Effect of nanofluid variable properties on natural convection in enclosures filled with a CuO–EG–Water nanofluid

International Journal of Thermal Sciences 49 (2010) 2339e2352

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

Effect of nanofluid variable properties on natural convection in enclosures filled with a CuOeEGeWater nanofluid Eiyad Abu-Nada a, *, Ali J. Chamkha b a b

Department of Mechanical Engineering, King Faisal University, Al-Ahsa 31982, Saudi Arabia Manufacturing Engineering Department, The Public Authority for Applied Education and Training, Shuweikh 70654, Kuwait

a r t i c l e i n f o

a b s t r a c t

Article history: Received 2 May 2010 Received in revised form 3 July 2010 Accepted 7 July 2010 Available online 8 August 2010

This work focuses on the study of natural convection heat transfer characteristics in a differentiallyheated enclosure filled with a CuOeEGeWater nanofluid for different published variable thermal conductivity and variable viscosity models. The problem is given in terms of the vorticityestream function formulation and the resulting governing equations are solved numerically using an efficient finite-volume method. Comparisons with previously published work are performed and the results are found to be in good agreement. Various results for the streamline and isotherm contours as well as the local and average Nusselt numbers are presented for a wide range of Rayleigh numbers (Ra ¼ 103e105), volume fractions of nanoparticles (0  4  6%), and enclosure aspect ratios (½  A  2). Different behaviors (enhancement or deterioration) are predicted in the average Nusselt number as the volume fraction of nanoparticles increases depending on the combination of CuOeEGeWater variable thermal conductivity and viscosity models employed. In general, the effects the viscosity models are predicted to be more predominant on the behavior of the average Nusselt number than the influence of the thermal conductivity models. The enclosure aspect ratio is predicted to have significant effects on the behavior of the average Nusselt number which decreases as the enclosure aspect ratio increases. Ó 2010 Elsevier Masson SAS. All rights reserved.

Keywords: Nanofluid Variable properties Natural convection Ethylene glycol

1. Introduction Natural convection heat transfer is an important phenomenon in engineering systems due to its wide applications in electronics cooling, heat exchangers, and various thermal systems [1e4]. Enhancement of heat transfer in such systems is essential from industrial and energy saving perspectives. The low thermal conductivity of conventional heat transfer fluids, such as ethylene-glycol (EG) or water is considered a primary limitation on the performance and the compactness of thermal systems. During the last years, an innovative technique for the improvement of heat transfer using nano-scale particles dispersed in a base fluid, known as nanofluid, has been studied 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 debate on the role of nanoparticles on heat transfer enhancement in natural convection applications.

* Corresponding author. Tel.: þ966 3 580 0000; fax: þ966 3 588 7211. E-mail address: [email protected] (E. Abu-Nada). 1290-0729/$ e see front matter Ó 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2010.07.006

Examples of these controversial results are found in those reported by Khanafer et al. [8] who studied Cuewater nanofluids in a two dimensional rectangular enclosure. They reported an augmentation in heat transfer with the increment percentage of the suspended nanoparticles at any given Grashof number. Recently, Oztop and Abu-Nada [9], Aminossadati and Ghasemi [10], as well as Ghasemi and Aminossadati and [11] showed similar trend, where an enhancement in heat transfers was registered by the addition of nanoparticles. Hwang et al. [12] studied free convection using rectangular cavity heated from below (Bénard convection) with nanofluids. They used convective heat transfer empirical formulas to estimate the heat transfer coefficient of nanofluids and reported an adverse effect of nanoparticles on heat transfer in natural convection regime. However, Kim et al. [13] studied analytically convective instability of nanofluids in natural convection via RayleigheBénard (RB) convection and reported an enhancement in heat transfer coefficient convection due to the presence of nanoparticles. Santra et al. [14] studied heat transfer characteristics of copperewater nanofluid in a differentially heated square cavity by treating the nanofluid as a non-Newtonian fluid and they reported a decrease in heat transfer by increasing the volume concentration of nanoparticles for a particular Rayleigh number. Ho et al. [15] reported a numerical simulation of natural convection of nanofluid in a square

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Nomenclature A cp CRM D G H H K lf Nu Pr PrT qw Ra Re T u, v U, V W x, y x’, y’

aspect ratio (W/H) specific heat at constant pressure (kJ kg1 K1) random motion velocity of a fluid particle diameter (m) gravitational acceleration (m s2) height of the enclosure (m) local heat transfer coefficient (W m2 K1) thermal conductivity (W m1 K1) mean free path of fluid particle Nusselt number, Nu ¼ hH/kf Prandtl number for the EGeWater mixture at reference condition, Pr ¼ nf o =af o Prandtl number for the EGeWater mixture at a given temperature heat flux, (W m2) Rayleigh number, Ra ¼ g bðTH  TL ÞH 3 =nmo amo C dp Reynolds number, Re ¼ RM nf dimensional temperature ( C) dimensional x and y components of velocity (m s1) dimensionless velocities, V ¼ vH=amo , U ¼ uH=amo width of the enclosure (m) dimensionless coordinates, x ¼ x0 =H, y ¼ y0 =H dimensional coordinates (m)

enclosure considering 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 the nanofluid, so that the heat transfer across the enclosure can be found to be enhanced or deteriorated with respect to the base fluid. On the other hand, the experimental findings reported by Putra et al. [16] and by Wen and Ding [17] highlighted deterioration in heat transfer by the addition of nanoparticles. Therefore, there is still a controversy on the role of nanofluids in natural convection heat transfer and the present study is intended to show that the use of nanofluid variable properties plays a role in clarifying this controversy between experimental and numerical simulations findings. Conceptually, natural convection heat transfer is affected by nanofluid properties and specifically by nanofluid viscosity and thermal conductivity [18]. Most of the previously mentioned numerical works relied on the viscosity and thermal conductivity models that are not sensitive to fluid temperature. Recently, Namburu et al. [19] studied experimentally the effect nanoparticles concentration and nanoparticles size on nanofluids viscosity under a wide range of temperatures. The nanofluid used in their experiment was a mixture of 60:40 (by weight) Ethylene-Glycol (EG) and water mixture containing CuO nanoparticles. They infer that viscosity drops with temperature using different concentration of nanoparticles. Moreover, the combined effect of temperature, nanoparticle size, and nanoparticles volume fraction on the thermal conductivity of nanofluids was confirmed experimentally by Jang and Choi [20]. Therefore, it is evident that the dependence of nanofluid properties on temperature and volume fraction of nanoparticles is very important from physical point of view and must be taken into account in order to predict the correct contribution of nanoparticles to the heat transfer enhancement. Recently, Abu-Nada [21,22] studied the effect of variable properties of Al2O3ewater and CuOewater nanofluids on natural convection in an annular region, respectively. He found that the heat transfer was

Greek symbols a fluid thermal diffusivity (m2 s1) b thermal expansion coefficient (K1) 3 numerical tolerance 4 nanoparticle volume fraction n kinematic viscosity (m2 s1) q dimensionless temperature, q ¼ ðT  TC Þ=ðTH  TC Þ j dimensional stream function (m2 s1) J dimensionless stream function, J ¼ j=af u dimensional vorticity (s1) U dimensionless vorticity, U ¼ uH 2 =af r density (kg m3) m dynamic viscosity (N s m2) Subscripts avg average C cold f fluid H hot m mixture nf nanofluid p particle w wall

reduced by increasing the concentration of nanoparticles for high Rayleigh numbers. However, there was an enhancement in heat transfer at low Rayleigh numbers. Also, Abu-Nada et al. [23] investigated the role of nanofluid variable properties in differentially heated enclosures and found that the effect of nanofluid variable properties plays a major role in the prediction of heat transfer enhancement. Also, they found that at high Rayleigh numbers the Nusselt number was deteriorated due to the presence of nanoparticles which is consistent with the experimental observations of Putra et al. [16] and Wen and Ding [17]. However, for low Rayleigh number an enhancement in heat transfer was registered. Therefore, the scope of the current work is to further examine the sensitivity of natural convection heat transfer to variable viscosity and thermal conductivity of nanofluids in natural convection using CuOeEGeWater nanofluid. The detailed experimental results reported by Namburu et al. [19] and the Jang and Choi model [20] will be used for the nanofluid viscosity and thermal conductivity, respectively. 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 is a schematic diagram of a differentially heated enclosure. The height and the width of the enclosure are given by H and W, respectively. The aspect ratio (A) is defined as the ratio of the width of the enclosure to the height of the enclosure (A ¼ W/H). The left wall is heated and maintained at a constant temperature (TH) higher than the right cold wall temperature (TC). The fluid in the enclosure is a mixture of 60:40 (by weight) Ethylene-Glycol (EG) and water mixture containing CuO nanoparticles. This nanofluid is assumed incompressible and the flow is conceived as laminar and two-dimensional. It is idealized that EGewater mixture and nanoparticles are in thermal equilibrium and no slip occurs between the two media. The density variation in the nanofluid is approximated by the standard Boussinesq model. The viscosity and

E. Abu-Nada, A.J. Chamkha / International Journal of Thermal Sciences 49 (2010) 2339e2352

2341

k

anf ¼  nf rcp nf

y

(6)

Also, the effective density of the nanofluid is given by:

rnf ¼ ð1  4Þrm þ 4rp

(7)

The heat capacitance of the nanofluid is expressed as:



TH

g

H

rcp

TC

nf

x

Re ¼

CRM dp

nf

Vorticity




    v vj v vj 1 v vu v vu u u m m  ¼ þ rnf vx0 nf vx0 vx0 vy0 vy0 nf vy0 vy0 vx0  
vT  þ 4bp þ ð1  4Þbf g vx0

Do ¼

!


 vu vv þ rnf vy0 vx0
 1 vmnf vu vmnf vu þ : ð1Þ þ rnf vx0 vx0 vy0 vy0 þ

(2)

Kinematics

v2 J v2 J þ ¼ u vx0 2 vy0 2

vj : vy0

vj v ¼  0: vx In Eq. (2), the thermal diffusivity is written as:

;

(11)

Do lf

(12)

kB T 3pdp mf

(13)

knf km



k ¼ 0:573 eff kf

! EG

k þð1  0:573Þ eff kf

! (14) Water

where knf is the nanofluid mixture thermal conductivity and subscript m denote the mixture. In general any property of the mixture z is evaluated as

zm ¼ 0:573zEG þ ð1  0:573ÞzWater (3)

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

u ¼

(10)

where the subscript f stands for fluid and the relation given by Eq. (9) is valid for both EG and water. As mentioned earlier that the fluid is a blend of EG (60%) and water (40%) by weight as experimental data for viscosity is available for this mixture following the experiment of Namburu et al. [19]. The corresponding volume fraction of this weigh percentage is 57.3%. The thermophysical properties of EG and water at 300 K are given in Table 1. Since we are dealing with a base fluid composed of two different fluids, then following relation is assumed to determine the mixture thermal conductivity of the nanofluid


Energy



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

(9)

where

CRM ¼

v2 mnf v2 mnf  vx0 2 vy0 2


 d þ 3C 4 f PrT Re2 dp

mf rf af

PrT ¼

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 in terms of the stream functionevorticity formulation using variable properties are given as:

1

!

where C ¼ 1.8  107 is a constant and b is the thermal expansion coefficient [20]. Here PrT and Re are defined by:

Fig. 1. Schematic of the enclosure.

4 v2 mnf vv rnf vx0 vy0 vy0

(8)

The effective thermal conductivity of the nanofluid calculated by the Jang and Choi model is derived for a pure fluid and is given as [20]:

kp keff ¼ ð1  4Þ þ b4 kf kf

W

þ

  ¼ ð1  4Þ rcp m þ4 rcp p

(4)

(5)

(15)

where z stands for any fluid property such as density, specific heat, thermal expansion coefficient, thermal diffusivity, etc.

Table 1 Thermo physical properties of fluid and nanoparticles [20,23]. Physical properties

CuO

Water

EG

Cp (J/kg K) r (kg/m3) k (W/mK) b  105 (1/K) dp (nm) lf (nm)

540 6510 18.0 0.85 29 27

4179 997.1 0.613 21 0.384 0.738

2415 1114 0.252 57 0.561 0.875

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The symbol kB is the Boltzmann constant ¼ 1.3807  1023 J/K, and lf is the mean path of fluid particles. The experimental correlation for the viscosity of CuOeEGeWater nanofluid is given by Namburu et al. [19].

 log mnf ¼ AeBT

Table 2 Grid independence study.

(16)

where

A ¼ 1:837542  29:6434 þ 165:65 B ¼ 4  106 42  0:0014 þ 0:0186

(17)

The previous relation is valid of 0  4  6.12%. The viscosity models, in Eqs. (13) and (14) is expressed in centi poise and the temperature in K. In the discussion section, the results using Eqs. (13) and (14) will be compared to the Brinkman model and MG models respectively given by [9]:

mnf ¼

mm ð1  4Þ2:5

(18)

 kp þ ðn  1Þkm  ðn  1Þ km  kp 4 knf  ¼ km kp þ ðn  1Þkm þ km  kp 4

Grid size

Nuavg

21  21 31  31 41  41 51  51 61  61 71  71 81  81 91  91 101  101

4.86443 4.56411 4.40201 4.39029 4.38812 4.38931 4.39015 4.38903 4.38897

v2 J v2 J þ 2 ¼ U vx2 vy

(23)

where the dimensionless numbers are

Ra ¼ Pr ¼

(19)

g bðTH  TL ÞH3

nmo amo

nmo amo

(24)

(25)

The dimensionless horizontal and vertical velocities are converted to:

where n ¼ 3. The following dimensionless variables are introduced to nondimensionalize the governing equations given by Eqs. (1)e(3):

a

u j x0 y0 v ; J ¼ ; V ¼ ; y ¼ ; U ¼ ; amo amo =H H H amo =H2 a m u T  TC k ; k ¼ nf ; a ¼ nf ; m ¼ nf ; ð20Þ ; q ¼ U ¼ amo =H amo mmo TH  TC kmo x ¼

where the subscript “o” stands for the reference temperature which is taken as 20  C in the current study. The temperature difference between the hot and the cold walls of the enclosure is fixed in this study to 20  C. The governing equations in dimensionless form are given as:



  v vJ v vJ U U  ¼ vx vy vy vx




Pr

rp


 v vU m vx vx

ð1  4Þ þ 4 rm 


bp v vU m þ þ Ra Pr 4 bm vy vy 
 vq þ ð1  4Þ vx ! Pr v2 m vV þ rp vxvy vy ð1  4Þ þ 4 rm ! Pr v2 m v2 m þ rp vx2  vy2 ð1  4Þ þ 4

rm




vU vV  þ vy vx



þ

Pr

rp

ð1  4Þ þ 4 rm
 vm vU vm vU  þ vx vx vy vy  

v vJ v vJ q q  ¼ vx vy vy vx

b

1

 



rcp p rcp m

ð1  4Þ þ 4

ð21Þ

 

v vq v vq þ k k vx vx vy vy (22)

Fig. 2. Code Validation: (a) Comparison between present work and other published data for the temperature distribution at the vertical mid section along the width of the enclosure (Ra ¼ 105, Pr ¼ 0.7), (b) Comparison against other work in literature for natural convection in partially heated enclosure filled with CueWater nanofluid.

E. Abu-Nada, A.J. Chamkha / International Journal of Thermal Sciences 49 (2010) 2339e2352

U ¼

2343

vJ ; vy

(26)

vJ V ¼  : vx

(27)

The dimensionless boundary conditions can be written as:

v2 J 1-On the left wall i:e:; x ¼ 0; J ¼ 0; U ¼  2 ; q ¼ 1 vx 2-On the right wall i:e:; x ¼ 1; J ¼ 0; U ¼ 

v2 J ; q¼0 vx2

3-On the top and bottom wall : J ¼ 0; U ¼ 

(28)

v2 J vq ; ¼0 vy2 vy

3. Numerical implementation Fig. 3. Grid independence test showing the effect of grid size on the temperature distribution at the vertical mid section along the width of the enclosure using CuOeEGeWater nanofluid (Ra ¼ 105, 4 ¼ 6%).

Equations (21) through (23), absorbing the variable properties given in Eqs. (14), and (16) along with the corresponding boundary conditions given in Eq. (28) are solved using a finite volume method

a

b

c

Fig. 4. Nusselt number distribution along the heated surface, A ¼ 1, non-dimensional (right column), dimensional (left column), (a) Ra ¼ 105, (b) Ra ¼ 104, (c) Ra ¼ 103.

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E. Abu-Nada, A.J. Chamkha / International Journal of Thermal Sciences 49 (2010) 2339e2352

-3 -5

0.8

0. 8

0.95

-1

0. 9

a -7

5

0.75 0.7

0.65

-9

0.6

-10

0.55

0.45 0.35 0.25

b

-1 -3 -5 -7 -9 -10

0.95

15 0.

0.05

9 0. 0.85

0.8

0.75 0.65

-11 0.55

0.45

0.35 0.25

c

0.95

-1 -3 -5 -7

5

0.05

0.1

9 0. 5 0.8 8 0.

0.75

-9

0.65

-10 -11

0.55

-12 0.45 0.35 0.25

d

0.95

0.1

-1

-3 -5 -7 -9 -10 -11

9 0. 0.85

5

0.05

0.8 0.75 0.65 0.55

-12

-12.2818 0.45 0.35

0.1

5

0.05

0.25

Fig. 5. Streamlines (on the left) and isotherms (on the right) for CuOeEGeWater nanofluid at Ra ¼ 105 (a) 4 ¼ 6%, (b) 4 ¼ 4%, (c) 4 ¼ 2%, (d) 4 ¼ 0%.

E. Abu-Nada, A.J. Chamkha / International Journal of Thermal Sciences 49 (2010) 2339e2352

-0.05

0.95 0.9

-0.15 -0.25 -0.35 -0.45

0.35

0.25

0.15 0.05

0.55

0.95

b

0.65

0.75

0.85

-0.55

0.45

a

2345

-0.1 -0.3

-0.5

0.75

0.85

-0.6 -0.7 -0.8

-0.0

5

0.05

0.95 -0.2

5 -0.45 -0.65 -0.85 -0.95 -1.05

0.85 0.7 5

c

0.15

0.35

0.25

0.45

0.55

0.65

-0.9

.1 0.2 4 -0. -0.6 -0.8 -1 -1.2

0.85

0.05

0.15

0.2 5

0.3 5

5

5 0.5

0.4

0.6

5

4 -1.

5

-0

0.7

d

0.95

0.15

0.05

0.25

0.35

0.4 5

0.5 5

0.6

5

-1.15

Fig. 6. Streamlines (on the left) and isotherms (on the right) for CuOeEGeWater nanofluid at Ra ¼ 103 (a) 4 ¼ 6%, (b) 4 ¼ 4%, (c) 4 ¼ 2%, (d) 4 ¼ 0%.

2346

E. Abu-Nada, A.J. Chamkha / International Journal of Thermal Sciences 49 (2010) 2339e2352

[24,25]. The diffusion term in the vorticity and energy equations is approximated by a second-order central difference scheme which is conducive to a stable solution. Furthermore, a second order upwind differencing scheme is adopted for the convective terms. For full details of the numerical implementation, the reader is referred to Oztop and Abu-Nada [9] and Abu-Nada et al. [23]. After solving for J, U, and q, more useful quantities for engineering applications are obtained. For example, the Nusselt number can be expressed as:

Nu ¼

hH km

(29)

The heat transfer coefficient is computed from:

h ¼

qw TH  TL

(30)

The thermal conductivity of the nanofluid is expressed as:

qw knf ¼  vT=vx

(31)

Substituting Eqs. (30) and (31) into Eq. (29), and using the dimensionless quantities, the local Nusselt number along the left wall can be written as:


 k vq Nu ¼  nf km vx

(32)

nanofluid by comparing the current code results against the work of Oztop and Abu-Nada [9] and Ghasemi and Aminossadati [11] for the range of 103 < R < 105 as shown in Fig. 2(b). The nanofluid used for validation is the CueWater nanofluid as used by Oztop and Abu-Nada [9] and Ghasemi and Aminossadati [11]. As shown from Fig. 2(b) a good agreement with previous studies in literature is established. 4. Results and discussion In this section, a representative set of graphical results are presented to illustrate the influence of the various physical parameters on the heat transfer characteristics in the CuOeEGeWater nanofluid cavity. The ranges of the magnitude of the Rayleigh numbers, volume fractions of nanoparticles, and the aspect ratios used in this study are Ra ¼ 103e105, 0  4  6%, and 0.5  A  2, respectively. It is worth mentioning that the right wall temperature of the enclosure is fixed to the reference temperature i.e., at 20  C, whereas the difference between the hot and the cold wall is fixed to 20  C, i.e., the hot wall temperature is set to 40  C. The Prandtl number at the reference temperature is calculated as 47.9. Fig. 4 displays the dimensional and non-dimension local Nusselt numbers (Nu and Nu*) distributions along the heated surface for various values of nanoparticles volume fractions (4 ¼ 0%, 2%, 4%, 6%)

100

a

80

where (knf/km) is calculated using Eq. (14). Finally, the average Nusselt number is determined from:

60

Z1 Nuavg ¼

40

NuðyÞdy

(33)

20

0

V

0 0

To evaluate Eq. (33), a 1/3 Simpson’s rule of integration is implemented. For convenience, a normalized average Nusselt number is defined as the ratio of Nusselt number at any volume fraction of nanoparticles to that of pure EGeWater that is:

Nuavg ð4Þ *Nuavg ð4Þ ¼ Nuavg ð4 ¼ 0Þ

0.2

0.4

0.6

0.8

1

-20 -40 -60

(34)

-80 -100 x

b

6

4

2

V

The normalized 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. An extensive mesh testing procedure was conducted to guarantee a grid independent solution. Seven different mesh combinations were explored for the case of Ra ¼ 105, Pr ¼ 0.7. The present code was tested for grid independence by calculating the average Nusselt number on the left wall. In harmony with this, it was found that a grid size of 51  51 ensures a grid independent solution. To check the sensitivity of the code to the grid size for the nanofluid case, another grid independence study was performed using CuOeEGeWater nanofluid. It was confirmed that the same grid size (51  51) ensures a grid independent solution as portrayed by Table 2. Moreover, Fig. 3 shows a comparison using five different grids for the temperature distribution along the x-axis at the mid height of the enclosure. As shown from Table 2 and Fig. 3, the grid size (51  51) ensures a grid independent solution. The present numerical solution was validated by comparing the present code results for Ra ¼ 105 and Pr ¼ 0.70 with the experimental results of Krane and Jessee [4], and the numerical simulation of Khanafer et al. [8]. It is evident that the outcome of present code is in excellent agreement with the work reported in the literature as reflected in Fig. 2(a). Another validation was carried out for the case of

0 0

0.2

0.4

0.6

0.8

1

-2

-4

-6 x Fig. 7. Variation of y-component of velocity for CuOeEGeWater nanofluid, A ¼ 1, y ¼ 0.5, Ra ¼ 105, (b) Ra ¼ 103.

E. Abu-Nada, A.J. Chamkha / International Journal of Thermal Sciences 49 (2010) 2339e2352

a

b

1.1

2347

1.1

1 1

Nuavg*

Nuavg*

0.9 0.9

0.8

Ra = 1E5

0.8

Ra = 1E5

Ra = 1E4

Ra = 1E4

0.7

Ra = 1E3

Ra = 1E3

0.7

0.6 0

0.02

c

0.04

0.06

0

0.02

0.04

0.06

0.04

0.06

1.3

1.2

Nuavg*

1.1

1 Ra = 1E5 Ra = 1E4

0.9

Ra = 1E3

0.8 0

0.02

0.04

0.06

Fig. 8. Non-dimensional Average Nusselt number, (a) A ¼ 1, (b) A ¼ 2, (c) A ¼ 0.5.

a

6

b

5

5 Ra = 1E5

4

Ra = 1E5

4

Ra = 1E4

Nua vg

Ra = 1E3

3

Ra = 1E3

2

1

1

0

Ra = 1E4

3

2

0 0

0.02

c

0.04

0.06

0

0.02

6 5 4

Nua vg

Nua vg

6

3 2 Ra = 1E5

1

Ra = 1E4 Ra = 1E3

0 0

0.02

0.04

0.06

Fig. 9. Average Nusselt number, (a) A ¼ 1 (b), A ¼ 2 (c) A ¼ 0.5.

2348

E. Abu-Nada, A.J. Chamkha / International Journal of Thermal Sciences 49 (2010) 2339e2352

and Rayleigh numbers (Ra ¼ 103, 104, 105) for a square cavity (A ¼ 1). For the cases of Ra ¼ 105 and Ra ¼ 104, it is predicted that an increase in the volume fraction of CuOeEGeWater nanoparticles results in reductions in the dimensional local Nusselt number Nu in most of the entire range 0 < y < 1. The locations where enhancement is taking place are clearly demonstrated by looking at the non-dimensional local Nusselt number Nu* where Nu* increases in the region y > 0.95 for Ra ¼ 105 and in the region y > 0.85 for Ra ¼ 104. However, for Ra ¼ 103 both the dimensional local Nusselt number Nu and the nondimensional local Nusselt number Nu* decrease in the lower half of the heated wall of the cavity while they increase in the upper half of the heated wall as the volume fraction of nanoparticles 4 increases. However, the distributions of Nu show a decreasing trend with the vertical distance y for all values of 4 while the distributions of Nu* show an increasing trend with the vertical distance y for values of 4 > 0% and show a constant behavior along the heated wall for 4 ¼ 0%. A similar type of behavior of the distributions of Nu was observed for CuOewater nanofluids as discussed previously by Abu-Nada et al. [23].

b

1.1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5 0

c

1.1

1

Nu*

Nu*

a

Fig. 5 shows the contour maps for the streamlines and isotherms for CuOeEGeWater nanofluid filled in a square cavity at Ra ¼ 105 for different values of the nanoparticles volume fraction 4. The streamlines contours show reductions in the values of the maximum stream function value and more confinement of the streamlines towards the core of the cavity as 4 increases. On the other hand, the isotherms are crowded close to the vertical walls and are more uniformly distributed in the core region of the cavity. Also, the isotherms contours illustrate the sensitivity of the thermal boundary layer thickness due increases in the volume fraction of nanoparticles which is related to the increased viscosity at high volume fraction of nanoparticles. High values of 4 cause the fluid to become more viscous which causes the movement of the fluid in the cavity to slow down resulting in a reduced convection effect. The reductions in the flow motion and convection effect tend to increases the thermal boundary layer thickness. The growth in the thermal boundary layer thickness is responsible for the lesser temperature gradients at the heated surface which lowers Nusselt number accordingly.

0.2

0.4

Y

0.6

0.8

1

d

1.1

0

0.2

0.4

0

0.2

0.4

0

0.2

0.4

Y

0.6

0.8

1

0.6

0.8

1

0.6

0.8

1

2.3 2.1

1

1.9 1.7

Nu*

Nu*

0.9 0.8 0.7

1.5 1.3 1.1 0.9

0.6

0.7 0.5

0.5 0

e

0.2

0.4

Y

0.6

0.8

1

f

1.5

1.3

1.3

1.2 1.1

Nu*

1.1

Nu*

1.4

1.4 1.2

Y

1 0.9

1 0.9 0.8

0.8 0.7

0.7

0.6

0.6

0.5

0.5 0

0.2

0.4

Y

0.6

0.8

1

Y

Fig. 10. Non-dimensional Nusselt number distribution along the heated surface (a) Ra ¼ 105, A ¼ 2, (b) Ra ¼ 105, A ¼ 0.5, (c) Ra ¼ 104, A ¼ 2, (d) Ra ¼ 104, A ¼ 0.5, (e) Ra ¼ 103, A ¼ 2, (f) Ra ¼ 103, A ¼ 0.5.

E. Abu-Nada, A.J. Chamkha / International Journal of Thermal Sciences 49 (2010) 2339e2352

In Fig. 6, the contour maps for the streamlines and isotherms for CuOeEGeWater nanofluid filled in a square cavity is presented at Ra ¼ 103 for different values of the nanoparticles volume fraction 4. By comparison with Fig. 5, it is clearly seen that the convection effect is less as depicted by the smaller value of the maximum stream function value and the flow is characterized by a single circular cell situated in the center of the cavity. Also, for the case of Ra ¼ 103, the addition of nanoparticles slows down the motion of the fluid and causes the thermal boundary layer thickness to increase for y < 1/2 and to decrease for y > 1/2. This is demonstrated by the behavior of the isotherms which start to straighten up near the heated wall as the volume fraction of nanoparticles increases. Furthermore, for relatively high volume fraction of nanoparticles (4 ¼ 6%) the isotherms become almost parallel to the heated wall. Actually, for 4 ¼ 6% and y > 1/2, the isotherms become closer to the wall and for y < 1/2 the isotherms spread away from the heated wall. The isotherms exhibit a trend almost similar to conduction in solids. This behavior leads to an enhancement in heat transfer for y > 1/2 and debilitation in the Nusselt number for y < 1/2. Fig. 7 presents typical profiles for the y-component of velocity V for CuOeEGeWater nanofluid at mid section of the square cavity (A ¼ 1 and y ¼ 0.5) for Ra ¼ 105 and Ra ¼ 103 and various values of the nanoparticles volume fraction 4. As expected, the vertical velocity increases close to the heated wall of the cavity due to the thermal buoyancy effects and decreases close to the cooled wall cavity. Also, as the nanoparticles volume fraction 4 increases, the magnitude of the vertical velocity decreases. It is concluded that the presence of the nanoparticles tends to slow down the movement of the fluid in the cavity. Also, it is shown

a

for both values of Rayleigh numbers that at the mid of the cavity the effect of nanoparticles is less pronounced due to the low level of velocity. Fig. 8 illustrates the influence of the Rayleigh number Ra and the nanoparticles volume fraction 4 on the non-dimensional average Nusselt number along the heated surface Nu*avg for different aspect ratios of the enclosure A. It is clearly observed that for A ¼ 2, the addition of nanoparticles cause the values of Nu*avg to decrease and the rate of decrease depends on the value of the Rayleigh number. For example, for Ra ¼ 104, the addition of 6% nanoparticles by volume, the non-dimensional average Nusselt number decreases by about 35%. For Ra ¼ 105, the decrease in the value of Nu*avg for 4 ¼ 6% is about 25%. However, for Ra ¼ 103, the value of Nu*avg decreases for 4 ¼ 6% about 5%. It is also evident that the values of Nu*avg decrease as Ra increases from 103 to 104 and then increases as Ra increases to 105 all non-vanishing values of nanoparticle volume fraction. This means that there is a critical value of Ra for which the addition of nanoparticles causes a reverse change in the behavior of the nondimensional average Nusselt number. For A ¼ 1, the same features are predicted except that for Ra ¼ 103, the addition of nanoparticles causes enhancements in the values of the non-dimensional average Nusselt number. This is also clearly portrayed by the average Nusslet number as shown in Fig. 9. For A ¼ 0.5, a different behavior is predicted compared to the case of A ¼ 2 for which the values of Nu*avg decrease as Ra increases from 104 to 105 for almost all nonvanishing values of 4. In addition, similar to the case of A ¼ 1, for the case of 103, the non-dimensional average Nusselt number increases with increasing values of 4 and that this increase is higher for A ¼ 0.5 than for A ¼ 1. For example, for A ¼ 0.5, the enhancement in Nu*avg for

b

1.3

2349

1.2

1.2 1.1

1.1

1

1

Nuavg*

0.8

MG & Brinkman MG & Namburu et al. [18] Brinkman & Jang and Choi [19]

MG & Namburu et al. [18] Brinkman & Jang and Choi [19]

0.6

Namburu et al. [18] & Jang and Choi [19]

0.6

0.8 0.7

MG & Brinkman

0.7

Namburu et al. [18] & Jang and Choi [19]

0.5

0.5 0

0.02

0.04

c

0.06

0

0.02

0.04

1.2 1.1 1 0.9

Nuavg*

Nuavg*

0.9

0.9

0.8 0.7

MG & Brinkman MG & Namburu et al. [18] Brinkman & Jang and Choi [19]

0.6

Namburu et al. [18] & Jang and Choi [19]

0.5 0

0.02

0.04

0.06

Fig. 11. Effect of the various conductivity and viscosity models on the Nusselt number, Ra ¼ 105, (a) A ¼ 2, (b) A ¼ 1, (c) A ¼ 0.5.

0.06

2350

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4 ¼ 6% is about 20% while it is only about 4% for A ¼ 1. This is further demonstrated by Fig. 9. Fig. 10 presents the effects of the nanoparticles volume fraction 4 on the non-dimensional Nusselt number distribution along the hot wall for different Rayleigh numbers (Ra ¼ 103, 104, 105) and two different enclosure aspect ratios (A ¼ 0.5 and A ¼ 2). In general, it is predicted that the non-dimensional Nusselt number is higher for A ¼ 0.5 than for A ¼ 2 for all values of the nanoparticles volume fraction 4 (except 4 ¼ 0%) and Rayleigh numbers. This is especially true in the lower part of the heated wall (0 < y < 0.5) for Ra ¼ 105 and in the entire range 0 < y < 1 for Ra ¼ 103 and Ra ¼ 104. For A ¼ 0.5, the most enhancement in heat transfer occurs for Ra ¼ 104 and all of the considered values of 4 compared with those corresponding to the other values of Ra. It is concluded that at high aspect ratios, the flow within the enclosure experiences deterioration in the values of the Nusselt number compared to an enclosure of lower aspect ratio. It can also be noted that enclosures with low aspect ratios would benefit more from the nanoparticles thermal conductivity enhancement especially at low Rayleigh numbers. Figs. 11e13 present comparisons between four different combinations of models for thermal conductivity and viscosity [9,18,20] on the normalized average Nusselt number for various volume fractions of CuOeEGeWater nanofluid (4) and different enclosure aspect ratios (A) at three Rayleigh numbers (Ra ¼ 105, 104, 103), respectively. It is clearly depicted from Figs. 11e13 that for all values of Ra, calculations using the Jang and Choi [20] and Brinkman models produce the highest values of the normalized average Nusselt number whereas the MG and Namburu et al. [19] models yield the lowest normalized average Nusselt number predictions for all values of 4 and A. However, from Figs. 11 and 12 for Ra ¼ 105 and

b

1.2

1.2 1.1

1

1

0.9

0.9

Nuavg*

1.1

0.8 0.7

MG & Brinkman MG & Namburu et al. [18] Brinkman and Jang and Choi [19]

Brinkman & Jang and Choi [19]

0.6

0.8 0.7

MG & Brinkman MG & Namburu et al. [18]

0.6

Namburu et al. [18] & Jang and Choi [19]

Namburu et al. [18] & Jang and Choi [19]

0.5

0.5 0

0.02

0.04

c

0.06

0

0.02

0.04

1.3 1.2 1.1 1

Nuavg*

Nuavg*

a

Ra ¼ 104, it is predicted that while the normalized average Nusselt number increases with increasing values of the nanoparticles volume fraction 4 using the MG and Brinkman and the Jang and Choi [20] and Brinkman models combination, it tends to decrease as 4 increases for the MG and Namburu et al. [19] and the Jang and Choi [20] and Namburu et al. [19] models combination. This behavior is true for all considered values of the enclosure aspect ratio A. The decreasing trend in the normalized average Nusselt number as 4 increases is associated to the Namburu et al. [19] viscosity model meaning that the effect of the thermal conductivity models is less significant than the viscosity models at high Rayleigh number. As a result, the prediction of the normalized average Nusselt number using Namburu et al. [19] is completely different than that using the Brinkman model. For low Rayleigh number (Ra ¼ 103), the same enhancement features in the normalized average Nusselt number as 4 increases are predicted upon using the combined models of MG and Brinkman and Jang and Choi [20] and Brinkman for values of the enclosure aspect ratio A as seen from Fig. 13. However, using the MG and Namburu et al. [19] and the Jang and Choi [20] and Namburu et al. [19] models combination produce totally-different predictions in the normalized average Nusselt number as 4 increases depending on the value of the aspect ratio. It is clearly seen from Fig. 13 that the normalized average Nusselt number decreases as 4 increases for A ¼ 2, decreases and then increases forming dips for A ¼ 1 and increases sharply for A ¼ 0.5 with smaller deviations from the predictions of the MG and Brinkman and Jang and Choi [20] and Brinkman models compared to the other values of A. It is worth mentioning that conclusions regarding the effect of models used for viscosity and thermal conductivity are limited to the models considered in this study.

0.9 0.8 MG & Brinkman

0.7

MG & Namburu et al. [18] Brinkman & Jang and Choi [19] Namburu et al. [18] & Jang and Choi [19]

0.6 0.5 0

0.02

0.04

0.06

Fig. 12. Effect of the various conductivity and viscosity models on the Nusselt number, Ra ¼ 104, (a) A ¼ 2, (b) A ¼ 1, (c) A ¼ 0.5.

0.06

E. Abu-Nada, A.J. Chamkha / International Journal of Thermal Sciences 49 (2010) 2339e2352

a

b

1.2

1.2 MG & Brinkman MG & Namburu et al. [18]

MG & Brinkman MG & Namburu et al. [18]

1.15

2351

Brinkman & Jang and Choi [19]

Brinkman & Jang and Choi [19]

Namburu et al. [18] & Jang and Chi [19]

1.15

Namburu et al. [18] & Jang and Choi [19]

1.1

1.1

Nuavg*

Nuavg*

1.05 1 0.95

1.05

0.9 1 0.85 0.8

0.95 0

0.02

0.04

c

0.06

0

0.02

0.04

0.06

0.04

0.06

1.25 MG & Brinkman MG & Namburu et al. [18] Brinkman & Jang and Choi [19]

1.2

Namburu et al. [18] & Jang and Choi [19]

Nua vg*

1.15

1.1

1.05

1

0.95 0

0.02

Fig. 13. Effect of the various conductivity and viscosity models on the Nusselt number, Ra ¼ 103, (a) A ¼ 2, (b) A ¼ 1, (c) A ¼ 0.5.

5. Conclusions The problem of steady natural convection heat transfer in a differentially-heated enclosure filled with a CuOeEGeWater nanofluid was investigated using different variable thermal conductivity and variable viscosity models. The governing equations were given in terms of the vorticityestream function formulation and were solved numerically using an efficient finite-volume method. Comparisons with previously published work were performed and the results were found to be in good agreement. Various representative numerical results for a wide range of Rayleigh numbers, volume fractions of nanoparticles and enclosure aspect ratios were presented. For high Rayleigh numbers, it was predicted that while the average Nusselt number increased with increasing values of the nanoparticles volume fraction using the MG and Brinkman and the Jang and Choi [20] and Brinkman models combination, it decreased as the nanoparticles volume fraction increased for the MG and Namburu et al. [19] and the Jang and Choi [20] and Namburu et al. [19] models combination. This behavior was found to be true for all considered values of the enclosure aspect ratio. The decreasing trend in the average Nusselt number as the nanoparticles volume fraction increased was related directly to the Namburu et al. [19] viscosity model which showed that the effect of the thermal conductivity models was less significant than the viscosity models at high Rayleigh number. However, at low Rayleigh numbers, the average Nusselt number based on the Namburu et al. [19] was enhanced by increasing the volume fraction of nanoparticles at low enclosure aspect ratios. For enclosures with high aspect ratios, the average Nusselt number was predicted to

experience more deterioration when compared to enclosures having low aspect ratios.

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