Natural convection in nanofluids: Are the thermophoresis and Brownian motion effects significant in nanofluid heat transfer enhancement?

International Journal of Thermal Sciences 57 (2012) 152e162

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

Natural convection in nanofluids: Are the thermophoresis and Brownian motion effects significant in nanofluid heat transfer enhancement? Zoubida Haddad a, b, Eiyad Abu-Nada c, Hakan F. Oztop a, *, Amina Mataoui b a

Department of Mechanical Engineering, Technology Faculty, Fırat University, TR-23119, Elazig, Turkey Department of Fluid Mechanics, Faculty of Physics, University of Sciences and Technology-Houari Boumediene, Algiers, Algeria c Department of Mechanical Engineering, King Faisal University, Al-Ahsa 31982, Saudi Arabia b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 1 August 2011 Received in revised form 19 January 2012 Accepted 20 January 2012 Available online 2 March 2012

Natural convection heat transfer and fluid flow of CuOeWater nanofluids is studied using the Rayleighe Bénard problem. A two component non-homogenous equilibrium model is used for the nanofluid that incorporates the effects of Brownian motion and thermophoresis. Variable thermal conductivity and variable viscosity are taken into account in this work. Finite volume method is used to solve governing equations. Results are presented by streamlines, isotherms, nanoparticle distribution, local and mean Nusselt numbers and nanoparticle profiles at top and bottom side. Comparison of two cases as absence of Brownian and thermophoresis effects and presence of Brownian and thermophoresis effects showed that higher heat transfer is formed with the presence of Brownian and thermophoresis effect. In general, by considering the role of thermophoresis and Brownian motion, an enhancement in heat transfer is observed at any volume fraction of nanoparticles. However, the enhancement is more pronounced at low volume fraction of nanoparticles and the heat transfer decreases by increasing nanoparticle volume fraction. On the other hand, by neglecting the role of thermophoresis and Brownian motion, deterioration in heat transfer is observed and this deterioration elevates by increasing the volume fraction of nanoparticles. Ó 2012 Elsevier Masson SAS. All rights reserved.

Keywords: Natural convection Brownian Thermophoresis Nanofluid

1. Introduction Enhancement of thermal conductivity of liquids is an extremely important topic from the energy efficiency point of view. The latest technique on this challenging subject is the using of addition of some particle into the base fluid. In this context, as Pioneer of these methods, Masuda [1] reported the liquid dispersions of submicron solid particles or nanoparticles, then, the term of “nanofluid” was first proposed by Choi [2]. Nanofluid becomes more attractive in recent years due to easy production methods and inexpensive price. Also, thermal conductivity of nanofluids relative to the base fluids is very high. Thus, nanofluids can be applied in many energetical systems such as cooling of nuclear systems, radiators, natural convection in enclosures. There are some review papers that show applications and detail solutions on nanofluids [3e5]. Number of experimental [6,7] and numerical studies on nanofluid is increased in recent years for different models and different

* Corresponding author. Tel.: þ90 424 237 0000x4222; fax: þ90 424 236 7064. E-mail addresses: [email protected], [email protected] (H.F. Oztop). 1290-0729/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2012.01.016

configurations. Abu-Nada et al. [8] studied on natural convection in horizontal annuli using different types of water based-nanofluids. They observed that addition of nanoparticles into base fluid enhances the heat transfer. Mahmoodi [9], Oztop and Abu-Nada [10], Abu-Nada and Oztop [11], Ögut [12], Aminossadati and Ghasemi [13], Abu-Nada et al. [14], Khanafer et al. [15] indicated that heat transfer enhances with addition of nanoparticle for constant viscosity. Number of studies on Brownian, Dufour and thermophoresis effects on natural convection in nanofluid filled enclosures is extremely limited. Among these, Buongiorno [16] investigated different slip mechanisms between nanoparticles and base fluid. He indicated that there are many slip mechanisms such as inertia, Brownian diffusion, thermophoresis, diffusiophoresis, magnus effect, fluid drainage, and gravity. He concluded that only Brownian diffusion and thermophoresis are important slip mechanisms in the absence of turbulent effects. A study has been performed by Nield and Kuznetsov [17] on the ChengeMinkowycz problem for natural convective boundarylayer flow in a porous medium saturated by a nanofluid. They included Brownian and thermophoresis effects and similarity solution is presented. In their similar work [18], they made an

Z. Haddad et al. / International Journal of Thermal Sciences 57 (2012) 152e162

Nomenclature A Cp D DB D*B DT D*T g Gr H hp jp kB k Le Nu P Pr Q Re Ra Sc T* V* U, V

aspect ratio (W/H) specific heat at constant pressure (J kg1 K1) diameter (m) dimensionless Brownian diffusion coefficient dimensional Brownian diffusion coefficient (m2/s) dimensionless thermal diffusion coefficient dimensional thermal diffusion coefficient (m2/s) gravitational acceleration (m s2) Grashof number, Gr ¼ gbf0 H 3 ðTH  TC Þ=n2fo height of the enclosure (m) specific enthalpy, (J/kg) total nanoparticle mass flux (kg/m2 s) Boltzmann constant (J/K) thermal conductivity (W/m$K) Lewis number, Le ¼ DTo =DBo Nusselt number, Nu ¼ qH/kfDT pressure (Pa) Prandtl number, Pr ¼ nfo =afo heat flux, (Wm2) Reynolds number, Re ¼ rf kB T=3pm2f lf Rayleigh number, Ra ¼ gbf0 H 3 ðTH  TC Þ=af nfo Schmidt number, Sc ¼ nfo =DBo dimensional temperature ( C) dimensional nanofluid velocity (ms1) dimensionless x- and y-components of velocity

analytical study at the onset of convection in a horizontal layer of a porous medium saturated by a nanofluid by including Brownian and thermophoresis. Again, Nield and Kuznetsov [19] presented a work on a linear stability analysis for the onset of natural convection in a horizontal nanofluid layer. The similar case is studied for natural convective boundary-layer flow of a nanofluid past a vertical plate by Kuznetsov and Nield [20]. Khan and Pop [21] published a paper on boundary-layer flow of a nanofluid past a stretching sheet as a first paper in that field. Their model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. They have taken into account the Pr number, Lewis number, Brownian motion numbers, Nb and thermophoresis number, Nt. They indicated that the reduced Nusselt number is a decreasing function of each dimensionless number and Sherwood number is an increasing function of higher Prandtl number and a decreasing function of lower Pr number for each Le, Nb and Nt numbers. Recently, Pakravan and Yaghoubi [22] worked the thermophoresis, Brownian and Dufour effect on natural convective heat transfer on nanofluids simultaneously. They showed that the effect of Dufour strongly decrease with temperature and Nusselt number of natural convection of nanofluids increases with mean temperature of mixture. The steady boundary-layer flow of a nanofluid past a moving semi-infinite flat plate in a uniform free stream is investigated by Bachok et al. [23]. In their case, the plate moves in the same or opposite directions to the free stream. They also included the effects of Brownian motion and thermophoresis parameters to the solution. Abu-Nada [24] studied the effect of variable thermal conductivity and variable viscosity of nanofluid on heat transfer enhancement in RayleigheBénard convection problem. He reported a decrease in heat transfer by increasing the volume fraction of nanoparticles for Ra > 103. 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.

x*, y* x, y

153

dimensional coordinates (m) dimensionless coordinates

Greek symbols a thermal diffusivity (m2 s1) b thermal expansion coefficient (K1) q dimensionless temperature k dimensionless thermal conductivity m dynamic viscosity (N sm2) r density (kg m3) s stress tensor (Pa) nanoparticle volumetric fraction f* f relative nanoparticle volumetric fraction U dimensionless vorticity u dimensional vorticity (s1) J dimensionless stream function j dimensional stream function (m2 s1) Subscripts avg average b bulk C cold f base fluid base fluid at reference temperature f0 H hot nf nanofluid P particle

The main aim of the present work is to discuss the effect of both thermophoresis and Brownian motion on RayleigheBénard problem. Based on author’s knowledge above literature survey, these effects are not taken into account for a mentioned problem. Thus, obtained results will be presented in this work via streamlines, isotherms, contours of nanoparticle distribution, local and average Nusselt numbers at different nanoparticle volume fractions and Rayleigh numbers. 2. Physical model Fig. 1 shows a schematic diagram of the RayleigheBénard (RB) problem. For the RB problem, the distance between the upper cold and lower hot plates is defined by H and the width of the top and bottom plates is defined by W. The plate’s 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 two-dimensional. It is idealized that

Fig. 1. Schematic of the problem with boundary conditions and coordinates.

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Z. Haddad et al. / International Journal of Thermal Sciences 57 (2012) 152e162

water and nanoparticles are in thermal equilibrium and only Brownian diffusion and thermophoresis are important slip mechanisms between the two media. The thermo-physical properties of the nanofluid are assumed to be constant and are listed in Table 1, whereas the density variation in the buoyancy force term is handled by the Boussinesq approximation. The viscosity and the thermal conductivity of the nanofluid are considered as variable properties; both vary with temperature and volume fraction of nanoparticles.

The thermal diffusivity of nanofluid is expressed as

k

anf ¼  nf rcp nf

(8)

The effective density of the nanofluid is given by






rnf ¼ 1  f* rf þ f* rp

(9)

3. Governing equations

The heat capacitance of the nanofluid is written as (Abu-Nada [24] and Khanafer et al. [15]):

The governing equations for the laminar, two-dimensional, steady state natural convection are written as: Continuity equation:



V$V * ¼ 0

1

rnf

Vp þ V$s þ g

 nf

¼




1  f*



rcp

 f

  þ f* rcp p

(10)

(1)

The effective thermal conductivity of the nanofluid calculated by the Chon et al. [25] model is given by:

(2)

!0:7476  0:3690 df kp knf ¼ 1 þ 64:7f0:7640 Pr0:9955 Re1:2321 T kf dp kf

Momentum equation:

V * $VV * ¼ 

rcp

(11)

where the stress tensor is given as [16]




*




s ¼ mnf VV þ VV

*

where PrT and Re are defined by:

t 

(3)

PrT ¼

Energy equation: The energy equation for the nanofluid can be written as [16]


rp cp 1 V $VT ¼ V k$VT * þ rnf cnf rnf cnf *

*

VT D*B Vf* $VT * þ D*T

* $VT *

!

TC* (4)

Nanoparticles conservation equation The continuity equation for the nanoparticles in the absence of any chemical reactions is given as [16]

V  $Vf* ¼ V$ D*B Vf* þ D*T

VT * TC

! (5)

Here f* is nanoparticle volume fraction, D*B is the Brownian diffusion coefficient given by the EinsteineStokes’s equation [16]:

D*B ¼

kB T * 3pmf dp

(6)

where mf is the viscosity of the fluid and dp is the nanoparticle diameter. The thermophoretic diffusion coefficient D*T as reported in [16] is given by

D*T

¼

mf rf

!

! kf f* 0:26 2kf þ kp

Re ¼

mf rf af

(12)

rf kB T

The symbol kB is the Boltzmann constant ¼ 1.3807  1023 J/K, and lf is the mean free 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. [27] 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 correlation for the dynamic viscosity of CuOeWater nanofluid is derived using available experimental data of Nguyen et al. [26]. The R2 value is 99.8% and a maximum error is 5%. The correlation for CuO nanofluids is given by:

15:937 1356:14 þ 1:238f þ T T2 f 19652:74  0:259f2  30:88  þ 0:01593f3 T T3

mCuO ðcpÞ ¼  0:6967 þ

þ 4:38206 (7)

In equation (7), kf and kp are the thermal conductivity of the fluid and particle materials, respectively. Table 1 Thermo physical properties of fluid and nanoparticles [24]. Physical properties

Fluid phase (water)

CuO

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

4179 997.1 0.613 21 0.384

540 6500 18.0 0.85 29

(13)

3pm2f lf

f2 T

þ 147:573

f T2

(14)

The viscosity given in Eq. (14) is expressed in centipoise and the temperature in  C. Fig. 2 presents a plot of the viscosity of CuOewater nanofluids as a function of temperature and concentration of nanoparticles calculated using Eq. (14). The figure also shows the measured data from Nguyen et al. experiments. It is very clear that for f ¼ 1% and f ¼ 7%, the current non-linear regression is in good agreement with the experimental measurements. 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]

mf ¼ 2:414  105  10247:8ðT140Þ

(15)

Z. Haddad et al. / International Journal of Thermal Sciences 57 (2012) 152e162

9% (Present Regression)

         v vJ v vJ 1 v vf v vf f f DB DB  ¼ pffiffiffiffiffiffi þ vx vy vy vy vx vx vy Sc Gr vx        DT 1 v Le vq v vq þ pffiffiffiffiffiffi þ (22) DT DT vx vy vy Sc Gr Tc fb vx

9% Nguyen et al. data [26]

Kinematics

12 1% (Present Regression)

10

4.5% (Present Regression) 7% (Present Regression)

μ (cp)

8

7% Nguyen et al. data [26]

6

v2 j v2 j þ ¼ U vx2 vy2

4.5 % Nguyen et al. data [26] 1% Nguyen et al. data [26]

4

155

(23)

where

gbf0 H 3 ðTH  TC Þ

2

Gr ¼

n2fo

0 20

30

40

50

60

70

80

T (°C) Fig. 2. Comparison between CuOewater viscosities calculated using Eq. (14) and those of Ref. [26].

x¼

u j x* y* ; y ¼ ; U ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; J ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; H H gb HDT =H H gb H DT f0

f* U* V T *  TC ;f¼ ; U ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; V ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; q ¼ fb TH  TC gbf0 H DT gbf0 HDT k¼

(16)

D* D* m a knf ; m ¼ nf ; a ¼ nf ; DB ¼ B ; DT ¼ T mfo a fo kfo DBo DTo

The non dimensional form of equations (1), (3) and (4) in terms of stream function-vorticity becomes:

         v vJ v vJ v vU v vU U U m m  ¼L þ vx vy vx vy vy vx vx vy ! !  ! 2 bp vq v m vV þ f þð1 fÞ þ4L bf0 vx vxvy vy !    v2 m v2 m vU vV vm vU vm vU þL  þ þL þ vy vx vx vx vy vy vx2 vy2

(17)

where

L ¼

1 ! pffiffiffiffiffiffi rp Gr ð1  fÞ þ f

(18)

rf

         x v vJ v vJ v vq v vq q q k k  ¼ pffiffiffiffiffiffi þ vx vy vy vy vx vx vy Pr Gr vx   gfb gLe vf vq vf vq þ pffiffiffiffiffiffiDB þ pffiffiffiffiffiffi þ vx vx vy vy Sc Gr Sc Gr    2  2  DT vq vq DT ð19Þ þ  Tc vx vy

rcp



!

f

x ¼

    ð1  fÞ rcp f þ f rcp p

g ¼

  ! rcp   p   ð1  fÞ rcp f þ f rcp p

(20)

(21)

(24)

(25)

vJ : vx

(26)

The dimensionless boundary conditions can be written as:

v2 J On the top plate : J ¼ 0; U ¼  2 ; q ¼ 0; vy

(27)

v2 J On the bottom plate : J ¼ 0; U ¼  2 ; q ¼ 1; vy

(28)

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. The literature has shown [17,20,22] that the type of boundary conditions used for f when incorporating both Brownian and thermophoresis effects, is the f-constant boundary conditions. Even though it was recognized that assuming that the volumetric fraction of the nanoparticles is constant on boundaries is somewhat arbitrary and it could be argued that zero particles flux on the boundaries is more realistic physically. It is therefore, pertinent to have a thorough knowledge of the flow adopting realistic boundary conditions for f. Thus, the divergence of the diffusion mass flux for the nanoparticles given as the sum of two diffusion terms (Brownian diffusion and thermophoresis) is set at zero and given by

V$

D*B Vf*

þ

VT * D*T TC

! ¼ 0

(29)

which can be expressed as

v2 f 1 DT ¼  NBT DB vy2

where



vJ ; vy

V ¼ 

f0

n fo nf DTo ; Sc ¼ o ; Le ¼ afo DBo DBo

where the subscript “o” stands for the reference temperature which is taken as 22  C in the current study, whereas the temperature difference between the bottom and top plates is fixed to 1  C. The dimensionless horizontal and vertical velocities are given as:

U ¼

Introducing the following dimensionless variables:

; Pr ¼

v2 q vy2

! (30)

where

NBT ¼

fb DBo TC DTo DT

ð ¼ Brownian diffusivity=thermophoretic diffusivityÞ

(31)

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Z. Haddad et al. / International Journal of Thermal Sciences 57 (2012) 152e162

In the linear theory, temperature change in the nanofluid is assumed small in comparison to Tc. Therefore, nanofluid temperature T in the denominator of Eq. (22) has been replaced by Tc [26].

Table 2 Grid independence study. Grid size 21 31 41 51 61 71 81

4. Numerical method Equations (17), (19) and (22), absorbing the variable properties along with the corresponding boundary conditions are solved using a finite volume method [28,29]. 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:

(32)

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 discretization 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:

Pj ¼ M Pi ¼ N nþ1  fn i ¼ 1 f j¼1 6 ε ¼ Pj ¼ M Pi ¼ N nþ1 310 i ¼ 1 f j¼1

U¼ 

8j1;j  j2;j 2ðDyÞ

2

qH kf D T

a

Present Work

0.8

Krane and Jesse [30]

0.6

0.5 0.4 0.3 0.2 0.1 0 0

b

θ (36)

Where cp and hp are the specific heat and enthalpy of the nanoparticle material, respectively. The Nusselt number on the hot wall can be defined as

(37)

0.2

0.4

x

0.6

1,0

0,6

0.8

1

Khanafer [15] present work

0,4 0,2 0,0 0,0

0,2

0,4

0,6

0,8

1,0

X

Where

k k ¼ nf kf

Khanafer et al. [15]

0.7

The energy flux relative to the nanofluid can be calculated as the sum of the conduction heat flux and the heat flux due to nanoparticle diffusion:

  vq DTo rp cp vf vq NBT DB þ DT  vy kf vy vy

(39)

0.9

0,8

Nu ¼ k

Nuavg ðfÞ Nuavg ðf ¼ 0Þ

1

(35)

q ¼ knf VT * þ hp jp

3.188343 3.146399 3.137510 3.135880 3.136542 3.137558 3.138772

The normalized average Nusselt number is used as an indicator of heat transfer enhancement where values greater than unity correspond to an enhancement in heat transfer.

(34)

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

Nu ¼

Nuavg * ðfÞ ¼

(33)

where ε 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 second order accurate formula is used for the vorticity boundary condition. For example, the vorticity at the bottom wall is expressed as

Nu

21 31 41 51 61 71 81

Note that if the second and the third term are zero, Eq. (37) becomes the familiar Nusselt equation for pure fluid. A normalized average Nusselt number is defined as the ratio of Nusselt number at any volume fraction of nanoparticles to that of pure water:

θ

ap fp ¼ aE fE þ aW fW þ aN fN þ aS fS þ b

      

(38)

Fig. 3. 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: Validation of the present code against Khanafer [15] for a square enclosure filled with a water-Cu nanofluid (f ¼ 5%, Gr ¼ 105, Pr ¼ 6.2).

Z. Haddad et al. / International Journal of Thermal Sciences 57 (2012) 152e162

5. Grid independency and code validation A grid independency study has been carried out for CuOewater nanofluid (Ra ¼ 105 and f ¼ 9%), taking into account thermophoresis and Brownian effects. Table 2 presents the average Nusselt number for five different grid sizes. A grid size of 51  51 is found to meet the requirements of both the grid independency study and the computational time limits. The present numerical solution is further validated by comparing the present code results for Ra ¼ 105 and Pr ¼ 0.70 against the experiment of Krane and Jessee [30] and numerical simulation of Khanafer et al. [15]. It can be seen that the results

157

agree well with the results reported in the literature as shown in Fig. 3a. Moreover, the present code is also validated against the work by Khanafer et al. [15] as shown in Fig. 3b. In this study, an enclosure filled with a water-Cu nanofluid (f ¼ 5%, Gr ¼ 105, Pr ¼ 6.2) is considered. It is clear that the results are in good agreement. The convergence criterion was set such that the relative error between two successive iterations was less than 106 [31]. 6. Results and discussion A numerical study has been performed in this work to investigate the effects of thermophoresis and Brownian motion in

Fig. 4. Isotherms (on the left) and streamlines (on the right) of CuO-nanofluid at Ra ¼ 104, (a) f ¼ 1% (Jmin ¼ 0.014, 0.03), (b) f ¼ 3% (Jmin ¼ 0.03), (c) f ¼ 5% (Jmin ¼ 0.02), (d) f ¼ 7% (Jmin ¼ 0.01), (e) f ¼ 9%, (——) absence of thermophoresis and Brownian effects, (d) with thermophoresis and Brownian effects.

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Z. Haddad et al. / International Journal of Thermal Sciences 57 (2012) 152e162

RayleigheBénard convection problem. The problem is solved for two cases. In the first case, thermophoresis and Brownian effects are taken into account and in the second case the problem is solved by neglecting the role of thermophoresis and Brownian effects. The Rayleigh number and the solid volume fraction are varied in the ranges of 104  Ra  106 and 0 f  9%. Water was chosen as base fluid and CuO nanoparticles are added into the base fluid. In this work, the results are presented for small temperature difference (fixed to 1  C). It is noteworthy that the conservation equations are strongly coupled. That is, v depends on f via viscosity; f depends on T mostly because of thermophoresis; T depends on f via thermal conductivity and also via the Brownian and thermophoretic terms

in the energy equation; f and T obviously depend on v because of the convection terms in the nanoparticle continuity and energy equations, respectively. Fig. 4 (a)e(e) present the isotherms (on the left side) and streamlines (on the right side) for the two different cases (with thermophoresis and Brownian effects and without thermophoresis and Brownian effects) at Ra ¼ 104 using different values of nanoparticles volume fraction. It must be noted that the results for Ra ¼ 105 and Ra ¼ 106 are not presented here as they show similar trends to the results of Ra ¼ 104. 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

Fig. 5. Nanoparticle distribution for different Rayleigh number as 105 (on the left) and 106 (on the right), (a) f ¼ 1%, (b) f ¼ 3%, (c) f ¼ 5%, (d) f ¼ 7%.

Z. Haddad et al. / International Journal of Thermal Sciences 57 (2012) 152e162

strength of streamlines contours is attenuated and the isotherms are stratified near the top and the bottom plates of the enclosure. This means weaker buoyant flow circulation is occurring in the enclosure, which demonstrates the dominance of the conduction heat transfer at high values of nanoparticle concentrations namely at f ¼ 9%. Comparison of overlapped isotherms and streamlines plots shows that the presence of thermophoresis and Brownian motion does not have a significant influence on the isotherms and streamlines shapes. However, it is clearly observed that there is a slight difference for f ¼ 1% where the minimum values of the stream function for the case with and without thermophoresis and

159

Brownian effects are Jmin ¼ 0.03 and Jmin ¼ 0.014, respectively. It is very interesting to note that this difference was observed for f ¼ 3% and f ¼ 5% at Ra ¼ 105 and Ra ¼ 106. This is attributed to a particulate buoyancy force, which is the density variation that is due to variable volume fraction of nanoparticles. This force helps nanofluid to have strong convective heat transfer for low concentration of nanoparticles in the presence of the combined effects of thermophoresis and Brownian motion. It is clear that the streamlines and isotherms are not sufficient to clarify the difference between the case of with and without thermophoresis and Brownian effects since T, v and f are strongly

Fig. 6. Variation of nanoparticle distribution along the cold wall (on the right) and hot wall (on the left), (a) Ra ¼ 104, (b) Ra ¼ 105, (c) Ra ¼ 106.

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Z. Haddad et al. / International Journal of Thermal Sciences 57 (2012) 152e162

dependent as it was explained earlier. Therefore, the spatial distribution of nanoparticles between the top and bottom plates is of great importance and it is a key parameter for studying the effect of nanoparticles on flow fields and temperature distributions. Therefore, to develop a better understanding of this point, Fig. 5 portrays

Fig. 7. Variation of local Nusselt number along the bottom wall for different cases and nanoparticle fractions, (a) Ra ¼ 104, (b) Ra ¼ 105, (c) Ra ¼ 106.

the nanoparticle distribution for Ra ¼ 105 (left side) and Ra ¼ 106 (right side) for different nanoparticle volume fraction. The general view of the figures shows that thermophoresis and Brownian effects make the nanoparticles distribution to become non-uniform throughout the domain. Nanoparticle distribution is more nonuniform for lower particle concentration and high Rayleigh number. However, it is more uniform for low Rayleigh number. For any case, nanoparticle concentration is higher at the top middle side and values are decreased with increasing of volume fraction. This decreasing is more clear at higher Rayleigh numbers. On the contrary, nanoparticle volume fraction values are very small near the left and right bottom corners. This indicates that in the analysis of thermal transport in nanofluid, one must be concerned about the near wall region which may have a lower or higher particle concentration, leading to higher or lower heat transfer rates. In order to obtain a better understanding of nanoparticles distribution within the enclosure, the nanoparticle volume fraction profiles along the hot bottom wall and cold top wall for different Rayleigh numbers and different volume fractions are presented in Fig. 6. Looking at Fig. 6, it is observed that the addition of nanoparticles causes the nanoparticle volumetric fraction to fall along the hot wall. Furthermore, higher values are obtained for higher Rayleigh number. It is also observed that that the nanoparticle volumetric fraction concentration along the cold wall is increased and along the hot wall is decreased. A maximum value is observed around X ¼ 1.0 and bell-shaped symmetrical distribution is obtained on cold wall. However, the nanoparticles volume fraction values are uniform around X ¼ 1.0 along the cold wall. In other words, the presence of Brownian and thermophoresis effects makes nanoparticles distribution, particularly at high values of Rayleigh number, to become non-uniform and enhances energy transport between the two walls, which supports the significance of the second and third terms on the right-hand side of Eq. (4). Fig. 7 presents Nusselt number variation along the bottom hot surface using various volume fractions of nanoparticles. Overall assessment indicates that for both cases, when Brownian and thermophoresis effects are neglected or considered, a reduction in Nusselt number is observed for all Rayleigh numbers by increasing the volume fraction of nanoparticles. It is worth noting that the heat transfer, by considering the role of Brownian and thermophoresis effects, is higher than the case without considering the role of thermophoresis and Brownian motion. There is one case registered where the heat transfer is pure conductive and this is for Ra ¼ 104 and f ¼ 9%. However, the rest of the cases convection heat transfer is taking place. Therefore, without considering the role of Brownian and thermophoresis effects, the presence of nanoparticles will make the nanofluid to be more viscous, which will reduce convection currents and accordingly diminish the temperature gradient and the Nusselt number at the heated surface (more explanation is given by Abu-Nada and Chamkha [32]). However, by considering the role Brownian and thermophoresis effects, the addition of nanoparticles will enhances heat transfer due to the relative motion of nanoparticles with respect to the base fluid (slip) which will cause more energy carried by these particles from places with high energy (hot wall) to the places with less energy (cold wall). This relative motion of nanoparticles almost double the heat transfer compared to the case of absence of Brownian motion and thermophoresis as shown in Fig. 7. Variation of average Nusselt number is presented in Fig. 8. As seen from the figure, average Nusselt number increases with Rayleigh number for both presence and absence of Brownian and thermophoresis effect. Also, it is decreased by increasing nanoparticle volume fraction for both cases. It is shown that higher average Nusselt number values are observed by considering the role of Brownian and thermophoresis effect. Besides, it is observed

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161

7. Conclusions

Nuavg

6

Presence of Brownian and Thermophoresis effects

Absence of Brownian and Thermophoresis effects

9 ϕ=0 ϕ=1% ϕ=3% ϕ=5% ϕ=7% ϕ=9%

RayleigheBénard problem is studied using CuoeWater nanofluid for different Rayleigh numbers and different nanoparticle volume fractions. The study is performed by considering the role of Brownian and thermophoresis effects and compared to the case where both effects are neglected. The main findings are listed as follows

3

0 4

5

10

6

10

10

Ra Fig. 8. Variation of average Nusselt number with nanoparticle volume fractions at different Rayleigh numbers.

from the figure that Nusselt number values are very close to each other for lower concentrations of nanoparticles for both cases since the viscosity of the nanofluid is relatively small. Finally, Fig. 9 illustrates the normalized average Nusselt number along the bottom heated surface. For the range of Ra ¼ 104e106, the trend if fairly similar for both cases where a decrease in Nusselt number occurs for an increase in volume fraction of nanoparticles. It is also worth mentioning that heat transfer experiences more deterioration for nanoparticles volume fraction greater than 5% and for all Rayleigh numbers. Also, the figures shows that by considering thermophoresis and Brownian there is always enhancement in heat transfer (since Nuavg* is greater than unity). But, from a heat transfer enhancement perspective, it is better to keep the concentration as mush low as possible to guarantee low viscosity of nanofluids. However, this conclusion is not true by neglecting the role of thermophoresis and Brownian motion where deterioration in heat transfer is taking place at large concentrations of nanoparticles.

2.5

2

Nu avg*

1.5

1

0.5 1E6 without

1E5 without

1E4 without

1E6 with

1E5 with

1E4 with

0

0

2

4

6

8

10

φ Fig. 9. Normalized Nusselt number using different volume fractions and different Rayleigh numbers.

 It is found that higher heat transfer is formed when Brownian and thermophoresis effects are considered.  By considering the role of thermophoresis and Brownian motion there is always an enhancement in heat transfer by the presence of nanoparticles. However, the enhancement is more pronounced at low volume fraction of nanoparticles.  When neglecting the role of thermophoresis and Brownian deterioration in heat transfer it is observed and this deterioration is increased by increasing the concentration of nanoparticles.  In general, increasing nanoparticle concentration has an adverse effect on heat transfer.  The presence of thermophoresis and Brownian motion does not have a significant influence on the isotherms and streamlines shapes.

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