and isotherms (right) for the enclo-sures filled with Cu-water nanofluid at Ra=10, 20, 100, 1000 and the referenced case is ϕ=0.1, B=1/3 and D=0.5.

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

0

10

20

30

40

50

60

70

80

90

Nus

X

Ra=10 Ra=50 Ra=100 Ra=500 Ra=1000

Fig. 6. Profile of local Nusselt number along the heat source for various Rayleigh numbers (Cu-water, B=1/3, D=0.5 and ϕ=0.1).

values of Rayleigh number at ϕ=0.1, B=1/3 and D=0.5. The streamlines show two counter rotating circulating cells for all values of Rayleigh number. It is found that, as Ra increases, the buoyancy force increases, which cause an increase in the fluid flow. Also, strong rotation of the nanofluid can be seen to increase the Raleigh number. It is, also found that the temperature distribu-tions strongly influenced by changes in the Raleigh number. The isotherms lines tend to crowd near the ver-

168

M.A. MANSOUR, S.E. AHMED, A.M. ALY, S.-W. LEE

tical walls as Ra increases. This causes an increase in the temperature gradients near the bottom and vertical walls. As a result, the rate of heat transfer represent by profiles of the local Nusselt number and average Nusselt number is enhanced by increase Ra. As it can be ob-served from Figs. 6, 7, 8 and 9, the local Nusselt num-ber along the heater and the average Nusselt number in-creases as Ra increases. Figure 10 displays the streamlines and the isotherms for the enclosures filled with Cu-water nanofluid at ϕ=0.1, B=1/3 and Ra=500. Two unsymmetrical circulat-ing cells with unequal strengths are observed when the heat source is located beside the left wall. As the heat

0.2 0.3 0.4 0.5 0.6 0.7 0.8

5

10

15

20

25

30

35

Num

B

Ra=10 Ra=50 Ra=100 Ra=500 Ra=1000

Fig. 7. Variation of average Nusselt number with heat source length for different values of Rayleigh number (Cu-water, φ=0.1 and D=0.5 ).

0.00 0.05 0.10 0.15 0.20

468

1012141618202224262830

Num

φ

Ra=10 Ra=50 Ra=100 Ra=500 Ra=1000

Fig. 8. Variation of average Nusselt number with solid volume fraction for different values of Rayleigh number (Cu-water, D=0.5 and B=1/3).

0.20 0.25 0.30 0.35 0.40 0.45 0.5046

810

1214

1618

2022

2426

2830

Num

D

Ra=10 Ra=50 Ra=100 Ra=500 Ra=1000

Fig. 9. Variation of average Nusselt number with heat source location for different values of Rayleigh number (Cu-water, ϕ=0.1 and B=1/3 ).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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Fig. 10. Streamlines (left) and isotherms (right) for the enclo-sures filled with Cu-water nanofluid at D=0.2, 0.3, 0.4 and 0.5 The referenced case is ϕ=0.1, B=1/3 and Ra=500.

0.0 0.2 0.4 0.6 0.8 1.0

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

V(Y=

0.5)

D=0.2 D=0.3 D=0.4 D=0.5

Fig. 11. Vertical velocity Profiles along the mid-section of the enclosure for various heat source locations (Cu-water, B=1/3, φ=0.1 and Ra=500).

source moves away from the left wall, the strength of both circulating cells increases until it reaches the mid-dle of the bottom wall, where two circulating cells with equal intensity appear in the enclosure. Temperature distributions, also, track the movement of the heat source until be symmetrical about the line X=0.5. Fig-ure11 presents the vertical velocity profiles along the mid-section of the enclosure. For the case, where the heater is not located in the middle, unsymmetrical ve-locity profiles indicate bun-balanced circulating cells within the enclosure. Moreover, the maximum velocity

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Latin American Applied Research 45:165-172 (2015)

increases as the heat source moves towards the middle. Figure 12 presents the effects of changing the heat source location on the local Nusselt number along the heater. It is clear that, at the part X≤0.22 from the heat source, as D increases, Nus decreases, whereas, at the remaining parts, Nus takes the inverse behavior.

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

0

10

20

30

40

50

60

70

Nus

X

D=0.2 D=0.3 D=0.4 D=0.5

Fig. 12. Profile of local Nusselt number along the heat source for various heat source locations (Cu-water, B=1/3, ϕ=0.1 and Ra=500).

0.0 0.2 0.4 0.6 0.8 1.0-200

-150

-100

-50

0

50

V(Y=

0.5)

X

φ=0.0 φ=0.05 φ=0.1 φ=0.15 φ=0.2

Fig. 13. Vertical velocity Profiles along the mid-section of the enclosure for various solid volume fractions(Cu-water, B=1/3, D=0.5 and Ra=500).

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

0

10

20

30

40

50

60

70

NUS

X

φ=0.0 φ=0.05 φ=0.1 φ=0.15 φ=0.2

Fig. 14. Profile of local Nusselt number along the heat source for various solid volume fractions(Cu-water, B=1/3, D=0.5 and Ra=500).

0.0 0.2 0.4 0.6 0.8 1.0

-350

-300

-250

-200

-150

-100

-50

0

50

100

V(Y=

0.5)

X

Cu Ag Al2O3

TiO2

Pure water

Fig. 15. Vertical velocity Profiles along the mid-section of the enclosure at different fluids (D=0.5, B=1/3, ϕ=0.1 and Ra=500).

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

0

10

20

30

40

50

60

70

80

90

Nus

X

Cu Ag Al2O3

TiO2

Pure water

Fig. 16. Profile of local Nusselt number along the heat source for different nanofluids (B=1/3, ϕ=0.1, D=0.5 and Ra=500).

100 200 300 400 5000

5

10

15

20

25

30

35

40

45

Num

Ra

Cu Ag Al2O3

TiO2

Pure water

Fig. 17. Variation of average Nusselt number with solid vol-ume fraction at various Rayleigh numbers for different nanofluids (B=1/3, ϕ=0.1 and D=0.5).

Solid volume fraction parameter φ is a key factor to study how nanoparticles affect the heat transfer of nanofluids. Figures13 and 14 show symmetrical pro-files of the velocity and local Nusselt number for differ-ent solid volume fraction. In general, increasing the vol-ume solid fraction causes the velocity and the local Nusselt number profiles to decreases. Figures 15, 16 and 17 show profile of the vertical velocity along the mid-section of the enclosure, local Nusselt number along the heat source and average Nusselt number with Ra, respectively, for different types of nanofluids. Pro-

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files are obtained for all nanofluids with the lowest Nusselt number for the middle of the heat source. It is found that, adding nanoparticles into the base fluid results in enhanced heat removal. TiO2 has the largest rate of heat transfer. On the other hand, Ag na-noparticles have the smallest values. The heat transfer performance in the nanofluid enclosure was clearly higher compared to pure water. All these behaviors are plotted in Figs. 15-17 with referenced cash D=0.5, B=1/3, ϕ=0.1 and Ra=500.

Table 1.Thermo-physical properties of water and nanoparticles.

Pure wa-ter

Cop-per(Cu)

Silver (Ag)

Alumina Al2O3

Titanium Ox-ide(TiO2)

ρ 997.1 8933 10500 3970 4250 Cp 4179 385 235 765 686.2 k 0.613 401 429 40 8.9538 β 21×10-5 1.67×10-5 1.89×10-5 0.85×10-5 0.9×10-5

Table 2. Grid independency study for Cu-water nanofluid at B=0.4, D=0.5, Ra=500, ϕ=0.1.

Grid 31×31 41×41 61×61 81×81 101×101 121×121 ymax 9.683 9.691 9.688 9.685 9.679 9.668

Table 3.Comparison of the average Nusselt number for pure fluid (φ=0).

Author Ra=10 Ra=100 Moya et al. (1987) 1.065 2.801

Baytas and Pop (1999) 1.079 3.16 Misirlioglu et al. (2011) 1.119 3.05 Mansour et al. (2005) 1.079 3.207

Present 1.079 3.207

Table 4. Comparison of Num, at B=0.4, φ=0.1, D=0.5. Ra Aminossadati and Ghasemi (2009) Present 103 5.451 5.450 104 5.474 5.475 105 7.121 7.204 106 13.864 14.014

VI. CONCLUSION Free convection cooling of heat source embedded on the bottom wall and two sinks embedded on the vertical walls of asquare cavity saturated porous medium filled with a nanofluid has been investigated in this paper. During this investigation, water is considered as based nanofluid and Cu, Ag, Al2O3 and TiO2 are considered as nanoparticles. Finite difference method is used to solve the dimensionless PDE governing the problem. The fol-lowing findings may be summarized from the present investigation: The local Nusselt number along the heat sources de-creases as the solid volume fraction increases Increasing either the heat source length or Rayleigh number leads to decrease the local Nusselt number along the heat source Adding Ag nanoparticles to the base fluid gives the largest heat transfer rate whereas adding TiO2 nanopar-ticles gives smallest heat transfer rate.

The average Nusselt number decreases, in linearly way, as the solid volume fraction increases. Also it de-creases as the heat source length increases.

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Received: April 2, 2014. Accepted: November 24, 2014. Recommended by Subject Editor: Walter Am-brosini.

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