free convection in a porous cavity filled with...

Free convection in a porous cavity filled with nanofluids GROSAN TEODOR1, REVNIC CORNELIA2, POP IOAN1

1Faculty of Mathematics and Computer Sciences

Babes-Bolyai University Cluj-Napoca ROMANIA

[email protected], [email protected]

2Faculty of Pharmacy University of Medicine and Pharmacy “Iuliu Hatieganu”, Cluj-Napoca

ROMANIA [email protected]

Abstract: The effect of different kind nanoparticles (cooper, alumina and titania) on free convection in a square cavity filled with a fluid-saturated porous medium have been investigated numerically. The top and bottom horizontal walls of cavity are considered adiabatic, while the vertical walls are kept at constant temperatures. The mathematical model consists in a set of partial differential equations along with the corresponding boundary conditions and these equations were solved numerically using a finite-difference scheme discretization and a Gauss-Seidel technique. The obtained results are presented in terms of streamlines, isotherms and local and averaged Nusselt numbers. Key-Words: free convection, porous media, nanofluid, numerical solution 1 Introduction Natural convective heat transfer in fluid-saturated porous media has occupied the centre stage in many fundamental heat transfer analyses and has received considerable attention over the last several decades. This interest is due to its wide range of applications in, for example, packed sphere beds, high performance insulation for buildings, chemical catalytic reactors, grain storage and such geophysical problems as frost heave. Porous media are also of interest in relation to the underground spread of pollutants, solar power collectors, and to geothermal energy systems. Porous materials, such as sand and crushed rock, underground saturated with water, which, under the influence of local pressure gradients, migrates and transports energy through the material. Literature concerning convective flow in porous media is abundant. Representative studies in this are may be found in the recent books by Nield and Bejan [1], Pop and Ingham [2], Ingham and Pop [3], Vadasz [4] and Vafai [5]. Analysis of natural convection heat transfer and fluid flow in enclosures filled with viscous fluids or porous media has been extensively made using numerical techniques and experiments because of its wide applications and interest in engineering such as nuclear energy, double pane windows, heating and

cooling of buildings, solar collectors, electronic cooling, etc. (see [6] ). An innovative technique to enhance heat transfer is by using nano-scale particles in the base fluid. Nanotechnology has been widely used in industry since materials with sizes of nanometers possess unique physical and chemical properties. Nano-scale particle added fluids are called as nanofluid which has been first introduced by Choi [7]. Some numerical and experimental studies on nanofluids have been done by Khanafer et al. [8], Tiwari and Das [9], Popa et al. [10], Grosan and Pop [11], Rosca et al. [12], etc. Detailed review studies on nanofluids are published in the book by Das et al.[13] and the review papers by Buongiorno [14], and Kakaç and Pramuanjaroenkij [15]. It is obvious from the foregoing review that most of the studies are performed considering the water-based nanofluids in cavities. Very little research is performed considering a porous medium filled with nanofluids. Recently, Nield and Kuznetsov [16] have studied the Cheng and Minkowycz’s [17] problem for natural convective boundary layer flow over a vertical flat plate embedded in a porous medium filled with nanofluid taking into account the combined effects of heat and mass transfer in the presence of Brownian motion and thermophoresis as proposed in [14], while Ahmad and Pop [18] have considered the steady mixed convection boundary

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layer flow over a vertical flat plate embedded in a porous medium saturated with a nanofluid using the nanofluid model proposed in [9]. In the present study, the problem of steady free convection heat transfer in a square cavity filled with a porous medium saturated with water-based nanofluid using the model proposed in [9]. 2 Basic Equations Consider the free convection in a two-dimensional porous square cavity filled with nanofluid based on water and different type of nanoparticles: Cu, Al2O3 and TiO2. A schematic geometry of the problem is shown in Fig. 1, where x and y are the Cartesian coordinates and L is the height of the cavity. It is assumed that the top and bottom walls of the cavity are adiabatic. Also, it is assumed that the left wall is maintain at hT while the right wall is keeps at cT , where we assume that ch TT > . Using the Darcy- Boussinesq approximation, the basic equations are (see [1]),

xTKg

yxK

ffss

nf

∂∂

−+−

=

∂∂

+∂∂

])1([

2

2

2

2

βρϕβρϕ

ψψµ

(1)

∂∂

+∂∂

=∂∂

+∂∂

2

2

2

2

yT

xT

yTv

xTu nfα (2)

Here T is the nanofluid temperature, K is the permeability of the porous medium and ψ is the stream function which is defined as yu ∂∂= /ψ and xv ∂∂−= /ψ .

Further, nfµ is the viscosity of the nanofluid,

nfα is the thermal diffusivity of the nanofluid which are given by

,)())(1()(

,)(

,)1( 5.2

spfpnfp

nfp

nfnf

fnf

CCCCk

ρϕρϕρ

ρα

ϕ

µµ

+−=

=−

=

)()2()(2)2(

sffs

sffs

f

nf

kkkkkkkk

kk

−++

−−+=

ϕϕ

(3)

where ϕ is the solid volume fraction, fµ is the dynamic viscosity of the fluid, nfpC )(ρ is the heat capacitance of the nanofluid, nfk is the thermal conductivity of the nanofluid, sk and fk are the solid and fluid thermal conductivities, sβ and fβ

are solid and fluid expansion coefficients, and sρ and fρ are solid and fluid densities. The viscosity of the nanofluid nfµ is approximated as viscosity of a base fluid fµ containing dilute suspension of fine spherical particles.

Fig. 1. Physical model and coordinate system.

Introducing the following dimensionless variables

TTT

LyYLxX

f ∆−==

==

/)(,/

,/,/

0θαψψ (4)

where ch TTT −=∆ , 2/)(0 ch TTT += with

ch TT > . Substituting (4) into Eqs. (1) and (2), we get

[ ]X

Ra

YX

fsfs ∂∂

+−−

=

∂∂

+∂∂

−

θββρρϕϕ

ψψϕ

)/)(/(1

)1(1

2

2

2

2

5.2 (5)

∂∂

+∂∂

=∂∂

∂∂

−∂∂

∂∂

2

2

2

2

YXYXXY f

nf θθααθψθψ (6)

where )/( fff TLgKRa ναβ ∆= is the Rayleight number. If we further perform the transformation

[ ] 5.2)1()/)(/(1

/

ϕαα

ββρρϕϕ

ααψ

−+−=Γ

Ψ=

nf

ffsfs

fnf

(7)

then, Eqs. (5) and (6) can be written as

XRa

YX ∂∂

Γ−=∂Ψ∂

+∂

Ψ∂ θ2

2

2

2

(8)

2

2

2

2

YXYXXY ∂∂

+∂∂

=∂∂

∂Ψ∂

−∂∂

∂Ψ∂ θθθθ (9)

The corresponding boundary conditions of these equations are

cT

L hT

ux,

vy,

g


ISBN: 978-1-61804-135-7 188

2/1,0 ==Ψ θ on 0=X 2/1,0 −==Ψ θ on 1=X (10)

0,0 =∂∂

=ΨYθ on 1,0=Y

It should be noticed that for 0=ϕ (regular fluid) then 1=Γ and Eq. (8) and (9) reduce to those of Bejan [19], Beckerman et al.[20], Manole and Lage [21] and Moya et al. [22]

Table 1. Thermophysical properties of fluid and nanoparticles (Oztop and Abu-Nada [23]).

Physical properties water Cu 2 3Al O 3TiO

pC (J/kg K) 4179 385 765 686.2 3(kg/m )ρ 997.1 8933 3970 4250

(W/mK)k 0.613 400 40 8.9538 )/(10 27 sm−×α 1.47 1163.1 131.7 30.7

510 (1/K)β −× 21 1.67 0.85 0.9 The physical quantities of interest is the local Nusselt number Nu which is defined as

fluidcondQQNu

,= (11)

where the heat transfer from the left hot wall, Q , is given by

0

)(=

∂

∂−=

xseff x

TkQ (12)

Using (11), (12) and (4), we get

0

)(

=

∂∂

−=Xf

seff

Xkk

Nu θ (13)

3 Numerical Method In order to solve the partial differential equations (8) and (9) with the boundary conditions (10) we used a central finite-difference discretization. The system of discretizated equation was solved using a non-uniform grid near the walls with the step sizes varying as a quadratic function ( =∆ minX

462.2min −=∆ eY ; 25.2maxmax −=∆=∆ eYX ) and with 8787× nodes. The both equations (8) and (9) have been solved using a Gauss-Seidel iterative scheme. The following criteria was used to check the convergence of the method

εφφ ≤− || oldji,

newji, (16)

where φ is either variable ψ or θ , and ε is a prescribe error, which is of order 710− .

Tabel 2. Validation of the code for 0=ϕ (regular fluid).

Authors Ra 10 100 1000 10000

Bejan [19] 4.2 15.8 50.80 Beckerman et al. [20]

3.113 48.9

Manole and Lage [21]

3.118 13.637 48.117

Moya et al. [22]

1.065 2.801

Present results

1.078 3.107 13.596 48.321

0 0.2 0.4 0.6 0.8 10

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 10

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 10

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 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 2. Streamline and isotherms for 08.0=ϕ (up)

and 2.0=ϕ (down) when 10=Ra 4 Results and discussions The effects of volume fraction of nanofluid parameter ϕ on the flow and heat transfer characteristics has been studied for three types of nanoparticles, namely, copper (Cu), alumina ( 32OAl ) and titania ( 3TiO ) (see Table 1) . The values of the governing parameters are the Reyleigh number Ra and the volume nanofluid friction ϕ , which vary in the ranges of 1000010 ≤≤ Ra and

2.00 ≤≤ ϕ . The present results are compared with those reported by [19], [20], [21] and [22] for a regular fluid ( 0=ϕ ) in Table 2. It is seen that the agreement is very good. Therefore, we are confident that the present numerical scheme is accurate. The Figs. 2 to 5 represent the streamlines and isotherms


ISBN: 978-1-61804-135-7 189

0 0.2 0.4 0.6 0.8 10

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 10

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 10

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 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


and 2.0=ϕ (down) when 100=Ra

0 0.2 0.4 0.6 0.8 10

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 10

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 10

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 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



patterns for Cu nanoparticles, Fig. 6 shows the horizontal velocity profiles for all materials nanoparticles, while Figs. 7 and 8 illustrate the variation of local and mean Nusselt numbers. It can be seen from Figs. 2 to 5 that for the values of Ra considered the conductive effect is more pronounced than that of the convective ones for all values of ϕ and type of nanoparticles. Figure 6 shows that the influence of the nanoparticles concentration is more present in the velocity profile

for Cu nanoparticles for small Rayleigh numbers, while for very large values of the Rayleigh number the velocity profiles are similar for copper, alumina and titania nanoparticles. The value of the local Nusselt number decreases with the increase of ϕ near the bottom wall ( 0=X ) while it increases near the top wall ( 1=X ) (see Fig. 7). In the case of cooper nanoparticles, the value of the mean Nusselt number increases with increase of ϕ , while the mean Nusselt number present a minimum for alumina and titania nanoparticles (see Fig. 8).

0 0.2 0.4 0.6 0.8 10

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 10

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 10

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 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



4 Conclusion The problem of free convection in a cavity saturated by a nanofluid was solved numerically. The characteristics of fluid flow (streamlines and velocity profiles) and heat transfer (isotherms and Nusselt numbers) were obtained for different kind of nanoparticles. The effect of volume fraction of the nanoparticles is more present for Cu nano-particles and small Rayleigh numbers. When 32OAl and 3TiO nanoparticles are used the effects on fluid flow and heat transfer characteristics are similar.

Acknowledgements This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS – UEFISCDI, project number PN-II-RU-TE-2011-3-0013.


ISBN: 978-1-61804-135-7 190

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

x

u

CuAl2O3

TiO3

a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-25

-20

-15

-10

-5

0

5

10

15

20

x

u

CuAl2O3

TiO3

b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100

-80

-60

-40

-20

0

20

40

60

80

x

u

CuAl2O3

TiO3

c) Fig.6. Variation of horizontal velocity u in the middle of the cavity for a) Ra = 100 b) Ra =1000 and c) Ra =10000 when 2.0=ϕ

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

Y

NuY

φ =0, 0.08, 0.16, 0.2

a)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

Y

Nu

φ =0, 0.08, 0.16, 0.2

b)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

Y

NuY

φ =0, 0.08, 0.16, 0.2

c)

Fig 7. Variation of local Nusselt number along the heated wall for a) Cu b) Al2O3 and c) TiO3

nanoparticles when Ra = 60


ISBN: 978-1-61804-135-7 191

0 0.05 0.1 0.15 0.22.6

2.7

2.8

2.9

3

3.1

3.2

3.3

φ

Nu

CuAl2O3TiO3

Fig 8. Variation of mean Nusselt numbers with

volume fraction for Ra = 60 References: [1] D. Nield and A. Bejan, Convection in Porous

Media (3rd ed), Springer, New York, 2006. [2] I. Pop and D.B. Ingham, Convective Heat

Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media, Pergamon, Oxford, 2001.

[3] D.B. Ingham and I. Pop (eds.), Transport Phenomena in Porous Media III, Elsevier, Oxford, 2005.

[4] P. Vadasz, Emerging Topics in Heat and Mass Transfer in Porous Media, Springer, New York, 2008.

[5] K. Vafai, Porous Media: Applications in Biological Systems and Biotechnology, CRC Press, New York, 2010.

[6] A. Bejan, Convection Heat Transfer (2nd ed), John Wiley & Sons, New York, 1995.

[7] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, In: Proceedings of the 1994 ASME International Mechanical Engineering, Congress and Exposition, 66, ASME, FED 231/MD, San Franciscos, USA, pp.99 – 105, 1995.

[8] K. Khanafer, K. Vafai and M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transfer 46 (2003) 3639–3653.

[9] R.K. Tiwari and M.K. Das, Heat transfer augmentation in a two sided lid-driven differentially heated square cavity utilizing nanofluids, Int. J. Heat Mass Transfer 50 (2007) 2002-2018.

[10] C.V. Popa, S. Fohanno, C.T. Nguyen and G. Polidori, On heat transfer in external natural convection flows using two nanofluids, Int. J. Thermal Sci. 49 (2010) 901-908.

[11] T. Grosan, I. Pop, Fully Developed Mixed Convection in a Vertical Channel Filled by a Nanofluid, J. Heat Transfer (ASME) 134 (2012) Art. No. 082501

[12] A. V. Rosca, N. C. Rosca, T. Grosan, I. Pop, Non-Darcy mixed convection from a horizontal plate embedded in a nanofluid saturated porous media, Int. Comm. Heat Mass Transfer 39 (2012) 1080-1085.

[13] S.K. Das, S.U.S. Choi, W. Yu and T. Pradet, Nanofluids: Science and Technology, Wiley, New York, 2007.

[14] J. Buongiorno, Convective transport in nanofluids, J. Heat Transfer (ASME) 128 (2006) 240-250.

[15] S. Kakaç and A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52 (2009) 3187-3196.

[16] D.A. Nield and A.V. Kuznetsov, The Cheng-Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid, Int. J. Heat Mass Transfer 52 (2009) 5792-5795.

[17] P. Cheng and W.J. Minkowycz, Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike, J. Geophys. Res. 82 (1977) 2040-2044.

[18] S. Ahmad and I. Pop, Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids, Int. Comm. Heat Mass Transfer 37 (2010) 987-991.

[19] A. Bejan, On the boundary layer regime in a vertical enclosure filled with a porous medium, Lett. Heat Mass Transfer 6 (1979) 93-102.

[20] C. Beckermann, R. Viskanta, S. Ramadhyani, A numerical study of non-Darcian natural convection in a vertical enclosure filled with a porous medium, Num. Heat Transfer 10 (1986) 557-570.

[21] D.M. Manole, J.L. Lage, Numerical benchmark results for natural convection in a porous medium cavity, Heat and Mass Transfer in Porous Media, ASME Conference HTD-vol. 216, pp. 55-60, 1992.

[22] S.L. Moya, E. Ramos, M. Sen, Numerical study of natural convection in a tilted rectangular porous material, Int. J. Heat Mass Transfer 30 (1987) 741-756.

[23] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326–1336.


ISBN: 978-1-61804-135-7 192

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