Effect of thermophoresis on natural convection in a Rayleigh–Benard cell filled with a nanofluid

International Journal of Heat and Mass Transfer 81 (2015) 142–156

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Effect of thermophoresis on natural convection in a Rayleigh–Benard cell filled with a nanofluid M. Eslamian a,⇑, M. Ahmed b, M.F. El-Dosoky b, M.Z. Saghir c a

University of Michigan – Shanghai Jiao Tong University Joint Institute, Shanghai 200240, China Department of Mechanical Engineering, Assiut University, Assiut 71516, Egypt c Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, ON M5B 2K3, Canada b

a r t i c l e

i n f o

Article history: Received 6 February 2014 Received in revised form 30 September 2014 Accepted 1 October 2014

Keywords: Nanofluids Thermophoresis Rayleigh–Benard convection Heat transfer augmentation

a b s t r a c t The objective of this paper is to clarify the role of thermophoresis in laminar natural convection in a Rayleigh–Benard cell filled with a water-based nanofluid and to study its relative importance compared to other effects, in an attempt to correct the present confusing interpretation of the magnitude of the thermophoresis coefficient in nanofluids. The major forces are introduced and the transport equations are solved using a two-phase lattice Boltzmann method (LBM) for a laminar flow with Ra numbers up to 106 with various particle loadings (particle volume fractions). The results indicate an increase in the average Nu number with an increase in the Ra number and particle loading. An increase in the Nu number for a 10% particle loading at Ra = 106 is less than 20%. When thermophoresis effect is taken into consideration, an increase in the Nu number is predicted, which is about 10%. Therefore, it is concluded that using a nanofluid in bottom-heated laminar natural convection results in a considerable increase in heat transfer rate and thermophoresis force is a significant contributor to heat transfer augmentation, particularly for high Ra numbers (Ra  106 and higher). It is observed that at low Ra numbers (weak convective flows), the nanofluid behaves homogenously, but at higher Ra numbers, it starts to behave heterogeneously. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction A nanofluid denotes a mixture comprising a base liquid, such as water, and a small amount of suspended inert nanoparticles with fairly high thermal conductivity, such as copper and copper oxide. Addition of such nanoparticles to the base fluid results in an anomalous change in some of the physical properties of the base fluid, such as thermal conductivity. Nanofluids may be used in various flow configurations, from natural convection to forced convection in turbulent pipe flows. In some heat transfer applications, a temperature gradient may exist in the domain. The force exerted on nanoparticles because of a temperature gradient in the fluid is called the thermophoresis force [1]. There is a difference in the response of a nanofluid to a temperature gradient in natural convection in a cavity compared to the forced convection in a laminar or turbulent flow. In natural convection, thermophoresis, Brownian and gravitational forces, which are external forces, considerably affect the composition and heterogeneity of the mixture due to creation of a slip or drift ⇑ Corresponding author. Tel.: +86 21 3420 7249; fax: +86 21 3420 6525. E-mail address: [email protected] (M. Eslamian). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.10.001 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.

velocity on nanoparticles surface with respect to the main flow. In forced convection, these small scale effects are less significant in the flow core and large scale effects, such as eddies and the fluid flow are more important [2,3]. As a result of neglecting, inadequate inclusion or sometimes overestimating the thermophoresis effect, theoretical studies in heat transfer augmentation in nanofluids are confusing and contradicting in several areas. In the experimental studies, particularly in natural convection, lack of control on the size, mono-dispersity and uniformity of the suspended particles as well as particle agglomeration may obscure the real contribution of nanoparticles leading to misleading results and conclusions. In forced convection, results are more reliable because particle agglomeration is reduced due to the flow momentum and agitation. The overall observation is that the heat transfer coefficient in forced convection of laminar and turbulent flow increases significantly compared to the base fluid, whereas in natural convection, the data are contradicting and the results are inconclusive. While the earlier measurements in the natural convection showed a decrease in the heat transfer rate, recent experimental and theoretical studies confirm minor increase in the heat transfer coefficient in natural convection. This will be discussed in the following literature review.

M. Eslamian et al. / International Journal of Heat and Mass Transfer 81 (2015) 142–156

143

Nomenclature A Abs c C ¼ DDxt Cr Cs cp dbf DT ei r f i ðx; tÞ r;eq

fi

Fb FB FD FH FP FT Fw F ri Fi g g ri H i j k kbf M n N Pr Ra Re t T

ðx; tÞ

coefficient in Eq. (21) absolute value nanofluid specific heat reference lattice velocity defined by Eq. (18) coefficient in Eq. (2) particle specific heat equivalent diameter of a molecule of the base fluid thermodiffusion (thermophoresis) coefficient lattice velocity vector density distribution function of the particles of component r at lattice position x along the direction i, and at time t the local equilibrium distribution function of the particles of component r at lattice position x along the direction i, and time t buoyancy force Brownian force drag force net gravitational force as a result of buoyancy and particle weight (FH = Abs(wp  Fb)) sum of all forces acting on a particle per unit volume thermophoresis force sum of all forces acting on the base fluid total inter-particle interaction forces driving force for natural convection gravitational acceleration energy distribution function dimension of the square cavity in x and y directions lattice velocity direction index of spatial coordinate in the y direction nanofluid thermal conductivity thermal conductivity of the base fluid molecular weight of the base fluid number of particle in a given lattice Avogadro’s number Prandtl number Rayleigh number Reynolds Number time fluid local temperature

There are numerous papers discussing various aspects of nanofluids. While many researchers have studied the heat transfer characteristics of nanofluids in various flow arrangements, there are few papers that address the fundamental physics behind it. The lack of the theoretical explanations has resulted in contradicting, or redundant studies with misleading conclusions. An increase in the heat transfer coefficient in nanofluids (to be discussed further) is only partially due to an increase in the thermal conductivity, supported by the fact that the heat transfer correlations developed for base fluids, such as the Dittus–Boelter’s equation used with the properties of nanofluids, underestimate the heat transfer coefficient in nanofluids [3]. This is because the presence of nanoparticles in a nanofluid may make it heterogeneous, i.e., as a result of external forces applied on nanoparticles, a drift or slip velocity may be created, which would enhance mixing. Buongiorno [3] theoretically studied the relative effect of seven forces that would potentially make a nanofluid heterogeneous by causing a slip or relative velocity between the base fluid and the nanoparticles. The considered forces include inertia (drag), Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect,

TH TC UT u v V V VP x y z

temperature of the hot wall (bottom wall) temperature of the cold wall (topwall) thermophoresis velocity horizontal velocity component vertical velocity component lattice volume flow velocity vector particle velocity vector x coordinate y coordinate z coordinate

Greek Symbols

a b Dt

e q qP qbf0 /

r l lbf sr

srh h

m xi

thermal diffusivity thermal expansion coefficient time step convergence criterion nanofluid overall density particle density density of the base fluid at 20 °C. particle volume fraction/particle loading components (r = 1, 2, water and nanoparticles) nanofluid viscosity base fluid viscosity dimensionless collision-relaxation time constant for the flow field of the r component dimensionless collision-relaxation time constant for the temperature field of the r component dimension-less temperature kinematic viscosity weight coefficients in Eq. (6)

subscripts/superscripts ⁄ dimensionless quantity (u⁄, v⁄) bf base fluid eq equilibrium i the lattice velocity direction P particle x x component y y component r component 1 or 2 for water and nanoparticles

fluid drainage, and gravity. Based on a time scale analysis, it was concluded that in nanofluids mostly thermophoresis and Brownian diffusion can cause slip, a relative velocity between the fluid and the nanoparticle (gravity and buoyancy forces were argued to be insignificant) [3]. Despite their importance, both of these effects have been sometimes neglected resulting in contradicting and erroneous results and conclusions. In this work, we numerically show the effect of each of the above mentioned forces in a bottom-heated cavity. We show that the thermophoresis force is as small as the gravitational forces, but can cause slip velocity. Our previous results verify the considerable effect of thermophoresis, Brownian and gravity forces in heat transfer enhancement in natural convection in a differentially-heated cavity [2]. In forced convection, our unpublished results show that thermophoresis and gravity forces are only considerable in creeping flow, whereas the Brownian force is several order of magnitude larger. Numerical and theoretical works on forced convection in conventional and micro-channels are abundant, e.g., [4–6]. This work is focused on natural convection in an enclosure or cavity. There are numerous studies in natural convection in enclosures using

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nanofluids, some of which in contradiction with one another. For instance, Glassl et al. [7], Elhajjar et al. [8], and Hwang et al. [9] concluded that with a positive Soret coefficient (particles forced by the temperature gradient to move to the cold side), the stability of the flow increases in the Rayleigh–Benard configuration, whereas Kim et al. [10] and Tzou [11] conclude the opposite. Experimental results of Wen and Ding [12] on water/titania nanofluid in a Rayleigh–Benard cell indicate a reduction in heat transfer coefficient with addition of nanoparticles, perhaps due to particle agglomeration observed in their cell. Alumina and copper oxide dispersed in water nanofluids were used by Putra et al. [13] in a differentially-heated enclosure, where a deterioration of the heat transfer rate was observed. Recent experimental data (only available on a differentially-heated enclosure), however, assert an increase in the heat transfer rate in low concentration nanofluids with respect to base fluids [14,15]. At low particle loadings (/ < 2%), heat transfer rate increases, whereas at higher particle loadings it declines. A reduction in heat transfer rate was attributed to a decrease in nanofluid Ra number as a result of an increase in nanofluid viscosity [14]. Similarly, results of Ho et al. [15] in a differentially-heated enclosure suggest that at particle volume concentration over 2%, the application of nanofluids is infeasible as it has a detrimental effect on the heat transfer rate. This may be due to particle agglomeration. At low particle volume fractions, on the other hand, an enhancement in the heat transfer rate was observed. Thermophoresis and Brownian diffusion play an important role in determining the level of heat transfer augmentation or deterioration [3]. This is because these two effects cause slip between the base fluid and particles making the nanofluid to behave heterogeneously. Neglecting these effects, i.e. assuming that the nanofluid is homogenous, may introduce errors in simulations, in which case the simulations may either predict an increase or decrease in heat transfer rate depending on the formulas used for the overall thermal conductivity and more importantly for the overall viscosity of the nanofluid [9]. In a numerical study, Haddad et al. [16] investigated the natural convection of a nanofluid in the Rayleigh–Benard cell with and without the effect of Brownian motion and thermophoresis. Without considering these two effects, heat transfer rate decreased. When these two effects were considered the heat transfer rate increased as a result of induction of a slip velocity between the base fluid and the nanoparticles. In their work, however, thermophoresis effect was overestimated, due to using the wrong equations. Oueslati et al. [17] numerically and analytically studied natural convection in a differentially- heated enclosure. With inclusion of thermophoresis effect heat transfer rate was enhanced, while without thermophoresis a decrease in the heat transfer coefficient was observed. They found an optimum nanofluid concentration for the maximum heat transfer augmentation. In their analysis, a random positive Soret coefficient of 0.02 was assumed, while in reality the Soret coefficient may be very different from one mixture to another. Using a single-phase homogenous model, Corcione [18] investigated heat transfer in a Rayleigh–Benard cell. Although thermophoresis was neglected, a slight heat transfer enhancement was observed, perhaps due to the choice of correlations used for the thermal conductivity and viscosity of the used nanofluids. However, the extent of the heat transfer enhancement compared to those investigations that considered the thermophoresis effect and Brownian force was significantly smaller. Abu-Nada and Chamkha [19] investigated natural convection in a differentiallyheated enclosure using a single phase homogenous approach. For a wide range of Ra numbers, and using various correlations for the overall thermal conductivity and viscosity, both heat transfer enhancement and deterioration was observed, depending on the choice of the correlations. Abouali and Ahmadi [20] performed a numerical analysis on natural convection in several enclosure configurations and

compared their results with those obtained from the correlations prescribed for pure fluids. They concluded that there would be no need for a separate numerical analysis for each geometry; instead a base fluid correlation linked with the overall properties of a nanofluid would provide the heat transfer coefficient. However, we note that in correlations prescribed for single fluids, the fluid is single component and homogenous and external forces such as thermophoresis and Brownian motion are therefore absent and irrelevant, whereas a nanofluid may become heterogeneous, because of the aforementioned effects. Therefore, results of Ref. [20] which predict an unconditional decrease in the heat transfer coefficient in natural convection are erroneous. In another controversial study, Pakravan and Yaghoubi [21] performed an analysis on the natural convection heat transfer, considering Soret effect (thermophoresis) and Brownian motion. Perhaps under the impression of dubious experimental data of Putra et al. [13] and Wen and Ding [12], which predict a decrease in the heat transfer coefficient, they presumed that the Soret effect has a deteriorating effect in heat transfer, subtracting a term they derived for the thermophoresis contribution from the general term for the Nu number of the base fluid. The forgoing literature review on natural convection in enclosures heated from sides or bottom reveals that thermodiffusion effect (as well as Brownian force) is present within the entire domain resulting in a slip or drift velocity on nanoparticles surface. Therefore, a change in the nanofluid physical properties, such as an increase in the thermal conductivity is not the sole reason responsible for an increase in the heat transfer coefficient; thermophoresis and Brownian diffusion are prominent, as well. The relative importance of these effects will be investigated in this paper. Thermophoresis has been taken into account in several studies, e.g., [16,17,22], although its effect has been considered qualitatively assuming approximate values for the Soret or thermophoresis coefficients. This is partly because of the lack of experimental data or a reliable theory that can be used to estimate thermophoresis coefficient (mobility) in nanofluids. Employing inaccurate values for thermophoresis coefficient may obscure, overestimate or underestimate its effect on the heat transfer rate. Motivated by the above-mentioned setback, in this work natural convection in an enclosure heated from below and filled with a water-based nanofluid is studied using a two-phase lattice Boltzmann method (LBM). Reliable values for the thermophoresis coefficient will be used to investigate the true effect of temperature gradient on heat transfer rate augmentation or deterioration. 2. The governing equations Instead of using the conventional transport equations, such as the Navier–Stokes, energy and continuity equations, the two-phase lattice Boltzmann method is used here. This is because from the microscopic point of view, the existing computational methods for conventional two-phase flows may fail to reveal the inherent nature of the flow and energy transport process inside a nanofluid [23]. A microscopic or mesoscaled approach should be introduced to describe the effect of interactions between suspended nanoparticles and liquid particle (molecules), as well as among the solid nanoparticles. The lattice Boltzmann equation for a multicomponent flow is written as follows [23]: r

r

f i ðx þ ei Dt; t þ DtÞ  f i ðx; tÞ

 r   1 r 2s  1 F ri :ei Dt r;eq þ DtF i ¼  r ðf i ðx; tÞ  f i ðx; tÞÞ þ 2 r s 2s Bi C

r ¼ 1; 2

ð1Þ

where F ri represents the total inter-particle interaction forces, and will be defined later (as FP, Fw, Eqs. (23) and (24)). F i is the natural convection driving force and is defined as follows:

M. Eslamian et al. / International Journal of Heat and Mass Transfer 81 (2015) 142–156

F i ¼ xi qgbDt:

ei

ð2Þ

C 2s

0

ð0; 0Þ i¼0  B  ði1Þ p ði1Þ p B i ¼ 1; 2; 3; 4 ei ¼ B C cos 2 ; sin 2 @   p p ; sin ð2i9Þ C cos ð2i9Þ i ¼ 5; 6; 7; 8 4 4 0 B Bi ¼ @ 13

ð3Þ

i¼8 1 X F r s r Dt r ur;eq ¼ r f i ei þ q qr

T r;eq ¼

2sr  1 2 C Dt 6

ð14Þ

ar ¼

2srh  1 2 C Dt 6

ð15Þ

The average kinematic viscosity coefficient and thermal diffusivity of nanofluid are defined as follows:

a¼

i ¼ 1; 2; 3; 4 i ¼ 5; 6; 7; 8

"

3ðei :ur;eq Þ C2

þ

9ðei :ur;eq Þ2 2C 4

þ

3ður;eq Þ2

# ð5Þ

2C 2

where xi, the weight coefficients are written as follows:

04

i¼0

9

xi ¼ B @ 19

i ¼ 1; 2; 3; 4

1 36

ð6Þ

i ¼ 5; 6; 7; 8

Similarly, one can introduce the lattice Boltzmann equation for the energy transport, by describing the energy equation as the energy distribution function g ri . The two-phase lattice Boltzmann energy equation based on neglecting the viscous dissipation is written as follows [24]:

  g ri ðx þ ei Dt; t þ DtÞ  g ri ðx; tÞ ¼  s1r g ri ðx; tÞ  g ri ;eq ðx; tÞ h

r ¼ 1; 2

ð7Þ

sr

where h is the dimensionless energy collision-relaxation time constant of the r component of the fluid. The energy distribution function at the equilibrium state is defined as follows:

" r;eq

gi

¼ xi T

r

1þ

3ðei :ur;eq Þ C2

þ

9ðei :ur;eq Þ2 2C 4

þ

3ður;eq Þ2 2C 2

# ð8Þ

The macroscopic density, velocity and temperature of each component are respectively calculated as follows:

qr ¼

i¼8 X r fi

ð13Þ

mr ¼

i¼0

¼ xi qr 1 þ

i¼8 X dT g ri þ Dtsrh dt i¼0

The kinematic viscosity m and the thermal diffusivity a of the r component are defined as follows:

where in Eq. (3), C ¼ DDxt is the reference lattice velocity. The equilibr;eq rium distribution function f i ðx; tÞ should be carefully selected to ensure that each of the components obeys the macro-scaled Navier–Stokes equation. The particle distribution functions at the equilibrium state for a multicomponent flow could be written as follows [24]: r;eq

ð12Þ

i¼0

m¼

fi

ð11Þ

By considering the internal and external forces, the macroscopic velocities, and temperatures for nanoparticles, and the base fluid are modified as follows [25–28]:

ð4Þ

1 12

i¼8 X g ri i¼0

In Eq. (1), sr is the dimensionless collision-relaxation time constant for flow field of the r component, ei is the lattice velocity vector, the 2 r subscript i represents the lattice velocity direction, C 2s ¼ C3 , f i ðx; tÞ is the density distribution function of the particles of component r r;eq with velocity ei at lattice position x and time t, and f i ðx; tÞ is the local equilibrium distribution function. One may select different forms of lattices, according to the hydrodynamic problem under study. Therefore, several types of the multi-speed lattice Boltzmann models are available in the literature. For a two-dimensional problem, the well-known D2Q9 model is widely used. It is a 9-speed model based on a two dimensional octagonal lattice in which the lattice velocity and parameter B in Eq. (1) is defined as follows [24]:

0

Tr ¼

145

ð9Þ

2

P

r r Cr s

1

C 2 Dt

ð16Þ

1

C 2 Dt

ð17Þ

6 2

P

r r C r sh

6

where

qr rq

Cr ¼ P

ð18Þ

The external forces exerted on nanoparticles considered here include the buoyancy force, the gravity force, the Brownian diffusion force, and the thermophoresis force. A drag force is also induced on nanoparticles by a relative velocity created by external forces between nanoparticles and base fluid (slip velocity). These forces may have a significant effect on the behavior of nanofluid and nanoparticles. Thermophoresis force FT exerted on a nanoparticle of diameter dp is related to thermophoretic velocity UT through the well-known Stokes equation:

F T ¼ 3plU T dp

ð19Þ

where UT is related to the thermophoresis or thermodiffusion or thermophoresis coefficient DT as follows [1]:

U T ¼ DT rT

ð20Þ

The thermophoresis coefficient may be estimated based on several approximate approaches [29]. Given that in nanofluids, nanoparticles are in the range of about 5–50 nm, the hydrodynamics-based theory is most suitable to estimate DT. For particles approaching the molecular size of the carrier liquid, the non-equilibrium thermodynamic equation is more suitable [29] unless the noncontinuum effects are taken into account to correct the hydrodynamic-based equation. For nanofluids where Kn number is sufficiently small and close to zero, the hydrodynamic equation written as follows is valid [29–33]:

DT ¼ 2A

k l 2k þ kp qT

ð21Þ

i¼0

where the physical properties are associated with the nanofluid (k, i¼8 X r ur ¼ f i ei i¼0

l, q) and the particles (kp). Coefficient A is a function of liquid physð10Þ

ical properties and temperature and is given in Ref. [32]. For water we have shown that A is equal to 0.0085 [29]. As mentioned before,

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M. Eslamian et al. / International Journal of Heat and Mass Transfer 81 (2015) 142–156

Eq. (21) was derived assuming that the non-continuum effects are absent. As the nanoparticle size decreases and approaches to 1 nm or so, the errors associated with Eq. (21) increases. Currently the erroneous expression of MacNab and Meisen [33] and even sometimes the gas-phase expression of Talbot et al. [34] are used to model thermophoresis in liquid nanofluids. Both of these expressions highly overestimate the thermophoresis effect [29]. The induced drag or friction force FD exerted on small particles in a creeping flow is obtained from the Stokes law (Re < 1) and for higher Re numbers from the modified forms. In the case of natural convection of spherical nanoparticles in a liquid, such as water, the Stokes law assumptions are satisfied. The Stokes law is written as follows:

FD ¼ 3pdP lðV  VP Þ

ð22Þ

Also, note that the liquid may be considered as a continuum, given that the mean free path of the water molecules (0.25 nm) is much smaller than the nanoparticle sizes (10 nm). Eq. (22) provides a reasonable prediction of the drag force for the conditions of this study. Eq. (22) reveals that the drag force is produced, if there is a difference between velocity of the fluid and that of nanoparticles (V  VP). This is the slip or drift velocity created by the external forces, such as Brownian and thermophoresis forces, acting on the nanoparticles in the nanofluid. Brownian motion is the random and fluctuating motion of particles caused by the collision of fluid molecules with the suspended particles. The Brownian motion is created by the Brownian force. The components of the Brownian force are modeled as a Gaussian white noise process. Details of calculating the Brownian force exerted on particles in a fluid can be found elsewhere, e.g., ([35] and references therein). Now back to the LBM simulation approach, the total force per unit volume acting on nanoparticles of a single lattice is written as follows:

FP ¼

n ½F H þ F D þ F B þ F T  V

ð23Þ

Table 1 Values of the average Nu number at the hot wall at different grid numbers for Ra = 105, Pr = 7.02. Number of grids Nu

50  50 4.282

100  100 4.096

150  150 4.1

200  200 4.12

The force FH is the summation of the buoyancy and gravity forces. The drag force, FD is given by Eq. (22), FB, is the Brownian force and is given in Ref. [35], and FT, is the thermophoresis force and is given by Eq. (19). The summation of the forces per unit volume, acting on the base fluid Fw is written as follows:

Fw ¼ 

n ½F D þ F B þ F T  V

ð24Þ

The nanofluid transport and physical properties vary with nanoparticle volume fraction and temperature. Here the effect of particle volume concentration is taken into account only give that the temperature change is insignificant. However, for a quantitative comparison, more sophisticated correlation need to be used [36]. The nanofluid density and heat capacity per unit volume are obtained based on the mixing rule and are expressed as follows:

q ¼ ð1  /Þqbf þ /qP

ð25Þ

qc ¼ ð1  /ÞðqcÞbf þ /ðqcÞP

ð26Þ

An empirical correlation obtained based on a large experimental database has been proposed for the viscosity of a nanofluid l normalized by the viscosity of the base fluid lbf [18]:

l 1 ¼ lbf 1  34:87dP =dbf 0:3 /1:03

ð27Þ

where dbf is the equivalent diameter of a molecule of the base fluid and is given by:

Fig. 1. Schematic of the physical domain with the thermal boundary conditions and the coordinate system (a) and the computational grid (b). All wall velocities are set to zero.

M. Eslamian et al. / International Journal of Heat and Mass Transfer 81 (2015) 142–156

" dbf ¼ 0:1

6M Npqbf 0

147

# ð28Þ

The following correlation by Maxwell is used for the effective thermal conductivity of a nanofluid k, normalized by the thermal conductivity of the base fluid kbf:

k¼

Fig. 2. Code validation: comparison of the predicted temperature profiles along the vertical direction of the enclosure obtained in this work with those obtained by D’Orazio et al. [39] at Ra from 103 to 106. Pr = 0.71, / = 0.0.

kP þ 2kbf þ 2/ðkP  kbf Þ kP þ 2kbf  /ðkP  kbf Þ

ð29Þ

The Maxwell equation is particularly applicable to low concentration nanofluids and with increase of the particle volume concentration the error increases. An in-house Fortran 2003 code was prepared to solve the above equations. A two-dimensional square Rayleigh–Benard enclosure (Fig. 1(a)) with the dimensions of 1.0 cm  1.0 cm filled with copper oxide (CuO)water nanofluid is chosen as the computational domain with the nanoparticle density of 6500 kg/m3, and nanoparticle specific heat capacity of 535.6 J/kg K. Thermal conductivity of copper oxide (CuO) nanoparticles is variable; here an average value of 20 W/m K is used [37]. The vertical walls are assumed adiabatic and the cell is heated from bellow to induce a boundary driven flow. All walls are impermeable with no inlet and velocities on all walls are zero at all the times. The dimensionless boundary conditions are written as follows:

Fig. 3. Code validation: comparison of streamlines (left) and isotherms (right) obtained in this work (a) and those obtained in Ref. [40] (b). Ra = 105, Pr = 7.02 and / = 0.1. Note that our results (a) show a counterclockwise circulation in the right circulation cell, whereas those of Ref. [40] (b) are in the clockwise direction. Both are considered as valid numerical solutions to this problem [41,42]. In our work, a two-phase LBM approach is used with all significant external forces considered, while in Ref. [40], a simplistic single phase fluid approach was assumed.

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M. Eslamian et al. / International Journal of Heat and Mass Transfer 81 (2015) 142–156

At y ¼ 0:5 u ¼ v  ¼ 0; h ¼ 1 At y ¼ 0:5 u ¼ v  ¼ 0; h ¼ 0:0 At x ¼ 0:5 u ¼ v  ¼ 0; At x ¼ 0:5 u ¼ v  ¼ 0;

@h ¼0 @x

@h ¼0 @x

ð30Þ

where

x ¼

x ; H

y ¼

y ; H

u ¼

uH

m

;

v ¼

vH m

;

h¼

T  TC TH  TC

Copper oxide nanoparticles of 10 nm diameter are initially uniformly distributed in water within the domain. In all runs, the bottom wall is kept at 310 K, while the top wall is kept at 300 K, i.e., a temperature gradient of 1000 K/m is applied in the vertical direction. In the simulations, the water expansion coefficient b is changed artificially to generate Ra numbers with desired magnitudes from 103 to 106 covering a range of conditions from the stable flow to the low Re number laminar flow. Nanoparticle volume fractions of 0.0–0.1 are examined to investigate the physics of the problem, while in practice the nanoparticle volume concentration is about 0.01. Ra number is defined as follows:

ð31Þ Ra ¼

gb

ma

ðT H  T C ÞH3

ð32Þ

Physical properties of the nanofluid are used towards calculation of the effective Ra number in the cell.

Fig. 4. Variation of the interaction forces between the nanofluid and nanoparticles versus x⁄ at the horizontal mid-plane (y⁄ = 0.0) at Ra = 106 for a single nanoparticle of diameter 10 nm. (a) The vertical component of the forces; (b) magnitude of the resultant forces. Note that in part (a), only the positive forces are shown given that a logarithmic scale is employed. In part (b) all resultant forces are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi positive and therefore shown. F T ¼ F 2Tx þ F 2Ty , F B ¼ F 2Bx þ F 2By , F D ¼ F 2Dx þ F 2Dy , FH = Abs(wp  Fb).

Fig. 5. Streamlines of (a) nanoparticles (particle path-lines) and (b) nanofluid for natural convection in a square cavity heated from below at Ra = 104. Pr = 7.02 and / = 0.1 with the thermophoresis effect considered.

M. Eslamian et al. / International Journal of Heat and Mass Transfer 81 (2015) 142–156

To test the mesh independence of the solution scheme for the given configuration with 1.0 cm  1.0 cm, four numerical experiments were performed with the mesh sizes of 50  50, 100  100, 150  150, and 200  200 in the case of Ra = 105 and / = 0.0, as shown in Table 1. For the best results, the mesh size of 200  200 was chosen (Fig. 1(b)). The criteria for the convergence of the numerical solution for the flow field and temperature field are defined as follows: ker1 ði; jÞk1 6 107 ; and ke2 ði; jÞk1 6 107 where:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  2  urx ði; j; t þ DtÞ  urx ði; j; tÞ þ ury ði; j; t þ DtÞ  ury ði; j; tÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi er1 ði; jÞ ¼  ffi 2 2 urx ði; j; t þ DtÞ þ ury ði; j; t þ DtÞ

149

difference in the orientation of the isotherms (Fig. 3). Thermophoresis and Brownian forces considered in our work may be responsible for this effect, as they act as a source of perturbation. Overall, the results of the current work is in line with the literature and indeed more accurate and the differences are due to the additional forces that have been considered in this work, as well as differences due to using alterative correlations used for the estimation of nanofluid properties. At Ra = 103, no flow was observed regardless of the particle loading and the temperature profile was linear (not shown). For a single phase fluid, the flow becomes unstable for Ra numbers greater than 1708, where the destabilizing upward buoyancy force overcomes the stabilizing gravity force. For nanofluid with a

ð33Þ

e2 ði; jÞ ¼

AbsðT r ði; j; t þ DtÞ  T r ði; j; tÞÞ AbsðT r ði; j; t þ DtÞÞ

ð34Þ

At each Ra number and particle volume fraction, the fluid and particle temperatures, velocities, and particle volume fraction are calculated within the cell, with and without thermophoresis effect considered. In all cases studied here, the Brownian motion is considered to be present. In order to study the heat transfer enhancement in the enclosure, the average Nu number on the bottom wall is computed from the knowledge of the temperature distribution within the enclosure and the temperature gradient on the hot wall [16,38]. The local Nu number along the bottom wall in x direction is defined as follows:

Nux ¼ 

knf @h kf @y

ð35Þ

The average Nu number on the bottom wall, Nu, is obtained by integration. Note that in Ref. [16], the Nu number has been considered as the summation of heat fluxes due to a temperature gradient (regular definition of Nu number), plus heat transfer due to diffusion of nanoparticles, where the latter is insignificant and neglected here. 3. Results and discussion All figures presented in this section show the steady state solution in a cavity filled with a water-based nanofluid (Pr = 7.02) with 10 nm sized copper oxide nanoparticles in a Rayleigh–Benard cell, unless stated otherwise. The code was first validated against standard cases for air [39] with no particles (Fig. 2) and for a waterbased nanofluid with / = 0.1 [40] (Fig. 3). Only numerical data were used toward code validation, owing to the absence of reliable experimental data for the current geometry and boundary conditions. For bottom heating natural convection of air in a cavity, fairly good and consistent agreement between our predictions and those of Ref. [39] is observed at various Ra numbers, as evidenced from Fig. 2. In the case of code validation for a nanofluid, our results for the streamlines and isotherms in a bottom heated enclosure as well as those of Abu-Nada and Oztop [40] are given and compared in Fig. 3. A single phase traditional finite volume numerical scheme is used in Ref. [40], whereas in the present work, the two-phase LBM approach has been adopted. In both works, two circulations have formed in the cell, although in opposite flow directions, which is acceptable since this problem has several numerical solutions. Depending on the magnitude of the Ra number and the extent of the flow perturbation, various solutions may be obtained for a bottom-heated enclosure filled with a pure fluid; these solutions include a single-cell circulation, double-cell horizontal or vertical circulations, either clockwise or counterclockwise [41,42]. The difference in the flow circulation directions in the current work with respect to Ref. [40] has resulted in a

Fig. 6. Streamlines of (a) nanoparticles (particle path-lines) and (b) nanofluid for natural convection in a square cavity heated from below at Ra = 106, Pr = 7.02 and / = 0.1 with the thermophoresis effect considered. Comparison of parts (a) and (b) indicates the induction of a relative velocity between particles and the flow (slip velocity).

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positive thermophoresis coefficient, suspended particles tend to segregate on the cold (top) side, if there is no flow to affect the direction of the temperature gradient. Once flow has developed, the direction of the thermophoresis force may also change leading to a mixed stabilizing or destabilizing effect. To study this effect further, different forces acting on 10 nm sized particles are shown in Fig. 4. In Fig. 4(a), the positive parts of the vertical components of the forces are shown in a logarithmic scale. A discontinuity in the graphs indicates that the missing parts have negative values, i.e., at those locations, the force is acting in the negative direction

or downward. The buoyancy force is positive and constant as shown and it is on the order of 1020 N. The vertical component of the Brownian force has a discontinuous curve, with the negative parts not shown. It is a random and oscillatory force and while its average value is close to zero (not shown), its local and random magnitude can be as large as 1016 N, which is significantly larger than the buoyancy and thermophoresis forces. The induced drag force is positive only on the right side of the cell, where the computed velocity vectors were found to act upward. It is noted that the direction of velocities may become reversed given than the

Fig. 7. Isotherms for natural convection in a square cavity heated from below at (a) Ra = 104 and (b) Ra = 106, Pr = 7.02 and / = 0.0.

Fig. 8. Isotherms for natural convection in a square cavity heated from below at Ra = 104. (a) Without the thermophoresis effect, and (b) with the thermophoresis effect. Pr = 7.02 and / = 0.10. At Ra = 104, thermophoresis has negligible effect on isotherms, although it may cause a perturbation in the flow and a change in the direction of the flow circulations.

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studied problem has more than one flow solution. The induced drag force as a result of external forces is the largest force acting on the particles and is of the order of 1012 N. The thermophoresis force is only positive and acting upward for a small portion of the horizontal axis shown. This is because in Eq. (20), the local temperature gradient is used to calculate the direction of the thermophoresis force. While the global temperature gradient between the bottom and top walls is always upward given that the bottom wall is heated and top wall is cooled, the local temperature gradients may change sign. A change in the direction of the thermophoresis

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force in the system is therefore only observed when a fluid flow is present. In the stationary systems subject to a temperature gradient, thermodiffusion or thermophoresis force always acts in a direction opposite to that of the temperature gradient, e.g., [1,43]. In Fig. 3b, the magnitude of the resultant forces in x and y directions are shown. This Figure can particularly demonstrate the relative magnitude of the forces acting on a 10 nm nanoparticle in the nanofluid. The dominant force is the induced particle drag, followed by the Brownian force, and at the end the gravitational and the thermophoresis forces. In spite of the small magnitude of

Fig. 9. Isotherms for natural convection in a square cavity heated from below at Ra = 106. (a) Without thermophoresis effect and (b) with thermophoresis effect. Pr = 7.02 and / = 0.10. At Ra = 106, the effect of thermophoresis is more profound compared to Ra = 104 (Fig. 8).

Fig. 10. Normalized horizontal velocity u⁄ profiles at the vertical mid-plane (x/H = 0.0) for different values of Ra = 103, 104 and 106. (a) Velocity profile without the thermophoresis effect and (b) velocity profile with the thermophoresis effect. Pr = 7.02 and / = 0.10. Thermophoresis has altered the velocity patterns at Ra = 106, particularly adjacent to the bottom wall.

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the thermophoresis force, it has some effects on the heat transfer augmentation, to be studied later in this paper. The nanofluid streamlines and nanoparticle path-lines for a nanofluid with / = 0.1 are standard flow characteristics and are shown in Figs. 5 and 6 for Ra = 104, and Ra = 106, respectively. At Ra = 104 (Fig. 5), a single circulation forms consistent with the results of a pure fluid and low Ra number studies on nanofluids performed by others, e.g., [16,44]. At Ra = 106 (Fig. 6), the flow is asymmetric due to the presence of a stronger flow field. Particle path-lines do not necessarily follow the flow streamlines, indicating that external forces affect the trajectory of the nanoparticles. This slip velocity is a source of better mixing and heat transfer enhancement.

As another general observation, Fig. 7 displays the isotherms, i.e., contours of constant temperatures, with pure water (/ = 0.0) for two Ra numbers. At Ra = 103, the no flow condition, the isotherms are a series of lines parallel to the top and bottom surfaces (not shown). At Ra = 104 and larger, a natural convection flow develops where the isotherms are bent and concentrated upward on the right side of the cell, under the influence of a positive vertical velocity on the right side, and similarly bent and concentrated downward on the left side because of a downward vertical velocity on the left side (counter clockwise solution of Rayleigh–Benard cell). At Ra = 106, the isotherms do not follow a specific pattern, as a result of a stronger induced flow. The temperature gradients

Fig. 11. Normalized temperature h profiles at the vertical mid-plane (x/H = 0.0) for different values of Ra = 103, 104, and 106. (a) Temperature profile without the thermophoresis effect and (b) temperature profile with the thermophoresis effect. Pr = 7.02 and / = 0.10. Thermophoresis affects the temperature profiles at Ra = 106.

Fig. 12. Normalized horizontal velocity profiles u⁄ of nanofluid at various particle loadings at the vertical mid-plane (x/H = 0.0). (a) Without the thermophoresis effect and (b) with the thermophoresis effect. Pr = 7.02 and Ra = 104. Thermophoresis has insignificant effect on velocity profiles at Ra = 104.

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and therefore the direction of the thermophoresis force are perpendicular to the isotherms. Figs. 8 and 9 display the isotherms for a cell with Ra = 104 and 6 10 , respectively, with the particle loading (/ = 0.10) for cases with and without thermophoresis effect. It is observed that the presence of thermophoresis in the cell may affect the pattern of the isotherms. At Ra = 104 (Fig. 8), the isotherms for cases with and without thermophoresis are identical. In some simulations with a coarser mesh (not shown here) it was observed that the isotherms for the case with thermophoresis was the mirror image of the case without thermophoresis, indicating a change in the direction of circulation. It was mentioned before that a Rayleigh–Benard problem

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may have several numerical solutions, including clockwise and counterclockwise circulations [41,42]. Inclusion of the thermophoresis force into the problem may result in switching the numerical solution from counterclockwise to clockwise. Thermophoresis force acts as a perturbation source. At Ra = 106 (Fig. 9), the flow is stronger when the thermophoresis force is considered, and the isotherms in the center of the cell are modified due to the fact that thermophoresis force enhances the flow mixing. Fig. 10 displays the normalized horizontal velocity u⁄ profiles of the nanofluid at the vertical mid-plane (x⁄/H = 0.0) versus y⁄ for / = 0.10 at different values of Ra, and for cases with and without the thermophoresis effect. At Ra = 103, the velocity is zero everywhere,

Fig. 13. Normalized temperature profiles h of nanofluid for various particle loadings at the vertical mid-plane (x/H = 0.0). (a) Without the thermophoresis effect and (b) with the thermophoresis effect. Pr = 7.02 and Ra = 104. Thermophoresis has insignificant effect on temperature profiles at Ra = 104.

Fig. 14. (a) Normalized horizontal velocity u⁄ of the base fluid, nanofluid and nanoparticles; (b) positive portion of the normalized horizontal velocity u⁄ in part (a) on the logarithmic scale. Plotted at the vertical mid-plane (x/H = 0.0) with the effect of thermophoresis considered. Pr = 7.02, Ra = 106 and / = 0.1.

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as expected. At Ra = 104, a weak velocity profile develops. At Ra = 106, a stronger natural convection flow develops, and thermophoresis has an influence on the direction of the flow close to the hot (bottom) wall. Similarly, Fig. 11 displays the dimensionless temperature profiles h of the nanofluid associated with the conditions of Fig. 10. At Ra = 103, the temperature profile is linear owing to the absence of flow in the cell. At Ra = 104, and Ra = 106, convective flow develops resulting in mixing in the center of the cell. As Ra number increases, the length of the mixing zone increases. In the mixing zone, the temperature profile tends to become uniform with some irregularities due to the presence of complex circulations. Close to the walls, the temperature changes linearly,

indicating that convection is weak at the vicinity of the walls and heat transfer is dominated by conduction. Comparison of Figs. 11(a) and (b) shows that thermophoresis has a clear effect when Ra = 106, where it results in a more uniform temperature profile in the mixing zone, which is an indication of a better mixing and heat transfer enhancement. The effect of particle loading on the normalized horizontal velocity profiles u⁄ at Ra = 104 is shown in Fig. 12. As the particle loading increases, the flow becomes more viscous and velocities decrease. Thermophoresis does not seem to have a significant effect on the velocity profiles. Fig. 13 shows the normalized temperature profiles h of the nanofluid for various particle loadings

Fig. 15. Nanoparticle volume fraction distribution at steady state. (a) Without thermophoresis, at Ra = 104, (b) with thermophoresis at Ra = 104, (c) without thermophoresis at Ra = 106 and (d) with thermophoresis at Ra = 106, Pr = 7.02 and / = 0.1. Thermophoresis is a stronger effect at Ra = 106.

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with and without thermophoresis effect at Ra = 104. Thermophoresis effect on the temperature profiles is insignificant. In Fig. 14, the horizontal velocity components of the nanoparticles, the base fluid and the nanofluid are plotted for Ra = 106 and / = 0.1. In Fig. 14(a) the velocities are displayed in a conventional scale. It is evident that the velocity of the nanoparticles is much smaller than those of the base fluid and the nanofluid. Nonetheless, this small velocity is important enough to affect the velocity of the base fluid resulting in a different overall velocity for the nanofluid, as shown. To better observe and investigate the relative magnitudes of the aforementioned velocities, a logarithmic scale is employed to illustrate the positive potion of the velocity profiles (Fig. 14(b)). The normalized velocity of the nanoparticles is on the order of 105 to 104, while that of the nanofluid is on the order of 102. It is also noted that the velocities are zero on the walls as well as on the center of the cell. The locations of the maximum positive velocities of the nanoparticles and the nanofluid are also shown in the figure. When computations are initialized, nanoparticles are evenly distributed within the cell. Under the influence of the existing external and drag forces, such as gravity, thermophoresis, Brownian and drag forces, nanoparticles may be displaced. Sever particle displacement, settlement and entrapment in the flow circulations have been observed for the case of micron-sized particle-laden flows, e.g., [45]. The response of nanoparticles to the existing forces may be different. The distribution of the local particle volume fraction within the cell at the steady state condition is computed and shown in Fig. 14, for Ra = 104 and 106, and for cases with and without the thermophoresis effect. The initial particle volume fraction is / = 0.1. Comparison of the cases with and without thermophoresis show that although it is expected that, at least in stationary systems, the thermophoresis force pushes the nanoparticles towards the cold (top) wall, its effect is not significant in the laminar natural convection problem studied here. Although, the nanoparticles may individually move and displace in the cell, the results show a nearly uniform particle distribution at the steady state. Similar observations were made in Ref. [25], although for different boundary conditions (differentially-heated); but the physical nature of the two problems is similar (see Fig. 15). Addressing issues regarding the heat transfer enhancement or deterioration by using nanofluids is an interesting and also a controversial matter. Some researchers in some cases even reported a deterioration of heat transfer when nanoparticles are added, through a decrease in the Nu number below 1, the conduction limit, which seems questionable [15]. Fig. 16 shows the variation of the average Nu number versus the nanoparticle volume fraction at different values of Ra number. Contribution of thermophoresis is also shown. The Nu number starts from 1, occurring at small Ra numbers, which warrants a no-flow and therefore the conduction condition. As Ra number increases, regardless of the particle loading or the thermophoresis effect, the Nu number increases due to the development of natural convection within the cell. Inclusion of nanoparticles to water, in the range of parameters studied here, shows an increase in the Nu number. Thermophoresis is also shown as a fairly important effect to be considered for the simulation purposes. Its effect is magnified at higher Ra numbers but its enhancing effect remains well below 10%. At Ra = 103, although the fluid is stable, Nu increases with particle volume concentration. This is because based on the definition of Nu given by Eq. (35), the thermal conductivity of the nanofluid increases leading to an increase in Nu number, although the temperature gradient remains constant. As the Ra number increases, addition of nanoparticles to the base fluid results in a higher increase in the Nu number. At Ra = 106 for a particle loading of 10% (/ = 0.1), 18% relative increase in the Nu number is observed. No optimum particle loading was

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Fig. 16. Variation of the average Nu number along the heated bottom plate versus nanoparticle volume fraction at different values of Ra, for two cases, with the thermophoresis effect, and without the thermophoresis effect. Thermophoresis has a considerable effect on the Nu number, and its effect increases as Ra increases.

observed for having a maximum heat transfer augmentation, as suggested by several other workers, e.g., [15]. It is noted that at larger Ra numbers, the physics of the problem may change significantly as a result of oscillatory behavior and development of turbulent flows in the cell, and therefore, the results obtained here may not be necessarily extendable to Ra numbers greater than 106. There is no reliable experimental data on Rayleigh–Benard cell. Results of Wen and Ding [12] in a Rayleigh–Benard cell are dubious due to particle agglomeration. More recent experimental data in a differentially-heated enclosure predict an increase in Nu number, but up to a volume concentration of 2%, whereas our numerical results, predict an increase in Nu number with particle volume concentration within the range of 0–10%. This may be due to two possible issues, one the inaccuracy of the experimental data for a nanofluid with a high particle volume fraction, which is quite likely as particle agglomeration may occur if the nanofluid is highly concentrated. The other possibility is inadequacy of our numerical simulation. Other factors may become important or different in a concentrated nanofluid. For instance, the Maxwell correlation used here to estimate nanofluid viscosity is valid for low particle concentration nanofluids. Nevertheless, within the range of 0–2% volume particle concentration, which is the practical range for the application of a nanofluid, the numerical results are in agreement with the experimental observations, although no direct and quantitative comparison could be made due to the lack of experimental data for the boundary conditions used here. 4. Conclusions Laminar natural convection in a square cavity heated from below and filled with a water-copper oxide nanofluid was studied at steady state using the two-phase lattice Boltzmann method. Cases with Ra numbers from 103 to 106 with 10 nm copper oxide nanoparticles and particle volume fractions from 0 to 0.1 were considered. The effect of external forces and in particular the thermophoresis force on the flow and heat transfer behavior was explored. The following main conclusions are made:

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1. The Nu number increases significantly from 1 for the no-flow condition at low Ra numbers to much higher values when Ra increases. The Nu number increases considerably with inclusion of nanoparticles as well. No optimum particle loading is observed. 2. Thermophoresis is a weak force in the laminar natural convection but its effect is intensified as Ra increases. It slightly affects the flow patterns, but considerably affects the isotherms. It is also responsible for an increase in the average Nu number up to about 10% at high Ra numbers (106). The gravitational forces including the particle weight and buoyancy force have also some role in the particle slip velocity and heat transfer enhancement. The largest external force affecting the particles is the random oscillatory Brownian force. The largest force in the system is the drag force, which is induced as a result of the presence of external forces that cause a slip/drift velocity on particles with respect to the main flow. 3. Velocities of the nanoparticles are several orders of magnitude smaller than those of the nanofluid, indicating a strong slip velocity on the particles surface created as a result of the large magnitude of external forces exerted on nanoparticles. In natural convection and within the range of parameters explored here, external forces were found large compared to the fluid flow momentum or force and affect and modify the trajectory of nanoparticles such that the nanoparticles path-lines may not be the same as the flow streamlines. 4. In the range of the parameters studied in this work, under the influence of the existing forces, the nanoparticles neither settle nor segregate from the main nanofluid. They remain suspended, although may undergo mixing and displacement. 5. An increase in the Ra number, results in an increase in the nanofluid velocity, whereas an increase in the particle loading results in a decrease in the nanofluid velocity. 6. Overall, it may be concluded that at low Ra numbers, the nanofluid (base fluid and nanoparticles) behaves homogenously, but as the Ra number increases, the flow starts to behave heterogeneously. Conflict of interest Authors declare no conflict of interest. References [1] M. Eslamian, Advances in thermodiffusion and thermophoresis (Soret effect) in liquid mixtures, Front. Heat Mass Transfer 2 (2011) 043001. [2] M. Ahmed, M. Eslamian, Natural convection in a differentially-heated square enclosure filled with a nanofluid: significance of the thermophoresis force and slip/drift velocity, Int. Commun. Heat Mass Transfer 58 (2014) 1–11. [3] J. Buongiorno, Convective transport in nanofluids, J. Heat Transfer 128 (2006) 240–250. [4] D.R. Ray, D.K. Das, R.S. Vajjha, Experimental and numerical investigations of nanofluids performance in a compact minichannel plate heat exchanger, Int. J. Heat Mass Transfer 71 (2014) 732–746. [5] M. Eslamian, M.Z. Saghir, Novel thermophoretic particle separators: numerical analysis and simulation, Appl. Therm. Eng. 59 (2013) 527–534. [6] Y.H. Diao, Y. Liu, R. Wang, Y.H. Zhao, L. Guo, X. Tang, Effects of nanofluids and nanocoatings on the thermal performance of an evaporator with rectangular microchannels, Int. J. Heat Mass Transfer 67 (2013) 183–193. [7] M. Glassl, M. Hilt, W. Zimmermann, Convection in nanofluids with a particleconcentration-dependent thermal conductivity, Phys. Rev. E 83 (2011) 046315. [8] B. Elhajjar, G. Bachir, A. Mojtabi, C. Fakih, M.C. Charrier-Mojtabi, Modeling of Rayleigh–Bénard natural convection heat transfer in nanofluids, C.R. Mec. 338 (2010) 350–354. [9] 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 Transfer 50 (2007) 4003–4010. [10] J. Kim, Y.T. Kang, C.K. Choi, Soret and Dufour effects on convective instabilities in binary nanofluids for absorption application, Int. J. Refrig. 30 (2007) 323– 328. [11] D.Y. Tzou, Thermal instability of nanofluids in natural convection, Int. J. Heat Mass Transfer 51 (2008) 2967–2979.

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