Rayleigh-Bénard convection in nanofluids: Effect of temperature dependent properties

International Journal of Thermal Sciences 50 (2011) 1720e1730

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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

Rayleigh-Bénard convection in nanofluids: Effect of temperature dependent properties Eiyad Abu-Nada a, b, * a b

Department of Mechanical Engineering, King Faisal University, Al-Ahsa 31982, Saudi Arabia Leibniz Universität Hannover, Institut für Technische Verbrennung, Welfengarten 1a, 30167 Hannover, Germany

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 October 2010 Received in revised form 2 April 2011 Accepted 5 April 2011 Available online 8 May 2011

In this work, heat transfer enhancement using CuO-water nanofluids in natural convection using the Rayleigh-Bénard convection problem is investigated. The main focus of the current study is on the effects of variable thermal conductivity and variable viscosity of nanofluids on heat transfer enhancement in natural convection. The results are presented over a wide range of Rayleigh numbers (Ra ¼ 103e106) and volume fractions of nanoparticles (0  4  9%). For Ra > 103, and using the variable properties of nanofluid, the average Nusselt number was reduced by increasing the volume fraction of nanoparticles. However, for Ra ¼ 103, the average Nusselt number was enhanced by increasing the volume fraction of nanoparticles. To study the significance of effects of temperature dependence of viscosity and thermal conductivity of nanofluid, the results obtained by using variable properties of nanofluid are compared with those based on constant property simulations and it was found that the temperature influence is small compared to the influence of high viscosity brought by the presence of high concentration of nanoparticles. The variable thermal conductivity and variable viscosity models were compared to both the Maxwell-Garnett (MG) model and the Brinkman model. It was found that for Ra > 103 the average Nusselt number was much more sensitive to the viscosity models than to the thermal conductivity models. Ó 2011 Elsevier Masson SAS. All rights reserved.

Keywords: Nanofluid Variable properties Natural convection Rayleigh-Bénard convection

1. Introduction Natural convection is found in many engineering applications such as electronics cooling, heat exchangers, and energy systems [1e4]. Enhancement of heat transfer in such systems is crucial from energy saving perspective. In recent years, nano-scale particles dispersed in a base fluid, known as nanofluid, has been used and researched extensively to enhance heat transfer. The presence of nanoparticles shows an enhancement in forced convection heat transfer applications [5e7]. However, for natural convection heat transfer enhancement using nanofluids is still controversial and there is a dispute on the role of nanoparticles on heat transfer enhancement in natural convection applications. Experimentally, the findings reported by Wen and Ding [8], Putra et al. [9] and Chang et al. [10], highlighted deterioration in heat transfer by the addition of nanoparticles. Also, Ho et al. [11] observed a deterioration in heat transfer for volume fraction of nanoparticles >2%; However, he reported an enhancement for low concentration at volume fraction ¼ 0.1% (18% enhancement) and * Department of Mechanical Engineering, King Faisal University, Al-Ahsa 31982, Saudi Arabia. Tel.: þ966 3580 0000. E-mail address: [email protected]. 1290-0729/$ e see front matter Ó 2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2011.04.003

did not give an explanation for such an enhancement. Similar observation where reported experimentally by Agwu Nnanna [12] and Li and Peterson [13]. Therefore, it seems that experimental studies observed deterioration in heat transfer at high volume fraction of nanoparticles. On other hand the computational studies were very dispute where the majority reported an enhancement of heat transfer due to the presence of nanoparticles and still some researcher were able to predict a reduction in heat transfer in agreement to what observed experimentally. Starting with the studies that observed augmentation in heat transfer, Khanafer et al. [14] reported an increase in heat transfer with the increase of suspended nanoparticles. Similarly, Oztop and Abu-Nada [15], Aminossadati and üt Ghasemi [16], Ghasemi and Aminossadati [17], Kim et al. [18], Ög [19] showed similar trend, where an enhancement in heat transfers was registered by the addition of nanoparticles. On the other hand, Hwang et al. [20] and Santra et al. [21] reported a decrease in heat transfer by increasing the volume concentration of nanoparticles for a particular Rayleigh number. Ho et al. [22] considered the effects due to uncertainties of viscosity and thermal conductivity by considering two viscosity and two thermal conductivity models. They reported that significant difference between enhancements in the viscosity models leads to contradictory heat transfer efficacy of

E. Abu-Nada / International Journal of Thermal Sciences 50 (2011) 1720e1730

Nomenclature A Cp d g H h k Nu Pr qw Ra Re T u, v U, V W x, y x 0 , y0

aspect ratio (W/H) specific heat at constant pressure (J kge1 Ke1) diameter (m) gravitational acceleration (m se2) distance between the bottom and top plates (m) local heat transfer coefficient (W me2 Ke1) thermal conductivity Nusselt number, Nu ¼ hH/kf Prandtl number, Pr ¼ nfo =afo heat flux, (W me2) Rayleigh number, Ra ¼ g bðTH  TC ÞH 3 =nfo afo Reynolds number, Re ¼ rf kb T=3pm2f lf dimensional temperature ( C) dimensional x- and y-components of velocity (m se1) dimensionless velocities, V ¼ vH/af, U ¼ uH/af width of the plates (m) dimensionless coordinates, x ¼ x0 /H, y ¼ y0 /H dimensional coordinates (m)

Greek Symbols a thermal diffusivity, (m2 se1) b thermal expansion coefficient (Ke1)

the nanofluid, so that the heat transfer can be found to be enhanced or deteriorated with respect to the base fluid. Recently, Abu-Nada [23,24] and Abu-Nada et al. [25] studied the effect of variable properties of Al2O3-water and CuO-water nanofluids on natural convection. They found that the heat transfer in natural convection is sensitive to the thermo-physical properties of the nanofluid (specifically viscosity and thermal conductivity). They found that the effect of viscosity is more dominant than the effect of thermal conductivity. The results were not sensitive to the thermal conductivity model employed where they used two models of thermal conductivity one which is the MG model and the other is the Chon et al. model [26]. However, the results were very sensitive to the viscosity model where they show that the Brinkman model underestimates the viscosity of the nanofluids. Also, the temperature dependence of viscosity plays a major role as the viscosity of the nanofluid drops sharply with temperature [27]. Therefore, AbuNada [23,24] and Abu-Nada et al. [25] gave an explanation to the deterioration in heat transfer observed experientially and they inferred this deterioration to the variable properties of nanofluids (particularly the viscosity). The research in heat transfer using nanofluids is still progressing toward a better understanding of heat transfer mechanisms. For example, different effects on mechanisms of heat transfer have been incorporated recently by researcher to such as the Brownian motion and the thermophoresis effects [28e31]. The current work will evaluate the impact of temperature dependent viscosity and thermal conductivity, derived from experimental data, on heat transfer in natural convection using the RB problem. The enhancement in heat transfer will be evaluated under a wide range of operating temperatures and a wide range of volume fraction of nanoparticles.

2. Governing equations and problem formulation Fig. 1 shows a schematic diagram of the Rayleigh-Bénard (RB) problem. For the RB problem, the distance between the upper

e z 4 n q k j J u U r m n x

1721

numerical tolerance non-dimensional thermal diffusivity nanoparticle volume fraction kinematic viscosity (m2 se1) dimensionless temperature, q ¼ (T e TC)/(TH e TC) non-dimensional thermal conductivity dimensional stream function (m2 se1) dimensionless stream function, J ¼ j=afo dimensional vorticity (se1) dimensionless vorticity, U ¼ uH 2 =afo density (kg me3) dynamic viscosity, (N s me2) kinematic viscosity, (m2 se1) no-dimensional dynamic viscosity

Subscripts avg average C cold f base fluid base fluid at reference temperature fo H hot nf nanofluid p particle w wall

cold and lower hot plates is defined by H and the width of the top and bottom plates is defined by W. The plates’ width W is considered infinite and treated by a periodicity boundary condition. The bottom plate is maintained at a hot temperature TH whereas the top plate is maintained at a cold temperature TC. The fluid enclosed between the plates is water based nanofluid containing CuO nanoparticles. The nanofluid is assumed incompressible and the flow is assumed as laminar and twodimensional. It is idealized that water and nanoparticles are in thermal equilibrium and no slip occurs between the two media. The thermo-physical properties of the nanofluid are listed in Table 1 [23,24]. The density of the nanofluid is approximated by the standard Boussinesq model. The viscosity and the thermal conductivity of the nanofluid are considered as variable properties; both vary with temperature and volume fraction of nanoparticles. The governing equations for the laminar, steady state natural convection, considering variable properties of viscosity and thermal conductivity, in terms of the stream function-vorticity formulation are given as Vorticity




    v vj v vj 1 v vu v vu u u m m  ¼ þ rnf vx0 nf vx0 vy0 vx0 vx0 vy0 vy0 nf vy0  
vT  4 v2 mnf vv þ 4bp þ ð1  4Þbf g þ 0 rnf vx0 vy0 vy0 vx !
 1 v2 mnf v2 mnf vu vv þ  þ rnf vx02 vy0 vx0 vy02
 1 vmnf vu vmnf vu ð1Þ þ þ 0 0 0 0 rnf vx vx vy vy Energy

1722

E. Abu-Nada / International Journal of Thermal Sciences 50 (2011) 1720e1730

6

Cold Plate (TC)

1% (Present Regression) 4% (Present Regression)

5 Left Boundary (Periodic)

H

g

Right Boundary (Periodic)

7% (Present Regression) 9.5% (Present Regression) 9.4 % Nguyen et al. data [27]

4

7% Nguyen et al. data [27]

μ (cp)

y Hot Plate (TH)

x

W

4.5 % Nguyen et al. data [27]

3

1% Nguyen et al. data [27]

2

Fig. 1. Schematic of the problem.

1



     v vj v vj v vT v vT a a T  T ¼ þ vx0 vy0 vy0 vx0 vx0 nf vx0 vy0 nf vy0

0

(2)

(3)

The horizontal and vertical velocities are given by the following relations,

vj ; vy0

(4)

vj v ¼  0: vx

(5)

In Eqs. (1) and (2), the thermal diffusivity is written as

knf

rcp

(6)

nf

Also, the effective density of the nanofluid is given by

(7)

The heat capacitance of the nanofluid is expressed as

rcp

nf

  ¼ ð1  4Þ rcp f þ4 rcp p

(8)

The effective thermal conductivity of the nanofluid calculated by the Chon et al. model [26] is given as


0:3690
0:7476 df kf knf ¼ 1 þ 64:740:7640 Pr0:9955 Re1:2321 T kf dp kp (9) Here PrT and Re are defined by

PrT ¼

Re ¼

50

rf kb T

60

70

80

(11)

3pm2f lf

The symbol kb is the Boltzmann constant ¼ 1.3807  10e23 J/K, and lf is the mean path of fluid particles given as 0.17 nm [26]. This model embraces the effect of nanoparticle size and temperature on nanofluid thermal conductivity encompassing a wide temperature range between 21 and 70  C. This model was further tested experimentally by Angue Minsta et al. [32] for the pair of Al2O3 and CuO nanoparticles and found suitable to predict the thermal conductivity of these nanoparticles up to a volume fraction of 9%. The results using Eq. (9) will be compared to the Maxwell-Garnett (MG) model given by [15,23,24]

  kp þ ðn  1Þkf  ðn  1Þ kf  kp 4 knf   ¼ kf kp þ ðn  1Þk þ k  kp 4 f

rnf ¼ ð1  4Þrf þ 4rp 

40

Fig. 2. Comparison between CuO-water viscosities calculated using Eq. (13) and those of Ref. [27].

v2 j v2 j þ ¼ u vx02 vy02

anf ¼ 

30

T (°C)

Kinematics

u ¼

20

mf rf af

(10)

where n ¼ 3. The correlation for the dynamic viscosity of CuO-water nanofluid is derived using available experimental data of Nguyen et al. [27]. In fact, no explicit correlation is given in Nguyen et al. [27] that define the viscosity of CuO-water nanofluid as a double function of temperature and volume fraction of nanoparticles simultaneously. Such a correlation was derived based on the two-dimensional regression performed on the experimental data of Nguyen et al. [27]. The R2 value is 99.8% and a maximum error is 5%. The correlation for CuO-nanofluids is given by [24,25]





15:937 1356:14  :25942 þ 1:2384 þ T T2 4 19652:74 42  30:88  þ :0159343 þ 4:38206 3 T T T

mCuO Cp ¼  0:6967 þ

þ 147:573

Table 1 Thermo-physical properties of fluid and nanoparticles [23,35]. Physical properties

Fluid phase (Water)

CuO

Cp (J/kg K) r (kg/m3) k (W/m K) b  10e5 (1/K) dp (nm)

4179 997.1 0.613 21 0.384

540 6500 18.0 0.85 29

(12)

f

4

T2

ð13Þ

The viscosity given in Eq. (13) is expressed in centi poise and the temperature in  C. Fig. 2 presents a plot of the viscosity of CuOwater nanofluids as a function of temperature and concentration of nanoparticles calculated using Eq. (13). The figure also shows the measured data from Nguyen et al. experiments [27]. It is very clear that the current non-linear regression is in good agreement with the experimental measurements. In the discussion section, the results, using Eq. (13) will be compared to the Brinkman model given by [15,23,24]

E. Abu-Nada / International Journal of Thermal Sciences 50 (2011) 1720e1730

1723

Table 2 Grid independence study.

mnf ¼

Grid size

Nuavg

21  21 31  31 41  41 51  51 61  61 71  71 81  81 101  101

1.948174 1.894624 1.879615 1.874776 1.874284 1.874622 1.8745311 1.874608

mf

(14)

ð1  4Þ2:5

It is worth mentioning that the viscosity of the base fluid (water) is considered to be variable with temperature and the flowing equation is used to evaluate the viscosity of water [26],

(15)

where the subscript “o” stands for the reference temperature which is taken as 22  C in the present study. To model a plate with infinite width a periodicity boundary condition is imposed at the right and the left sides of the domain and the aspect ratio (A) is defined as the ratio of the width of the plate to the distance between the plates (A ¼ W/H). The aspect ratio is set to 2 in the current study. The governing equations are re-written in dimensionless form as:

(21)

The dimensionless horizontal and vertical velocities are converted to:

U ¼

a

vJ ; vy

(22)

1

0.15

0.65

0.5

1  ! rCp p pffiffiffiffiffiffiffiffiffiffiffi Pr Ra ð1  4Þ þ 4  rCp f  

v vq v vq k k þ vx vx vy vy

0.5

0.4 0.5

0

b

5 0.8 0.95

0

0.5

1

1.5

2

1

0.05 0.1

¼

5

0.35

55 0.

45 0.

5

where the dimensionless numbers are

0.5 0.7

v2 J v2 J þ 2 ¼ U vx2 vy

0.05

5

rm



v vJ v vJ q q  vx vy vy vx

nfo afo

0.65




(20)

n fo a fo

25 0.



Pr ¼

g bðTH  TC ÞH3

5




Ra ¼

0.2

pffiffiffiffiffi   



v vJ v vJ v vU Pr  U U x  ¼ pffiffiffiffiffiffi
rp vx vx vy vx vx vy Ra ð1  4Þ þ 4 rm 



 bp v vU vq x þ 4 þ ð1  4Þ þ bm vx vy vy ! pffiffiffiffiffi 2 v x vV Pr  þ pffiffiffiffiffiffi
rp vxvy vy Ra ð1  4Þ þ 4 rm !
pffiffiffiffiffi  Pr v2 x v2 x vU vV  þ pffiffiffiffiffiffi
 þ rp vy vx vx2 vy2 Ra ð1  4Þ þ 4 rm pffiffiffiffiffi
 vx vU vx vU Pr  ð17Þ þ pffiffiffiffiffiffi
þ rp vx vx vy vy Ra ð1  4Þ þ 4

Fig. 3. Code validation against the work of Ghasemi and Aminossadati [17] using Cuwater nanofluid.

0. 45

u j x0 y0 x ¼ ; y ¼ ; U ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; J ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; H H gbðTH  TC ÞH =H H g bðTH  TC ÞH u v T  TC ; U ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; V ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; q ¼ TH  TC g bðTH  TC ÞH g bðTH  TC ÞH a m k k ¼ nf ; z ¼ nf ; x ¼ nf ; ð16Þ afo mfo kfo

5

The following dimensionless variables are introduced:

0.7

mf ¼ 2:414  105  10247:8=ðT140Þ

ð18Þ

0.85

(19)

0

0.95

0

0.5

1

1.5 5

2 4

Fig. 4. Isotherms for pure water (Pr ¼ 6.57) (a) Ra ¼ 1 10 , (b) Ra ¼ 1  10 .

1724

V ¼ 

E. Abu-Nada / International Journal of Thermal Sciences 50 (2011) 1720e1730

vJ : vx

The dimensionless boundary conditions can be written as: 1 On the top plate: J ¼ 0, U ¼ e (v2J/vy2), q ¼ 0 2 On the bottom plate: J ¼ 0, U ¼ e (v2J/vy2), q ¼ 1

(23)

3 On the left and right boundaries, a periodic boundary condition is used: J0,j ¼ JN,j, U0,j ¼ UN,j, and q0,j ¼ qN,j, where 0 and N represent the left and the right boundary of the computational domain, respectively (see Fig. 1). Also, the symbol j stands for the grid location along the left or right boundary.

Fig. 5. Nusselt number distribution along the heated surface (right column), Non-dimensional Nusselt along the heated surface (left column), (a) Ra ¼ 106, (b) Ra ¼ 105, (c) Ra ¼ 104 (d) Ra ¼ 103.

E. Abu-Nada / International Journal of Thermal Sciences 50 (2011) 1720e1730

3. Numerical implementation Equations (17) through (19), absorbing the variable properties given in Eqs. (9) and (13) along with the corresponding boundary conditions are solved using a finite volume method [33,34]. The diffusion term in the vorticity and energy equations is approximated by a second-order central difference scheme which gives a stable solution. Furthermore, a second-order upwind differencing scheme is adopted for the convective terms. The algebraic finite volume equations for the vorticity and energy equations are written into the following form:

0

0.5

0.7

0.4

2

5 0.4

5

(26)

0.55

0.5 0.8

0

c

0.2 0.4

0.6

0.5

5

0.95

0

0.5

1

1.5

1

0.1

2

0.25 0.35

0.65 0.6 0.7

0.5

0 .5

5

(27)

qw TH  TC

d

0

0.5

knf kf

vq vy

(30)

0.65

0.05 0.3

0.6

0.5

5

0

(31)

0.2

0.5

0

NuðxÞdx

2

0.55

0.7

Nuavg ¼

1.5

0.15

where (knf/kf) is calculated using Eq. (9) when using the Chon et al. model [26] and calculated from Eq. (12) when using the MG model. Finally, the average Nusselt number is determined from

Z2

1

1

Substituting Eqs. (28) and (29) into Eq. (27), and using the dimensionless quantities, the local Nusselt number along the bottom plate can be written as

!

0.4

0. 8

0

(29)

0.4

0.9

(28)

qw knf ¼  vT=vy

5

0.5

The thermal conductivity of the nanofluid is expressed as

Nu ¼ 

1.5 0.1

The heat transfer coefficient is computed from

h ¼

1

1

After solving for J, U, and q, more useful quantities for engineering applications are obtained. For example, the Nusselt number can be expressed as

hH Nu ¼ kf

0. 45

5

(25)



2ðDyÞ2

0.5

0.95

0

jfnþ1 j

8J1;j  J2;j

0.3

0.8

0.55

0.7

U¼ 

0.6

0.65

where e is the tolerance; M and N are the number of grid points in the x and y directions, respectively. An accurate representation of vorticity at the surface is the most critical step in the stream function-vorticity formulation. A secondorder accurate formula is used for the vorticity boundary condition. For example, the vorticity at the bottom wall is expressed as



0.05 0.2 0.7

j¼1 i¼1

< 10

6

(32)

1

0.5

b

jfnþ1  fn j

j¼ ¼N PM i P

a

Nuð4Þ Nuð4 ¼ 0Þ

75 0.

e ¼

j¼1 i¼1

*

Nuð4Þ ¼

(24)

where P, W, E, N, S denote cell location, west face of the control volume, east face of the control volume, north face of the control volume and south face of the control volume respectively. Similar expression is also used for the kinematics equation where only central difference is used for the discritization at the cell P of the control volume. The resulted algebraic equations are solved using successive over/under relaxation method. Successive under relaxation was used due to the non-linear nature of the governing equations especially for the vorticity equation at high Rayleigh numbers. The convergence criterion is defined by the following expression: j¼ ¼N PM i P

To evaluate Eq. (31), a 1/3 Simpson’s rule of integration is implemented. For convenience, a normalized local and average Nusselt numbers are defined as the ratio of Nusselt number at any volume fraction of nanoparticles to that of pure water that is

0. 65

aP fP ¼ aE fE þ aW fW þ aN fN þ aS fS þ b

1725

45 0.

5 0.8 0.95

0

0.5

1

1.5

2

Fig. 6. Isotherms of CuO-nanofluid at Ra ¼ 1 105, (a) 4 ¼ 9% (b) 4 ¼ 7% (c) 4 ¼ 5% (d) 4 ¼ 3%.

1726

E. Abu-Nada / International Journal of Thermal Sciences 50 (2011) 1720e1730

*

Nuavg ð4Þ ¼

Nuavg ð4Þ Nuavg ð4 ¼ 0Þ

(33)

The average Nusselt number is used as an indicator of heat transfer enhancement where an increase in Nusselt number corresponds to an enhancement in heat transfer.

a

1 0.05

0.1 0.2

To check the sensitivity of the code to the grid size, a grid independence study was performed using CuO-water nanofluid with 4 ¼ 9%. It was confirmed that the grid size (51  51) ensures a grid independent solution as portrayed by Table 2. The present numerical solution was further validated for the case of nanofluid by comparing the current code results against the work Ghasemi and Aminossadati [17] for the range of 103 < Ra < 105 as shown in Fig. 3. The nanofluid used for validation is the Cu-Water nanofluid as used by Ghasemi and Aminossadati [17]. As shown from Fig. 3 a good agreement is observed.

0.3

4. Results and discussion

0.4 0.5 0.65

0.5

0.75 0.8

0.7

0.85

0.9 0.95

0

b

0

0.5

1

1.5

1

2

0.05

0.1 0.2

0.4

0.7

0.85

3 0.

0.5

6 0.

0.8

75 0.

0.5

0.95

0

c

0

0.5

1

1.5

1

2

0.05

0.1 0. 2

0.3 5 0.4

0. 55

0.65

5

0.75

0.5

0.85

0

d

The range of Rayleigh number and volume fraction of nanoparticles are Ra ¼ 103e106 and 0  4  9%, respectively. The top plate temperature is maintained at the reference temperature i.e., at 22  C, whereas the temperature difference between the bottom and top plates is fixed to 30  C. The Prandtl number, at the reference temperature, is calculated as 6.57. The RB thermo convection problem is driven by buoyancy effects when a fluid is confined between two horizontal (hot and cold) plates having a temperature difference DT. For pure fluid, if the Ra number exceeds 1707, then thermo convective fluid motion is initiated between the plates and the heat transfer is no longer dominated by conduction. This convective motion develops a thermal plume with two adjacent fluid rolls rotating in opposite directions. The strength of the convection currents increases with Rayleigh number and the shape of the plume becomes very clear as the Rayleigh number increases. Fig. 4 presents isotherms for the RB convection problem for the pure fluid case. As shown from these isotherms the basic feature of RB problem are depicted accurately, such as the appearance of the thermal plume and the two thermal boundary layers at the top and the bottom plates. Besides, it is shown that next to the top and bottom plates the temperature isotherms are almost horizontal, which demonstrates the dominance of the conduction heat transfer whereas in the region between the plates the temperature isotherms are no longer horizontal due to the dominance of convection. Fig. 5 presents Nusselt number (left column) and normalized Nusselt number (right column) variation along the bottom hot surface using various volume fractions of nanoparticles. For 4 > 1, overall observation indicates that for Ra  104 an increase in the volume fraction of nanoparticles leads to a reduction in Nusselt number. However, for this range some minor enhancement is observed in the plume region at some volume fraction of nanoparticles. Also, it is observed that for 4 ¼ 1 some enhancement in

0.95

0

0.5

1

1.5

1

2 0.05

0.1

5

0.55

5

0. 45

0.35

0.65

25 0.

0.7

0.5

0.85

0.95

0

0

0.5

1

1.5

2

Fig. 7. Isotherms of CuO-nanofluid at Ra ¼ 1  104, (a) 4 ¼ 9% (b) 4 ¼ 7% (c) 4 ¼ 5% (d) 4 ¼ 3%.

Fig. 8. Effects of nanoparticles volume fraction on the y-component of velocity at y ¼ 0.5, Ra ¼ 105.

E. Abu-Nada / International Journal of Thermal Sciences 50 (2011) 1720e1730

heat transfer is taking place. In general, the influence of nanoparticles has two opposite effects on the Nusselt number: a positive effect that is determined by the presence of high thermal conductivity nanoparticles, and an adverse effect promoted by the high level of viscosity experienced due to the existence of nanoparticles.

a

In effect, the heat transfer in RB natural convection is dominated by convection at high Rayleigh numbers while at low Rayleigh numbers is dominated by conduction. Therefore, for Ra  104, the heat transfer is dominated by convection and the presence of nanoparticles raise the nanofluid viscosity, which deteriorates

1

0.02

0.5

0

b

1727

0

-0.0 2

0.5

1

1.5

2

1

-0.028

0.028

0.5

0

c

0

0.5

1

1.5

2

1

0.0 36

-0.036

0.5

0

d

0

0.5

1

1.5

2

1

0.0 4

-0.04

0.5

0

e

0

0.5

1

1.5

2

1

0

-0.048

0.048

0.5

0

0.5

1

1.5

2

Fig. 9. Streamlines of CuO-nanofluid at Ra ¼ 1  10 , (a) 4 ¼ 9% (b) 4 ¼ 7% (c) 4 ¼ 5% (d) 4 ¼ 3% (e) 4 ¼ 0%. 5

Fig. 10. Streamlines of CuO-nanofluid at Ra ¼ 1  104, (a) 4 ¼ 9% (b) 4 ¼ 7% (c) 4 ¼ 5% (d) 4 ¼ 3% (f) 4 ¼ 0%.

1728

E. Abu-Nada / International Journal of Thermal Sciences 50 (2011) 1720e1730

a

6

1.4

b

1.3

5

1.2 1.1

4

Nuavg*

Nuavg

1

3

2

0.9 0.8 0.7 0.6

1 0.5 0.4

0 0

0.02

0.04

0.06

0.08

0

0.1

0.02

0.04

0.06

0.08

0.1

Fig. 11. Effect of volume fraction of nanoparticles and Rayleigh number on average Nusselt number (a) Average Nusselt number (b) Normalized average Nusselt number.

convection and consequently reduces the temperature gradient and the Nusselt number at the bottom heated surface. Also, this increase in nanofluid viscosity is accompanied by an increase in thermal boundary layer thickness outside the plume and this increase is interpreted as a reduction in temperature gradients. This influence of high viscosity of nanofluid is accompanied by some augmentation in heat transfer due to the high thermal conductivity of nanoparticles but, such enhancement is small compared to the drop promoted by viscosity. This is related to the deceleration of convection currents next to the bottom heated surface which reduces the role of Brownian motion in enhancing thermal conductivity because thermal conductivity of nanofluid is inversely proportional to viscosity (with the proportionality power of 1.2321) as portrayed in Eq. (9). On the other hand, for 4 ¼ 1, the role of viscosity is less significant and the adverse effect of viscosity is less than that of the enhancement brought by the thermal conductivity,

b

1.1

1.1

1

1

0.9

0.9

Nu*avg

Nu*avg

a

which leads to enhancement in Nusselt number. This is similar to what observed experimentally by some researchers at low volume fraction of nanoparticles such as Ho et al. [11]. The previous discussion, with the help of Eq. (30), is helpful in explaining the partial enhancement observed in the plume region for Ra  105. Equation (30) shows that the Nusselt number is influenced by two parameters, which are the temperature gradient at the bottom hot surface and the thermal conductivity ratio. It is very interesting to see from Figs. 6 and 7 how the plume is influenced by increasing the volume fraction for nanoparticles where for the case of 105 (as shown in Fig. 6) the plume strength (or plume height) decreases. Also, for the case of Ra ¼ 104, the plume strength drops severely and almost disappears for the case of 4 ¼ 9 (see Fig. 6). The effect of volume fraction of nanoparticles on the plume behavior is related to the increased value of viscosity, which causes the fluid to become more viscous and the velocity to attenuate

0.8 0.7

Variable

0.6

0.8 0.7

0.5

Variable

0.6

To=22

To=22

0.5

To=37

0.4

To=37

0.4

0

0.02

0.04

0.06

0.08

0.1

0

0.02

0.04

ϕ

0.06

0.08

0.1

ϕ

c

1.1 1

Nu*avg

0.9 0.8 0.7

Variable

0.6

To=22

0.5

To=37

0.4 0

0.02

0.04

0.06

0.08

0.1

ϕ Fig. 12. Comparison between estimating the viscosity and thermal conductivity as variable with temperature and those properties evaluated at reference temperatures (cold temperature To ¼ 22 and average temperature To ¼ 37) (a) Ra ¼ 106 (b) Ra ¼ 105 (c) Ra ¼ 104.

E. Abu-Nada / International Journal of Thermal Sciences 50 (2011) 1720e1730

accordingly (see Fig. 8). This leads to a reduction in convection (reflected by shorter plume height). However, even though the plume height is decreased, but such a decrease is considered small as depicted in Fig. 6 for the case of Ra ¼ 105. In effect the temperature gradients will not be affected significantly and therefore the role of thermal conductivity enhancement is considered in the same order of the temperature gradient and from Eq. (30) the Nusselt number increases in certain regions along the hot plate which explains the partial enhancement in the plume region. However, for the case of Ra ¼ 104 and 4 ¼ 9% the plume disappearance corresponds to a severe reduction in temperature gradient and therefore the enhancement brought by high thermal conductivity nanoparticles will not balance the severe drop in temperature gradient and therefore a reduction in Nusselt number is observed in the plume region. Looking at Fig. 5(d), it is observed that for Ra ¼ 103, an enhancement in heat transfer is constantly observed with steady value and this is attributed to the fact that for this low Rayleigh number the RB convection is absent and therefore the heat transfer is dominated by conduction. Consequently, there is no influence of viscosity and the addition of high thermal conductivity nanoparticles will lead to an enhancement in heat transfer. Figs. 9 and 10 portrays the streamlines for the case of Ra ¼ 105 and 104, respectively. It is evident from the streamlines the existence of the two circulations RB rolls and it is clear that by elevating the volume fraction of nanoparticles the maximum strength of streamlines contours is attenuated (i.e., circulation rolls strength) due to the higher viscosity of the nanofluids as mentioned earlier.

a

1729

Fig. 11 illustrates the average Nusselt number and the normalized average Nusselt number along the heated surface. It is shown that for the range of Ra ¼ 104e106, a decrease in Nusselt number occurs for an increase in volume fraction of nanoparticles. It is also worth mentioning that the case of Ra ¼ 104 experiences more deterioration in Nusselt number when compared to Ra ¼ 105 and 106 cases. In fact, for Ra ¼ 104, the inertia forces are smaller than those of Ra ¼ 105 and 106. This causes the adverse effect of nanoparticles to become more severe at Ra ¼ 104, which causes more reduction in Nusselt number compared to higher Rayleigh numbers. On the other hand, for low Rayleigh numbers, i.e., Ra ¼ 103, the heat transfer is only due to conduction. Hence, by adding more nanoparticles, the conduction is enhanced due to primarily the high thermal conductivity of nanoparticles, and accordingly the heat transfer is enhanced. Fig. 12 shows the significance of the effect of temperature dependence of the thermal conductivity and viscosity. This is done by comparing the results obtained by using variable properties of nanofluid (i.e., variable viscosity and variable thermal conductivity with temperature) with those based on constant property simulations. The same equations used for variable thermal conductivity and viscosity (given by Eqs. (9) and (13) respectively) are still used for the constant property, but are evaluated at constant temperature reference conditions. Two sets of reference conditions are used one at the cold temperature condition (i.e., at 22  C) and the other at the mean temperature of the hot and the cold enclosure walls (i.e., at 37  C). It can be shown from the figure that there is deviation between the simulations obtained using variable and constant

b

1.3

1.3

1.2 1.2

1.1 1.1

1

Nu*avg

Nu*avg

1

0.9 0.8

0.8 MG & Brinkman

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Chon et al.[26] & Brinkman

MG + Nguyen et al. [27]

0.6

MG & Brinkman

0.7

Chon et al. [26] & Brinkman

MG + Nguyen et al. [27]

0.6

Chon et al. [26] & Nguyen et al. [27]

Chon et al. [26] & Nguyen et al. [27]

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0.5 0

c

0.9

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0.04

ϕ

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0

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ϕ

d

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1.2 1.25

1.1

1.2

1

1.15

Nu*avg

Nu*avg

0.9 0.8

1.1 1.05

0.7 MG & Brinkman

0.6

Chon et al. [26] & Brinkman

MG + Nguyen et al. [27]

0.5

MG & Brinkman

1

Chon et al. [26] & Brinkman

MG + Nguyen et al. [27]

0.95

Chon et al. [26] & Nguyen et al. [27]

Chon et al. [26] & Nguyen et al. [27]

0.9

0.4 0

0.02

0.04

0.06

ϕ

0.08

0.1

0

0.02

0.04

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0.08

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ϕ

Fig. 13. Effect of the various conductivity and viscosity combinations on the Nusselt number, CuO-nanofluid (a) Ra ¼ 106 (b) Ra ¼ 105 (c) Ra ¼ 104 (d) Ra ¼ 103.

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E. Abu-Nada / International Journal of Thermal Sciences 50 (2011) 1720e1730

temperature (especially the case of cold temperature condition). However, the same trend is still observed using the both approaches and generally the evaluation of properties at cold temperature increases the viscosity and accordingly reduces the heat transfer. Thus, the use of constant properties over estimates the reduction in heat transfer which means variable properties gives more realistic results. Also, this figure shows the heat transfer enhancement of nanofluids is more sensitive to volume fraction of nanoparticles than the temperature effects (note that temperature effects are pronounced as Brownian effects as given by Eqs. (9)e(11)). Thus, the temperature influence is small compared to the high viscosity brought by the presence of high concentration of nanoparticles. Fig. 13 presents an assessment of the different models used for thermal conductivity and viscosity on the average Nusselt number of CuO-water nanofluid. As shown from the figure, four different combinations of viscosity and thermal conductivity models are used. Fig. 12(aec) shows that the difference between average Nusselt number calculated using the MG model and the Chon et al. model is relatively minute. However, the difference in Nusselt number when using the Brinkman model and the Nguyen et al. data is substantial. This proves that the effect of the thermal conductivity models is less important than the viscosity models at high Rayleigh number. Accordingly, the Nguyen et al. data gives completely different prediction of Nusselt number compared to the Brinkman model. Therefore, the Brinkman model is shown to underestimate the viscosity of the nanofluid which leads to an enhancement in heat transfer, which contradicts with the experimental measurements reported in literature. This explains the enhancement in heat transfer observed by previous researchers in literature who used the Brinkman model [14e17,19]. On the other hand for low Rayleigh number, i.e., Ra ¼ 103 the effect of viscosity model is absent due to the absence of convection. Also, Fig. 12 shows that the difference between the Chon et al. model and the MG model is more distinct for Ra ¼ 103 especially for high volume fractions of nanoparticles. It can be concluded that the MG model over estimates the enhancement in heat transfer compared to the Chon et al. model at high volume fractions of nanoparticles. 5. Conclusions For the Rayleigh-Bénard convection problem, the results showed deterioration in heat transfer at high Rayleigh number by increasing the volume fraction of nanoparticles. However, at low Rayleigh number, an enhancement in heat transfer was observed due to the presence of nanoparticles. The temperature dependent viscosity and thermal conductivity models were compared to the temperature independent models namely the Maxwell-Garnett model (MG) and the Brinkman model. It was observed that for high Rayleigh number (Ra  104) the heat transfer number was mainly affected by the viscosity model employed and the heat transfer predictions were insensitive to the thermal conductivity employed. For Ra ¼ 103, the MG model over predicted the enhancement in heat transfer when compared to the Chon et al. model. References [1] G. De Vahl Davis, I.P. Jones, Natural convection in a square cavity: a bench mark numerical solution, Int. J. Numer. Meth. Fluid. 3 (1983) 227e248. [2] G. Barakos, E. Mistoulis, Natural convection flow in a square cavity revisited: laminar and turbulent models with wall functions, Int. J. Numer. Meth. Heat Fluid Flow 18 (1994) 695e719. [3] T. Fusegi, J.M. Hyun, K. Kuwahara, B. Farouk, A numerical study of threedimensional natural convection in a differentially heated cubical enclosure, Int. J. Heat Mass Transf. 34 (1991) 1543e1557.

[4] R.J. Krane, J. Jessee, Some detailed field measurements for a natural convection flow in a vertical square enclosure, in: 1st ASME-JSME Thermal Engineering Joint Conference, vol. 1, 1983, pp. 323e329. [5] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: D.A. Siginer, H.P. Wang, (Eds.), Developments and Applications of NonNewtonian Flows, FED-vol. 231, MD-vol. 66, (1995) 99-105. [6] W. Daungthongsuk, S. Wongwises, A critical review of convective heat transfer of nanofluids, Renew. Sustain. Energy Rev. 11 (2007) 797e817. [7] V. Trisaksri, S. Wongwises, Critical review of heat transfer characteristics of nanofluids, Renew. Sustain. Energy Rev. 11 (2007) 512e523. [8] D. Wen, Y. Ding, Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions, Int. J. Heat Mass Transf. 47 (2004) 5181e5188. [9] N. Putra, W. Roetzel, S.K. Das, Natural convection of nano-fluids, Heat Mass Transf. 39 (2003) 775e784. [10] B.H. Chang, A.F. Mills, E. Hernandez, Natural convection of microparticles suspension in thin enclosures, Int. J. Heat Mass Transf. 51 (2008) 1332e1341. [11] C.J. Ho, W.K. Liu, Y.S. Chang, C.C. Lin, Natural convection heat transfer of alumina-water nanofluid in vertical square enclosures: an experimental study, Int. J. Therm. Sci. 49 (2010) 1345e1353. [12] A.G. Agwa Nnana, Experimental model of temperature-driven nanofluid, Asme J. Heat Transf. 129 (2007) 697e704. [13] C.H. Li, G.P. Peterson, Experimental studies of natural convection heat transfer of Al2O3/DI water nanoparticle suspensions (nanofluids), Adv. Mech. Eng. 2010 (2010) 742739. [14] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transf. 46 (2003) 3639e3653. [15] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosure filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326e1336. [16] S.M. Aminossadati, B. Ghasemi, Natural convection cooling of a localised heat source at the bottom of a nanofluid-filled enclosure, Eur. J. Mech. B/Fluids 28 (2009) 630e640. [17] B. Ghasemi, S.M. Aminossadati, Brownian motion of nanoparticles in a triangular enclosure with natural convection, Int. J. Therm. Sci. 49 (2010) 931e940. [18] J. Kim, Y.T. Kang, C.K. Choi, Analysis of convective instability and heat characteristics of nanofluids, Phys. Fluids 16 (7) (2004) 2395e2401. üt, Natural convection of water-based nanofluid in an inclined enclo[19] E.B. Ög sure with a heat source, Int. J. Therm. Sci. 48 (2009) 2063e2073. [20] K.S. Hwang, J.H. Lee, S.P. Jang, Buoyancy-driven heat transfer of water-based Al2O3 nanofluids in a rectangular cavity, Int. J. Heat Mass Transf. 50 (2007) 4003e4010. [21] A.K. Santra, S. Sen, N. Chakraborty, Study of heat transfer characteristics of copper-water nanofluid in a differentially heated square cavity with different viscosity models, J. Enhanced Heat Transf. 15 (4) (2008) 273e287. [22] C.J. Ho, M.W. Chen, Z.W. Li, Numerical simulation of natural convection of nanofluid in a square enclosure: effects due to uncertainties of viscosity and thermal conductivity, Int. J. Heat Mass Transf. 47 (2008) 4506e4516. [23] E. Abu-Nada, Effects of variable viscosity and thermal conductivity of Al2O3ewater nanofluid on heat transfer enhancement in natural convection, Int. J. Heat Fluid Flow 30 (2009) 679e690. [24] E. Abu-Nada, Effects of variable viscosity and thermal conductivity of CuOewater nanofluid on heat transfer enhancement in natural convection: mathematical model and simulation. ASME J. Heat Transfer 132 (2010) 052401. [25] E. Abu-Nada, Z. Masoud, H. Oztop, A. Campo, Effect of nanofluid variable properties on natural convection in enclosures, Int. J. Therm. 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