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Nano Fluids
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Natural convection of nanofluidsflow with “nanofluid-oriented”models of thermal conductivityand dynamic viscosity in the
presence of heat sourceGeorge C. Bourantas
Department of Medical Physics, School of Medicine,University of Patras, Patras, Greece
Eugenios D. SkourasDepartment of Chemical Engineering, University of Patras, Patras, Greece andInstitute of Chemical Engineering and High Temperature Chemical Processes,
Foundation for Research and Technology, Patras, Greece
Vassilios C. LoukopoulosDepartment of Physics, University of Patras, Patras, Greece, and
George C. NikiforidisDepartment of Medical Physics, School of Medicine,
University of Patras, Patras, Greece
Abstract
Purpose – The purpose of this paper is to make a numerical study of natural convection ofwater-based nanofluids in a square cavity when a discrete heat source is embedded on the bottom wall,applying a “nanofluid-oriented” model for the calculation of the effective thermal conductivity(Xu-Yu-Zou-Xu’s model) and the effective dynamic viscosity ( Jang-Lee-Hwang-Choi’s model). Anothermotivation is the numerical solution of the equations of the flow with a meshless method.
Design/methodology/approach – A meshless point collocation method with moving least squares(MLS) approximation is used. A test validation study of the numerical method takes place for purewater flow, as well for water/Al2O3 nanofluids. The influence of pertinent parameters such as Rayleighnumber (Ra), the non-uniform nanoparticle size keeping the mean nanoparticle diameter fixed, thevolume fraction of nanoparticles and the location of heat source on the cooling performance arestudied.
Findings – The presence of a discrete heat source, as well as the various thermal boundaryconditions affects the characteristics of the nanofluid flow and heat transfer. When the ratio ofminimum to maximum nanoparticle diameter is increased, the local Nusselt number is increased andthe heat source temperature is decreased. The increase of solid volume fraction of nanoparticles causesthe heat source maximum temperature to decrease and the Nusselt Number to increase.
Originality/value – The present study constitutes an original contribution to the nanofluid flow andheat transfer characteristics when a discrete heat source is presence. “Nanofluid-oriented” models areused for the calculation of the effective thermal conductivity and dynamic viscosity.
Keywords Flow, Viscosity, Thermal conductivity, Mathematical analysis, Nanofluids,Effective thermal conductivity, Effective dynamic viscosity, Meshfree point collocation method
Paper type Research paper
The current issue and full text archive of this journal is available at
www.emeraldinsight.com/0961-5539.htm
Received 15 August 2010Revised 9 March 201128 May 2011Accepted 8 June 2011
International Journal of NumericalMethods for Heat & Fluid FlowVol. 23 No. 2, 2013pp. 248-274q Emerald Group Publishing Limited0961-5539DOI 10.1108/09615531311293452
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1. IntroductionA nanofluid is a mixture consisting of nanometer-sized particles and fibers dispersedin a liquid. This mixture has as result the alteration of physical properties of the basefluid, such as viscocity, density and heat transfer, among others. The concept ofnanofluid, in its primitive form, was first reported in the middle of the eighteenthcentury, but it was possible to put it into practice only after the tremendous developmentof nanotechnologies during the last decades (Choi, 1995).
Choi was the first who introduced the thermal conductivity enhancement and,measured the thermal conductivity of nanofluids. Since the pioneer work of Choi, a largenumber of experimental and theoretical studies have been carried out by numerousresearchers (Xie et al., 2002; Xuan and Le, 2000; Keblinski et al., 2002). Compared to aconventional liquid and a conventional two-phase mixture, nanofluids have someinteresting properties, namely, they have higher thermal conductivity, they do not blockflow channels and, they induce a very small pressure drop. Furthermore, nanoparticlesresist sedimentation, as compared to larger particles, due to Brownian motion andinter-particle forces.
A great interest appeared for natural convection heat transfer characteristics ofnanofluids, due to the wide applications in electronic cooling, heat exchangers, doublepane windows, etc. Nanofluids are used in a wide range of engineering applications,although the development of the field faces several challenges such as the lack ofagreement between experimental (or between experimental and theoretical) resultsprovided by different groups, the poor performance of suspensions and, the lack oftheoretical understanding of the mechanisms. In order to identify innovating applicationsfor these fields, further theoretical and experimental research investigations are neededto understand the heat transfer characteristics of nanofluids (Wang and Mujumdar,2007). Regarding the lack of agreement between experimental (or even experimentaland theoretical) results provided by different groups, a representative case is thefollowing. Authors in Khanafer et al. (2003) presented a 2D numerical simulation ofnatural convection of nanofluids in a vertical rectangular enclosure, utilizing a dispersionmodel similar to that for the flow through porous media. They concluded that as thevolumetric fraction of the copper nanoparticles in water was increased, at any Grashofnumber, heat transfer across the enclosure was increased also. However, contradictoryresults have been observed in the experimental studies (Pruta et al., 2003; Wen andDing, 2005).
Over the last years, numerous theoretical investigations and several models havebeen proposed for the effective thermal conductivity to include various mechanisms,so that to predict the anomalously high thermal conductivity of nanofluids. Moreprecisely, many possible mechanisms for the anomalous increase in nanofluids heattransfer have been consider, including mechanisms as, Brownian motion of thenanoparticles, liquid layering at the liquid/particle interface, nature of the heat transportin the nanoparticles, and, the effect of nanoparticle clustering (Murshed et al., 2008).Furthermore, the increased surface area due to suspended nanoparticles, the increasedthermal conductivity of the fluid, the interaction and collision among particles, theintensified mixing fluctuation and turbulence of the fluid, and finally, the dispersion ofnanoparticles, are some physical properties that may play crucial role to thedetermination of the thermo-physical properties of nanofluids (Wang and Mujumdar,2007). Additional to these factors the non-uniform nanoparticle size and mean
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nanoparticle diameter were taken into account in Xu et al. (2006). Besides thesemechanisms, authors in Murshed et al. (2008), believe that the effects of particlesurface chemistry and particles interaction for nanometer-sized particles could besignificant in enhancing the thermal conductivity of nanofluids. Since the “classical”models were found to be unable to predict the anomalously high thermal conductivityof nanofluids, many theoretical studies have been carried out to predict theanomalously increased thermal conductivity of nanofluids (“modern” or “nanofluids-oriented” models). A detailed summary of all classical and recently developed modelsfor the prediction of the effective thermal conductivity of nanofluids is provided inMurshed et al. (2008).
The need for validation of theoretical results and the need of less expensiveexperiments led to the development of sophisticated numerical methods in order tofacilitate the so-called numerical experiments. On the other hand, theoretical scientistneeded numerical methods that could provide accurate results with low computationalcost. In the field of nanofluids, the most popular numerical method used is the wellestablished finite volume method. In recent years, research on meshless (or meshfree)methods has made significant progress, particularly in the area of computationalmechanics (Liu, 2002). Meshfree methods are a particular class of numerical simulationalgorithms for the simulation of physical phenomena. Meshfree methods eliminatesome or all of the traditional mesh-based view of the computational domain and relyon a particle (either Lagrangian or Eulerian) view of the physical problem (Atluri andShen, 2002; Liu and Gu, 2005). A goal of meshfree methods is to facilitate thesimulation of increasingly demanding problems, arising in engineering and scienceand, that require the ability to treat large deformations, advanced materials, complexgeometry, nonlinear material behavior, discontinuities and singularities. A surveyon some of the most significant studies which use different types of meshlessnumerical methods on the solution of the Navier-Stokes equations has been given inBourantas et al. (2010).
Regarding of numerical simulations, as it is mentioned in Khanafer et al. (2003) andWang and Mujumdar (2007), two approaches have been adopted in the literature in orderto investigate the heat transfer characteristics of nanofluids, a continuum and atwo-phase model approach. In details, the first approach assumes that the continuumassumption is still valid for fluids with suspended nanosize particles, while the otherapproach uses a two-phase model for better description of both the fluid and the solidphases. Additionally, the single phase model is much simpler and computationally moreefficient and it is common in the open literature. Nevertheless, the great number ofpossible mechanism, which maybe explains the anomalously high thermal conductivityof nanofluids, is difficult to be described (Wang and Mujumdar, 2007). In Aminossadatiand Ghasemi (2009) presented a numerical study of natural convection cooling of a heatsource embedded on the bottom wall of an enclosure filled with nanofluids. Therein, thetop and vertical walls of the enclosure were maintained at a relatively low temperature.The transport equations for a Newtonian fluid were solved numerically with a finitevolume approach using the simple algorithm. In this study Brinkman’s model had beenused for the effective dynamic viscosity, while Maxwell’s model was used for theeffective thermal conductivity. The results indicate that adding nanoparticles into purewater improves its cooling performance especially at low Rayleigh numbers.Additionally, in Jang et al. (2007), thermal characteristics of natural convection
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in a rectangular cavity heated from below with water-based nanofluids containingalumina (Al2O3 nanofluids) were theoretically investigated with Jang and Choi’s modelfor predicting the effective thermal conductivity of nanofluids and various models forthe effective viscosity. The theoretical results were compared with experimental resultspresented there. It was shown that the experimental results were between a theoreticalline derived from Jang and Choi’s model and Einstein’s model and a theoretical line fromJang and Choi’s model and Pak and Cho’s correlation. Authors in Lin and Violi (2010)analyzed the heat transfer and fluid flow of natural convection in a cavity filled withAl2O3/water nanofluid that operates under differentially heated walls. TheNavier-Stokes and energy equations were solved numerically, coupling Xu’s model(Xu et al., 2006) for calculating the effective thermal conductivity and Jang’s model( Jang et al., 2007) for determining the effective dynamic viscosity, with the slipmechanism in nanofluids. The heat transfer rates were examined for parameters ofnon-uniform nanoparticle size, mean nanoparticle diameter, nanoparticle volumefraction, Prandtl number, and Grashof number. Enhanced and mitigated heat transfereffects due to the presence of nanoparticles were identified and highlighted. Based onthese insights, they determined the impact of fluid temperature on the heat transfer ofnanofluids.
In the present work we investigate numerically the natural convection of water-basednanofluids in a vertical cavity when a discrete heat source is embedded on the bottomwall, applying a “nanofluid-oriented” model for the calculation of the effective thermalconductivity (Xu-Yu-Zou-Xu’s model), (Xu et al., 2006) and for the determination of theeffective dynamic viscosity ( Jang-Lee-Hwang-Choi’s model), ( Jang et al., 2007). The type ofnanoparticles that is taken into consideration is the aluminium oxide (Al2O3). We examinethe influence of pertinent parameters such as Rayleigh number (Ra), the non-uniformnanoparticle size keeping the mean nanoparticle diameter fixed, the volume fraction ofnanoparticles and the location of the heat source on the cooling performance. The effects ofvarious thermal boundary conditions on the nanofluid flow formations for differentlocations of heat source are also studied. Moreover, we solve the equations of the flow witha meshless method. To our attention, the meshless methods have not been used until nowfor the calculation of natural convection of nanofluid (in rectangular enclosures). One ofour goals is to use the meshless point collocation method (MPCM) with moving leastsquares (MLS) approximation, in order to obtain numerical results for nanofluids flow andverify the accuracy and the efficiency of the method. For validation purposes results arecompared with previous published results.
The rest of the paper is organized as follows. Section 2 is referred to the physicalproblem description. In Section 3, we describe the main steps of the meshless pointcollocation numerical procedure, which is used and an algorithm validation is realized.Numerics for nanofluids flow in the presence of a heat source take place in Section 4,where a natural convection problem in a rectangular enclosure filled with nanofluids isinvestigated. Finally, at Section 5 the conclusions are presented.
2. Problem descriptionFigure 1 shows the spatial domain of the 2D enclosure considered in the present study.A heat source is located on the thermally insulated bottom wall of the enclosure. In thisfigure, b is the length of heat source and d is the distance of heat source from the leftwall. The horizontal upper wall is adiabatic and impermeable to mass transfer.
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The vertical walls of the enclosure are maintained at a relatively high temperature(TH) the left wall and a relatively low temperature (TC) the right wall. Furthermore,another set of boundary conditions are also applied and, in order to demonstrate thecrucial role of thermal boundary conditions to the thermal properties of nanofluids.The thermo-physical properties of the base fluid and the nanoparticles are given inTable I.
As far as the nanofluid flow boundary conditions, the no-slip and no penetrationassumptions are imposed on the walls (Lin and Violi, 2010). Furthermore, severalassumptions are made regarding the mathematical equations that describe thephysical model. In details:
. The nanofluid in the cavity is Newtonian, incompressible, and the flow is steady,laminar and non-isothermal.
. The thermo-physical properties are constant except for the density in thebuoyancy force (Boussinesq’s hypothesis).
. The fluid phase and nanoparticles are in a thermal equilibrium state.
. It is assumed that the nanoparticles of Al2O3 are uniformly suspended in water;there is no aggregation of nanoparticles in the fluid medium, as demonstrated
r (kg/m3) Cr ( J/kgK) k (W/mK) b £ 102 s (K21)
Pure water 997.1 4,179 0.613 21Alumina (A12O3) 3,900 850 46 1.67
Source: Lin and Violi (2010)
Table I.Thermo-physicalproperties of waterand nanoparticles
Figure 1.A schematic diagram ofthe physical model
L
L
TH TCg
Nanofluid
d q''
b
x
y
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experimentally in Jang et al. (2009) with the use of transmission electronmicroscopy (TEM). Actually, there are few agglomerated nanoparticles and mostnanoparticles suspended in a nanofluid would not aggregate.
. Nanoparticles are spherical.
Fluid flow is governed by the Navier-Stokes equations, a well-known coupled set ofnon-linear partial differential equations expressing the conservation of mass, linearmomentum, and energy. The governing equations in terms of the primary variablesformulation are:
7 ·u* ¼ 0; ð1Þ
ðu* ·7 Þu* ¼1
rnf
ðrb Þnf g T * 2 T*/
� �� �2
1
rnf
7p * þmnf
rnf
72u*; ð2Þ
ðu* ·7 ÞT * ¼ anf72T *; ð3Þ
where u* is the dimensional velocity vector, g is the acceleration of gravity, p* isthe pressure, T * is the temperature, while the effective density of the nanofluid isgiven as:
rnf ¼ ð1 2 wÞrf þ wrs ð4Þ
and w is the solid volume fraction of nanoparticles. The thermal diffusivity of thenanofluid is:
anf ¼knf
ðrcpÞnf
; ð5Þ
where, the heat capacitance of the nanofluid is given as:
ðrcpÞnf ¼ ð1 2 wÞðrcpÞf þ w ðrcpÞs: ð6Þ
Additionally the thermal expansion coefficient of the nanofluid can be determined by:
ðrbÞnf ¼ ð1 2 wÞðrbÞf þ w ðrbÞs: ð7Þ
The subscript f is referred to the base fluid, while the subscript s to the solidnanoparticles.
In order to explain the observed phenomena, many theoretical studies on the effectivethermal conductivity in nanofluids have been proposed, with the various modelsgrouped in two main categories (Murshed et al., 2008). As far as the first category,it considers stationary nanoparticles in multiphase systems, including a static modelfor heat conductivity and the second category include a dynamic model for heatconductivity.
Recently, Xu et al. (2006) derived a new model for predicting the thermalconductivity of nanofluids by taking into account the fractal distribution ofnanoparticle sizes and heat convection between nanoparticles and liquids due to theBrownian motion of nanoparticles in fluids. The proposed model is expressed as afunction of the average size of nanoparticles, fractal dimension, concentration of
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253
nanoparticles, temperature and properties of fluids. In this way, the total dimensionlessthermal conductivity of nanofluids is expressed as:
knf
kf
¼ks þ 2kf 2 2wðkf 2 ksÞ
ks þ 2kf þ wðkf 2 ksÞþ c
Nupdf
Pr
ð2 2 Df ÞDf
ð1 2 Df Þ2
½ðdmax =dmin Þ12Df 2 1�
½ðdmax =dmin Þ22Df 2 1�
1
dp: ð8Þ
The aforementioned model of Xu et al. (2006) has been chosen in the present study forthe description of thermal conductivity of nanofluids. The first term is the Hamiltonand Crosser model (Hamilton and Crosser, 1962), which considers the nanoparticlesin the liquid as stationary, and the second term is the thermal conductivity based onheat convection due to Brownian motion. c is an empirical constant, which is relevantto the thermal boundary layer and dependent on different fluids (e.g. c ¼ 85 for thede-ionized water and c ¼ 280 for ethylene glycol) but independent of the type ofnanoparticles. Nup is the Nusselt number for liquid flowing around a sphericalparticle and equal to Nup ¼ 2 for a single particle in this work. The fluid moleculardiameter df is taken as 4.5 £ 102 10 m for water in present study, as in Lin andVioli (2010). Pr is the Prandtl number, w and dp are the nanoparticle volume fractionand mean nanoparticle diameter, respectively. The fractal dimension Df isdetermined by:
Df ¼ 2 2lnw
lnðdp;min =dp;max Þ; ð9Þ
where dp,max and dp,min are the maximum and minimum diameters of nanoparticles,respectively. With the given/measured ratio of (dp,min /dp,max ) ¼ R, the minimum andmaximum diameters of nanoparticles can be obtained with mean nanoparticlediameter dp from the statistical property of fractal media:
dp;max ¼ dpDf 2 1
Df
dp;min
dp;max
� �21
; dp;min ¼ dpDf 2 1
Df
ð10Þ
On the other hand, regarding the effective dynamic viscosity of water/Al2O3
nanofluids, the model of Jang and Choi is used ( Jang et al., 2007), respectively. Thismodel was presented by Jang and Choi for a fluid, which contain a dilute suspensionof small rigid spherical particles and it accounts for the slip mechanism in nanofluids,capturing the new rheological features of nanofluids. Thus, the effective dynamicviscosity of nanofluids in the above equations is:
mnf ¼ mf ð1 þ 2:5wÞ 1 þ hdp
H
� �221
w ð2=3Þð1þ1Þ
" #: ð11Þ
The empirical constant 1 and c are 20.25 and 280 for Al2O3, respectively.Introduce the following dimensionless quantities:
x ¼x *
L; y ¼
y *
L; u ¼
u *L
af
; v ¼v *L
af
; p ¼p *L 2
ra2f
; u ¼T * 2 T *
C
T*H 2 T*
C
; ð12Þ
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where L is the dimensional cavity width. Then, the continuity, momentum, and energyequations (1)-(3), in non-dimensional form, in Cartesian system of coordinates aregiven as:
7 ·u ¼ 0; ð13Þ
ðu ·7 Þu ¼21
ðð1 2 wÞ þ w ð rs=r f ÞÞ7p
þð1 þ 2:5wÞ½1 þ h ðdp=H Þ221w ð2=3Þð1þ 1Þ�Pr
ðð1 2 wÞ þ w ðrs=rf ÞÞ72u
þðð1 2 wÞ þ w ðrsbs=rfbf ÞÞRaPr
ðð1 2 wÞ þ w ð rs=rf ÞÞuj;
ð14Þ
ðu ·7 Þu ¼knf =kf
ðð1 2 wÞ þ w ððrcÞs=ðrcf ÞÞÞ72u: ð15Þ
Taking the curl of the two sides for the above equation we get:
7 £v ¼ 272u; ð16Þ
2ðv ·7 Þu þ ðu ·7 Þv ¼ð1 þ 2:5wÞ½1 þ h ðdp=H Þ221w ð2=3Þð1þ 1Þ�Pr
ðð1 2 wÞ þ w ðrs=rf ÞÞ72v
þðð1 2 wÞ þ w ðrsbs=rfbf ÞÞRaPr
ðð1 2 wÞ þ w ðrs=rf ÞÞ
›u
›xk;
ð17Þ
ðu ·7Þu ¼knf =kf
ðð1 2 wÞ þ w ððrcÞs=ðrcÞf ÞÞ72u; ð18Þ
where the Reynolds, Prandtl and Rayleigh numbers of the flow are defined as:
Re ¼rf UD
mf
; Pr ¼mf
af rf
; Ra ¼gbH 3ðTH 2 TLÞ
na: ð19Þ
For 2D, v ¼ vk and u ¼ (u, v) and the above equations can be written:
72u ¼ 2›v
›y; ð20Þ
72v ¼›v
›x; ð21Þ
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255
u›v
›xþ v
›v
›y¼ð1 þ 2:5wÞ½1 þ hðdp=H Þ221w ð2=3Þð1þ 1Þ�Pr
ðð1 2 wÞ þ w ðrs=rf ÞÞ
›2v
›x 2þ
›2v
›y 2
� �
þðð1 2 wÞ þ w ðrsbs=rfbf ÞÞRaPr
ðð1 2 wÞ þ w ðrs=rf ÞÞ
›u
›x;
ð22Þ
u›u
›xþ v
›u
›y¼
knf=kf
ðð1 2 wÞ þ w ððrcÞs=ðrcÞf ÞÞ
›2u
›x 2þ
›2u
›y 2
� �: ð23Þ
The boundary conditions, used to solve the equations (20)-(23) are as follows:
u ¼ v ¼ 0 and u ¼ 1 for x ¼ 0 and 0 # y # 1;
u ¼ v ¼ 0 and u ¼ 0 for x ¼ 1 and 0 # y # 1;
u ¼ v ¼ 0 and›u
›y¼ 0 for y ¼ 0 and 0 # x # ðD 2 0:5BÞ;
u ¼ v ¼ 0 and›u
›y¼ 2
kf
knf
for y ¼ 0 and ðD 2 0:5BÞ # x # ðD þ 0:5BÞ;
u ¼ v ¼ 0 and›u
›y¼ 0 for y ¼ 0 and ðD þ 0:5BÞ # x # 1;
u ¼ v ¼ 0 and›u
›y¼ 0 for y ¼ 1 and 0 # x # 1:
ð24Þ
The local Nusselt number on the heat source surface can be defined as:
Nus ¼hL
kf
; ð25Þ
where, h is the convection heat transfer coefficient:
h ¼q
Ts 2 Tcð26Þ
and Ts is the dimensional temperature of the heat source. Rearranging the local Nusseltnumber by using the dimensionless parameters (equation (12)) yields:
NusðxÞ ¼1
usðxÞ; ð27Þ
where, us is the dimensionless heat source temperature. The average Nusselt number(Num) is determined by integrating Nus along the heat source:
Num ¼1
B
Z Dþ0:5B
D20:5B
NusðxÞ dx: ð28Þ
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The thermal performance of the nanofluid-filled enclosure under different thermalboundary conditions is studied for a range of solid volume fractions (0 # w # 0.2).
3. Algorithm validationAn iterative scheme is utilized for the solution of the velocity-vorticity formulation ofthe Navier-Stokes equations (conservation of mass, linear momentum and energy).In the majority of the incompressible flow problems modeled with Navier-Stokesequations the most natural boundary conditions arises when the velocity is prescribedall over the boundaries of the problem. The vorticity boundary conditions aredetermined iteratively from computations. Thus, the solution algorithm used for thediscretized fluid flow set of equations (20)-(23), must ensure coupled satisfaction of allthe equations at convergence. For the numerical solution of the equations of the flow,a MPCM with MLS approximation for the field variables is applied. The method isdeveloped and presented analytically in Bourantas et al. (2010). Moreover, we followthe approach of the so-called MAC algorithm proposed in Harlow and Welch (1965) andused in Bourantas et al. (2010). It permits us to attain solutions for high values ofcharacteristic numbers of the flow.
Two representative examples are executed for the validation of the meshlessscheme.
First validation – steady 2D square cavity flows of buoyancy-driven laminar heattransferThe numerical method was implemented in a MATLAB program. The effect of gridresolution was examined in order to select the appropriate grid density. Thus, an extensivemesh testing procedure was conducted to guarantee a grid independent solution bycomparisons with the results in the literature on steady 2D square cavity flows ofbuoyancy-driven laminar heat transfer. In order to ensure the accuracy and the validity ofthe proposed scheme, we analyze a system composed of pure fluid in an enclosure withPr ¼ 0.7 and different values of Rayleigh number. Thus, the present numerical code wasvalidated against the results of other natural convection studies in enclosures developedby Lin and Violi (2010), Tiwari and Das (2007), Davis (1983), Markatos and Pericleous(1984) and Hadjisophocleous et al. (1998). Table II shows the comparison between theresults obtained with the MPC method and the values presented in the literature.
Second validation – steady 2D square cavity flows of buoyancy-driven laminar heattransfer of nanofluids without the presence of heat sourceDifferent uniform meshes of 41 £ 41, 81 £ 81, 121 £ 121, 161 £ 161 and 201 £ 201 gridpoints, were examined for Gr ¼ 105, Pr ¼ 6, w ¼ 0.05, dp ¼ 5 nm and R ¼ 0.001 or0.007. It was found that a grid size of 161 £ 161 ensures a grid independent solution.The numerical results were compared with those obtained using the Non-StaggeredArtificial Pressure for Pressure-Linked equation (NAPPLE) algorithm (Lee and Tzong,1992). Furthermore, the present code was tested against the code used in Lin and Violi(2010). Results are shown in Figure 2.
4. Numerical resultsThe main objective of the present investigation is to explore the heat transfer behaviorof natural convection inside a cavity with water/Al2O3 nanofluid with a heat source
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located at the bottom of the enclosure, applying a “nanofluid-oriented” model for thecalculation of the effective thermal conductivity and the effective dynamic viscosity.Specifically, we will analyze steady state flow fields, temperature fields, and heattransfer rates for various values of the Rayleigh number, the ratio of minimum tomaximum nanoparticle diameter keeping the mean nanoparticle diameter fixed, thenanoparticle volume fraction and the locations of the heat source. Moreover, the effectsof various thermal boundary conditions on the nanofluid flow and heat transfercharacteristic are examined. In all calculations the Prandtl number is constant and equalto Pr ¼ 6.0.
4.1 Effect of Rayleigh number (or the effect of nanofluid temperature)Figure 3 shows the streamlines (left) and isotherms (right) for the nanofluid (solid line)with solid volume fraction w ¼ 0.05 and the pure water (dashed line) at differentRayleigh numbers, where the heat source is located in the middle of the bottom wall(D ¼ 0.5), with D being the dimensionless distance of heat source from the left wall (d/L),while the dimensionless length of the heat source B ¼ (b/D) is B ¼ 0.4. The streamlinesflow patterns show a primary re-circulating clockwise vortex that occupies the bulk ofthe cavity, while the isotherms display different behaviours as Rayleigh number isincreased. When conduction dominates the flow regime, that is for Ra ¼ 103 andRa ¼ 104, the isotherms tend to be vertical, with a light curvature along the heat source.
MPCLin and Violi
(2010)Tiwari andDas (2007)
Davis(1983)
Markatos andPericleous (1984)
Hadjisophocleouset al. (1998)
(a)Ra ¼ 103 3.652 3.597 3.642 3.649 3.544 3.544umax 0.812 0.819 0.804 0.813 0.832 0.814y 3.700 3.690 3.7026 3.697 3.593 3.586vmax 0.175 0.181 0.178 0.178 0.168 0.186x/Nu 1.118 1.118 1.0871 1.118 1.108 1.141(b)Ra ¼ 104 16.191 16.158 16.1439 16.178 16.18 15.995umax 0.825 0.819 0.822 0.823 0.832 0.814y 19.662 19.648 19.665 19.617 19.44 18.894vmax 0.118 0.112 0.110 0.119 0.113 0.103x/Nu 2.234 2.243 2.195 2.243 2.201 2.29(c)Ra ¼ 105 34.93 36.732 34.30 34.73 35.73 37.144umax 0.887 0.858 0.856 0.855 0.857 0.855y 63.517 68.288 68.7646 68.59 69.08 68.91vmax 0.068 0.063 0.05935 0.066 0.067 0.061x/Nu 4.508 4.511 4.450 4.519 4.430 4.964(d)Ra ¼ 106 68.569 66.46987 65.5266 64.63 68.81 66.42umax 0.937 0.86851 0.839 0.85 0.872 0.897y 220.824 222.33950 219.7361 217.36 221.8 226.4vmax 0.0375 0.03804 0.04237 0.0379 0.0375 0.0206x/Nu 8.8852 8.757933 8.803 8.799 8.754 10.39
Table II.Comparison of pure fluidsolutions with previousworks in an enclosure forPr ¼ 0.7 with differentRayleigh numbers
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Figure 2.Streamlines (left) and
isotherms (right) for theenclosures filled with
nanofluids at differentGrashof numbers and
different values of R
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
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0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
R =
0.0
01G
r =
104
R =
0.0
07G
r = 1
04R
= 0
.001
Gr =
105
R =
0.0
07G
r =
105
(a)
(b)
(c)
(d)
y = –11.6994
y = –13.1188
y = –23.0660
y = –25.8972
Notes: (a) Gr = 104, (knf /kf) = 1.424 and R = 0.001;(b) Gr = 104, (knf /kf) = 1.684 and R = 0.007; (c) Gr = 105,(knf /kf) = 1.424 and R = 0.001; (d) Gr = 105, (knf /kf) = 1.684and R = 0.007 for Pr = 6, dp = 5 nm and j = 0.05
Naturalconvection of
nanofluids flow
259
Figure 3.Streamlines (left) andisotherms (right) for theenclosures filled with purefluid (dashed lines) andnanofluid (solid lines)at different Rayleighnumbers
(a)
(b)
(c)
(d)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.1
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ra
= 10
3R
a =1
04R
a =1
05R
a =1
06
ymin, nf = –5.6171, ymin, nf = –6.4593
ymin, nf = –13.5387, ymin, nf = –14.5594
ymin, nf = –26.1625, ymin, nf = –28.3153
ymin, nf = –0.9394, ymin, nf = –1.2353
Notes: (a) Ra = 103; (b) Ra = 104; (c) Ra = 105; (d) Ra = 106,when (knf /kf) = 1.424, R = 0.001, D = 0.5 and B = 0.4; themaximum value of streamfunction is presented
HFF23,2
260
As the Rayleigh number is increased, streamlines exhibit stronger flow patterns andisotherms display more distinguished boundary layers near the vertical left and rightedges, while in the middle, isotherms tend to become horizontal. In fact the flow andtemperature patterns are influenced by the presence of nanoparticles comparing themwith corresponding of pure fluid.
Figure 4(b) shows the surface temperature at the bottom wall where the heat source islocated. It can be observed that it is not uniform. It has a maximum value located at the leftend of the heat source, which is near the hot wall. At this exact point the local Nusseltnumber is minimum, as it can be seen at Figure 4(a). More precisely, Figure 4(a) shows theprofile of local Nusselt number along the bottom wall of the enclosure and along the heatsource at different Rayleigh numbers, when the solid volume fraction w is constant andequal to w ¼ 0.05 and the ratio of the minimum to maximum nanoparticles diameter R isalso constant and equal to R ¼ 0.001. As we can see from Figure 4(a) and (b), when theRayleigh number increases the temperature maximum decreases, while the Nusseltnumber maximum increases. As it can be observed the reduction in the heat sourcemaximum temperature is a result of the enhancement in the heat removal process.
4.2 Effect of non-uniform nanoparticle diameter keeping the mean nanoparticlesdiameter fixedIn this part we examine the effect of fractal distributions on the heat transfer effect interms of minimum to maximum nanoparticle diameter R on the steady-state variationof the streamlines and isotherms for Ra ¼ 105 and Ra ¼ 106, while the mean diameter,nanoparticle volume fraction and the Prandtl number are fixed at 5 nm, 0.05 and 6,respectively. The heat source is located in the middle of the bottom wall (D ¼ 0.5),while the dimensionless length of the heat source is B ¼ 0.4.
Figure 5 shows the effect of R on the steady-state variation of the streamlines andisotherms for Ra ¼ 105 and Ra ¼ 106. The intensity of flow patterns are depicted sincethe values of streamline contours are presented. In this physical model, the flowpatterns are characterized by a primary re-circulating clockwise vortex that occupiesthe bulk of the cavity. As R is increased from R1 ¼ 0.001 to R2 ¼ 0.007, the flowpatterns remain quite the same while the absolute circulation strength is enhanced dueto relatively high velocity of the fluid flow.
Figure 4.The profile of local Nusselt
number and the localtemperature along
the bottom wall of theenclosure and along
the heat source at differentRayleigh numbers, whenw ¼ 0.05, (knf /kf) ¼ 1.424
and R ¼ 0.001(a) (b)
7
6Ra = 103
Ra = 104
Ra = 105
Ra = 106
5
4Nu
3
2
10.3 0.35 0.4 0.45
X
0.5 0.55 0.6 0.65
0.8
0.6
0.5
0.4
0.3
0.2
0.7Ra = 103
Ra = 104
Ra = 105
Ra = 106
θ
0.10.3 0.35 0.4 0.45
X
0.5 0.55 0.6 0.65
Naturalconvection of
nanofluids flow
261
Figure 5.Streamlines and isothermcontours for R1 ¼ 0.001and R2 ¼ 0.007 with(knf /kf) ¼ 1.424 and(knf /kf) ¼ 1.684,respectively, at differentRayleigh numbers
y = –13.5387
y = –26.1625
(a)
(b)
(c)
(d)
y = –15.1083
y = –29.2995
R =
0.0
01R
a =
105
R =
0.0
01R
a =
106
R =
0.0
07R
a =
105
R =
0.0
07R
a =
106
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
HFF23,2
262
The local Nusselt number for these conditions is calculated using equation (27) and theresult reported in Figure 6 shows that the higher the Ra number, the larger the heattransfer. It can be shown in Figure 6(a) that, as the R is increased the local Nusseltnumber is increased. As a consequence, nanofluids enhance heat transfer in both largeand small buoyancy conditions (Figure 6(b)).
4.3 The effects of solid volume fractionIn this part of the study, an enclosure filled with water/Al2O3 nanofluid with a heatsource (B ¼ 0.4) located in the middle of the bottom wall (D ¼ 0.5) is considered.Furthermore, we analyze the effect of the nanoparticle volume fraction (0 # f # 0.2)on the heat transfer characteristics.
Figure 7 shows the streamlines (left) and isotherms (right) for the nanofluid, withRa ¼ 105, Pr ¼ 6.0, volume fraction having values 0.0 # f # 0.2 and R ¼ 0.001. It isobvious that the flow and temperature patterns are strongly influenced by the presenceof nanoparticles. Adding nanoparticles to the base fluid (pure water) reduces thestrength of flow field, as in Ho et al. (2008). This reduction is more pronounced at lowRayleigh numbers where conduction heat transfer dominates. It is also apparent thatas nanoparticles are added, the maximum dimensionless temperature is increasedwhich is an indication of enhancing the enclosure thermal performance.
At Figure 8 the profiles of the local Nusselt number and the temperature values areplotted along the heat source for different volume fraction of the nanoparticles. Thelowest and the highest local Nusselt number are obtained at both ends of the heatsource, respectively. As it can be observed, as the volume fraction increases the localNusselt number increases thus, increased cooling performance is observed withincreasing volume fraction.
4.4 The effect of heat source location and various thermal boundary conditionsIn this part of the study, an enclosure filled with water/Al2O3 nanofluid (w ¼ 0.05)with a heat source (B ¼ 0.4) is considered. A numerical study has been conductedaltering the location of the heat source. Two sets of thermal boundary conditions areapplied.
First set of boundary conditions. The first set of thermal boundary conditions is thatshown in Figure 1. That is, a heat source is located on the thermally insulated bottom
Figure 6.Local Nusselt number andtemperature distributionsalong the heat source for
R1 ¼ 0.001 andR2 ¼ 0.007
6.5Ra = 105 R1Ra = 105 R7Ra = 106 R1Ra = 106 R7
5.5
4.5Nu
3.5
0.3 0.35 0.4 0.45 0.5 0.55x
0.6 0.65 0.3
0.34
0.32
0.3
0.28
0.26
0.24θ
0.22
0.2
0.18
0.16
0.35 0.4 0.45 0.5x
0.55 0.6 0.652.5
3
4
5
6Ra = 105 R1Ra = 105 R7
Ra = 106 R1Ra = 106 R7
Naturalconvection of
nanofluids flow
263
Figure 7.Streamlines (left) andisotherms (right) for theenclosures filled withwater/Al2O3 nanofluid,when Ra ¼ 105,R ¼ 0.001, D ¼ 0.5 andB ¼ 0.4 at differentvolume fractions
(a)
(b)
(c)
y = –14.5594
y = –14.0046
y = –15.0026
j =
0.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
j =
0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
j =
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4
(continued)
0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
HFF23,2
264
Figure 8.Local Nusselt number andtemperature distributions
along the heat source atdifferent volume fractions
6
5.5
4.5
ϕ = 0.00ϕ = 0.05ϕ = 0.1ϕ = 0.15ϕ = 0.2
ϕ = 0.00ϕ = 0.05ϕ = 0.1ϕ = 0.15ϕ = 0.2
4
3.5
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
x
0.3
0.36
0.34
0.32
0.3
0.28
0.26θ
0.24
0.22
0.2
0.18
0.160.35 0.4 0.45 0.5 0.55 0.6 0.65
x
3
2.5
Nu
5
Figure 7.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(d)
(e)
y = –15.9367
y = –18.8820
j =
0.15
j =
0.2
Notes: (a) j = 0.0; (b) j = 0.05; (c) j = 0.1; (d) j = 0.15; (e) j = 0.2
Naturalconvection of
nanofluids flow
265
wall of the enclosure. The horizontal upper wall is adiabatic and impermeable tomass transfer. The vertical walls of the enclosure are maintained at a relatively hightemperature (TH) the left wall and a relatively low temperature (TC) the right wall.
Figures 9 and 10 show the streamlines (left) and the isotherms (right) for differentlocations of the heat source on the bottom wall, while Figures 11 and 12 show thelocal Nusselt number and the temperature variation at the heat source, considering aRayleigh number Ra ¼ 105, volume fraction w ¼ 0.05, R ¼ 0.001 or R ¼ 0.007 and((knf/kf) ¼ 1.424) or ((knf/kf) ¼ 1.684), respectively. In both cases, the flow patternsare characterized by a primary re-circulating clockwise vortex that occupies thebulk of the cavity. Even though the flow patterns are the same, their strength ischanged when R is varied. Concerning the isotherms, they show that as the heatsource moves away from the left hot wall, the maximum flow temperature increases,for both cases. This can be explained by the distance that the fluid needs to travel inthe circulating cell to exchange the heat between the heat source and the left hotwall. It is clear that the isotherms follow the movement of the heat source. FromFigures 11 and 12 we see that the closer the heat source is to the left hot wall, thelower heat removal and the higher heat source maximum temperature is achieved.The maximum heat removal is at the right end of heat source, while the maximumheat source temperature is at the left end of the heat source for all locations of theheat source.
Second set of boundary conditions. Since no important difference appeared to thestreamlines due to the different location of the heat source, different boundary conditionsare used and examined for water/Al2O3 nanofluid. By this, it can be obtained theinfluence of the heat source location to the cooling properties of the nanofluid used. Theboundary conditions used and the non-dimensionalization procedure are that depictedin Aminossadati and Ghasemi (2009), were Cu-nanoparticles are used and all the wallshave steady temperature, while the heat source is located at the bottom wall, beingadiabatic.
Figure 13 shows the streamlines (left) and the isotherms (right) for different locationsof the heat source on the bottom wall, considering a Rayleigh number Ra ¼ 105,w ¼ 0.05, R ¼ 0.001 and (knf/kf) ¼ 1.424. Results are comparable with them obtained inAminossadati and Ghasemi (2009), where “classical” models, for the thermalconductivity and dynamic viscosity, were used. Namely, Brinkman’s model was usedfor the effective dynamic viscosity and Maxwell’s model was used for the effectivethermal conductivity. As it can be seen, when the heat source is located close to theleft wall two unsymmetrical circulating cells with unequal strengths are formatted.The strength of the circulating cells increases as the heat source moves away fromthe left wall. When the heat source reaches the middle of the bottom wall, the twocirculating cells in the enclosure have the same intensities. As far as the isotherms, wesee that the maximum flow temperature increases as the heat source moves away fromthe cold wall.
In Figure 14 the vertical velocity profiles along the midsection of the enclosure ispresented. The results are much closed to them in Aminossadati and Ghasemi (2009),and verify the existence of two counter-rotating circulating cells for all differentlocations of the heat source. Unsymmetrical velocity profiles indicate unbalancedcirculating cells within the enclosure for the cases, where the heat source is not located inthe middle. As the heat source is moving towards the middle, the maximum velocity
HFF23,2
266
Figure 9.Streamlines (left) and
isotherms (right) for theenclosures filled with
water/Al2O3 nanofluidfor different locations
of the heat source
D =
0.3
(a)
(b)
(c)
ymax = –13.5207
ymax = –13.5445
ymax = –13.5387
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
D =
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
D =
0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Notes: (a) D = 0.3; (b) D = 0.4; (c) D = 0.5 for Ra = 105, j = 0.05, B = 0.4,R = 0.001 and (knf /kf) =1.424
Naturalconvection of
nanofluids flow
267
Figure 10.Streamlines (left) andisotherms (right) for theenclosures filled withwater/Al2O3 nanofluid fordifferent locations of theheat source
Ymax = –15.0869
D =
0.3
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(a)
Ymax = –15.1127
D =
0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b)
Ymax = –15.1083
D =
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(c)Notes: (a) D = 0.3; (b) D = 0.4; (c) D = 0.5, for Ra = 105, j = 0.05, B = 0.4,R = 0.007 and (knf/kf) = 1.684)
HFF23,2
268
increases and the profile becomes symmetrical. Finally, Figure 15 shows the localNusselt number and the temperature variation along the heat source, respectively. As theheat source moves towards the middle of the bottom wall, the maximum temperatureincreases while the local Nusselt number decreases. This happens because the heatsource is moving away from the cold wall.
5. ConclusionsNatural convection in a partially heated enclosure from below, filled with water/Al2O3
nanofluid has been numerically investigated. “Nanofluid-oriented” models were appliedfor the calculation of the effective thermal conductivity (Xu-Yu-Zou-Xu’s model) andthe effective dynamic viscosity (Jang-Lee-Hwang-Choi’s model) of the nanofluid. Theinfluence of pertinent parameters such as the Rayleigh number, the non-uniformnanoparticle size keeping the mean nanoparticle diameter fixed, the volume fraction ofnanoparticles and the location of the heat source on the cooling performance, were studied.
For the first set of boundary conditions, we conclude the following:. When Rayleigh number is increased the thermal properties of the nanofluid are
increased. The surface temperature, at the bottom wall where the heat source islocated, has a maximum value located at the left end of the heat source, which isnear the hot wall. At this exact point the local Nusselt number present a minimum.Although in Aminossadati and Ghasemi (2009), different boundary conditions
Figure 11.Nusselt and temperature
along the heat sourcefor R ¼ 0.001 and
(knf/kf) ¼ 1.4240.11.5
2
2.5
3
3.5Nu
4
4.5
5
5.5 0.65
0.6
0.55
0.5
0.45
0.4θ
0.35
0.3
0.25
0.2
D = 0.3D = 0.4D = 0.5
D = 0.3D = 0.4D = 0.5
0.2 0.3 0.4 0.5x
0.6 0.7 0.8 0.1 0.2 0.3 0.4 0.5x
0.6 0.7 0.8
Figure 12.Nusselt and temperature
along the heat sourcefor R ¼ 0.007 and
(knf/kf) ¼ 1.684
Nu
6
5.5
4.5
3.5
2.5
1.5
2
3
4
5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
D = 0.3D = 0.4D = 0.5
x
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
D = 0.3D = 0.4D = 0.5
x
θ
Naturalconvection of
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269
Figure 13.Streamlines (left) andisotherms (right) fordifferent locations of theheat source for Ra ¼ 105,w ¼ 0.05, B ¼ 0.4,R ¼ 0.001 and(knf/kf) ¼ 1.424
Yleft = 0.4601, |Yright| = 4.0144
D =
0.3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
qmax = 0.1570
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Yleft = 1.1099, |Yright| = 4.1329
D =
0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Yleft = 2.8947, |Yright| = 2.8947
D =
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
qmax = 0.1642
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
qmax = 0.1667
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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and “classical” models for the thermal conductivity and dynamic viscosity, areapplied, conclusions similar to that presented there can be derived. That is,increasing the Rayleigh number strengthens the natural convection flows,increasing the Nusselt number and decreasing the heat source temperature.
. When the ratio of minimum to maximum nanoparticle diameter, R, is increased from0.001 to 0.007, the heat transfer characteristics of the nanofluid can be enhanced, asin Lin and Violi (2010) where there is not a discrete heat source embedded on thebottom wall. As the R is increased the local Nusselt number is increased and the heatsource temperature is decreased for both values of Rayleigh number.
. The flow and temperature patterns are influenced by the presence of nanoparticles.Adding nanoparticles to the base fluid (pure water) can reduce the strength of flowfield. It is also apparent that the increase of solid volume fraction of nanoparticlescauses the heat source maximum temperature to decrease and the Nusselt Number
Figure 14.Vertical velocity profilesalong the mid-section of
the enclosure for differentlocations of the heat source
20
15
10
5
0v
–5
–10
–15
–200 0.1 0.2 0.3 0.4 0.5
x0.6 0.7 0.8
D = 0.3D = 0.4D = 0.5
0.9 1
Note: Ra = 105, j = 0.05, B = 0.4, R = 0.001, (knf /kf) = 1.424
Figure 15.Nusselt and temperaturealong the heat source for
R ¼ 0.001 and(knf /kf) ¼ 1.424
13
12
11
10
Nu
9
8
7
60.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
D = 0.3D = 0.4D = 0.5
x
0.17
0.16
0.15
0.14
0.13
0.12
0.11
0.1
0.09
0.1
0.08
0.2 0.3 0.4 0.5 0.6 0.7 0.8
D = 0.3D = 0.4
D = 0.5
x
θ
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to increase. It means that adding nanoparticles into water improves its coolingperformance, applying the first set of boundary conditions.
The location of the heat source proved to significantly affect the heat source maximumtemperature, as well as the nanofluid flow and heat transfer characteristics, for both setof thermal boundary conditions which are applied. More specific, for the first set ofboundary conditions the closer the heat source is to the left hot wall, the lower heatremoval and the higher heat source maximum temperature is achieved. The maximumheat removal is at the right end of heat source, while the maximum heat sourcetemperature is at the left end of the heat source for all locations of the heat source. For thesecond set of boundary conditions as the heat source moves towards the middle of thebottom wall, the maximum temperature increases while the local Nusselt numberdecreases. Finally, for the second set of boundary conditions, the heat source maximumtemperature continuously increases as the heat source moves from the left wall towardsthe middle of the bottom wall of the enclosure.
The MPCM and MLS approximation were used for the solution of the equations of theflow. A velocity-vorticity formulation, utilizing a velocity-correction method wasapplied in order to solve the governing equations. The efficiency and the validity ofthe proposed numerical scheme were demonstrated, since the numerical results obtainedwere compared with those obtained using the finite volume method. A relative densegrid of 161 £ 161 regular distributed nodes was used, providing accurate numericalresults. Various grids were used ensuring the grid independence of the solution.A velocity-correction method was implemented, ensuring the continuity equation.
Since nanofluids are used or could be used in a wide range of engineeringapplications, although the development of the field faces several challenges, furthertheoretical, numerical and experimental research investigations are needed, so thatinformation to be transformed from one way of research to other and to understandthus the mechanism of the heat transfer characteristics of nanofluids.
References
Aminossadati, S.M. and Ghasemi, B. (2009), “Natural convection cooling of a localised heatsource at the bottom of a nanofluid-filled enclosure”, European Journal of MechanicsB/Fluids, Vol. 28, pp. 630-40.
Atluri, S.N. and Shen, S.P. (2002), The Meshless Local Petrov-Galerkin (MLPG) Method, TechScience Press, Encino, CA, p. 440.
Bourantas, G.C., Skouras, E.D., Loukopoulos, V.C. and Nikiforidis, G.C. (2010), “Meshfree pointcollocation schemes for 2D steady state incompressible Navier-Stokes equations invelocity-vorticity formulation for high values of Reynolds numbers”, CMES: ComputerModeling in Engineering & Sciences, Vol. 59, pp. 31-64.
Choi, S. (1995), “Enhancing thermal conductivity of fluids with nanoparticles”, ASME FluidsEngineering Division, Vol. 231, pp. 99-105.
Davis, G.D.V. (1983), “Natural convection of air in a square cavity: a benchmark solution”, Int.J. Numer. Meth. Fluids, Vol. 3, pp. 249-64.
Hadjisophocleous, G.V., Sousa, A.C.M. and Venart, J.E.S. (1998), “Predicting the transient naturalconvection in enclosures of arbitrary geometry using a nonorthogonal numerical model”,Numerical Heat Transfer, Part A Applications, Vol. 13, pp. 373-92.
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Hamilton, R.L. and Crosser, O.K. (1962), “Thermal conductivity of heterogeneous two componentsystems”, Indus. Eng. Chem. Fund., Vol. 1, pp. 187-91.
Harlow, F.H. and Welch, J.E. (1965), “Numerical calculation of time-dependent viscousincompressible flow of fluid with free surface”, Physics of Fluids, Vol. 8, pp. 2182-89.
Ho, C.J., Chen, M.W. and Li, Z.W. (2008), “Numerical simulation of natural convectionof nanofluid in a square enclosure: effects due to uncertainties of viscosity andthermal conductivity”, International Journal of Heat and Mass Transfer, Vol. 51,pp. 4506-16.
Jang, S.P., Lee, J.H., Hwang, K.S. and Choi, S.U.S. (2007), “Particle concentration and tube sizedependence of viscosities of Al2O3-water nanofluids flowing through micro- andminitubes”, Applied Physics Letters, Vol. 91, pp. 243112/1-243112/3.
Jang, S.P., Lee, J.-H., Hwang, K.S., Moon, H.J. and Choi, S.U.S. (2009), “Response to ‘comment onparticle concentration and tube size dependence of viscosities of Al2O3-water nanofluidsflowing through micro- and minitubes’”, Applied Physics Letters, Vol. 94,pp. 066102/1-066102/3.
Keblinski, P., Phillpot, S.R., Choi, S.U.S. and Eastman, J.A. (2002), “Mechanisms of heat flow insuspensions of nanosized particles nanofluids”, International Journal of Heat and MassTransfer, Vol. 45, pp. 855-63.
Khanafer, K., Vafai, K. and Lightstone, M. (2003), “Buoyancy-driven heat transfer enhancementin a two-dimensional enclosure utilizing nanofluids”, International Journal of Heat andMass Transfer, Vol. 46, pp. 3639-53.
Lee, F.H. and Tzong, R.Y. (1992), “Artificial pressure for pressure linked equation”, InternationalJournal of Heat and Mass Transfer, Vol. 35, pp. 2705-16.
Lin, K.C. and Violi, A. (2010), “Natural convection heat transfer of nanofluids in a vertical cavity:effects of non-uniform particle diameter and temperature on thermal conductivity”,International Journal of Heat and Fluid Flow, Vol. 31, pp. 236-45.
Liu, G.R. (2002), Mesh Free Methods, Moving Beyond the Finite Element Method, CRC Press,Boca Raton, FL.
Liu, G.R. and Gu, Y.T. (2005), An Introduction to Meshfree Methods and Their Programming,Springer, Dordrecht.
Markatos, N.C. and Pericleous, K.A. (1984), “Laminar and turbulent natural convection inan enclosed cavity”, International Journal of Heat and Mass Transfer, Vol. 27,pp. 755-72.
Murshed, S.M.S., Leong, K.C. and Yang, C. (2008), “Thermophysical and electrokineticproperties of nanofluids – a critical review”, Applied Thermal Engineering, Vol. 28,pp. 2109-25.
Pruta, N., Roetzel, W. and Das, S.K. (2003), “Natural convection of nanofluids”, Heat MassTransfer, Vol. 39, pp. 775-84.
Tiwari, R.K. and Das, M.K. (2007), “Heat transfer augmentation in a two-sided lid-drivendifferentially heated square cavity utilizing nanofluids”, International Journal of Heat andMass Transfer, Vol. 50, pp. 2002-18.
Wang, X.Q. and Mujumdar, A.S. (2007), “Heat transfer characteristics of nanofluids: a review”,International Journal of Thermal Sciences, Vol. 46, pp. 1-19.
Wen, D. and Ding, Y. (2005), “Formulation of nanofluids for natural convective heat transferapplications”, International Journal of Heat and Fluid Flow, Vol. 26, pp. 855-64.
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Xie, H., Wang, J., Xi, T., Liu, Y., Ai, F. and Wu, Q. (2002), “Thermal conductivity enhancementof suspensions containing nanosized alumina particles”, Journal of Applied Physics, Vol. 91,pp. 4568-72.
Xu, J., Yu, B., Zou, M. and Xu, P. (2006), “A new model for heat conduction of nanofluids based onfractal distributions of nanoparticles”, Journal of Physics D: Applied Physics, Vol. 39,pp. 4486-90.
Xuan, Y. and Le, Q. (2000), “Heat transfer enhancement of nanofluids”, International Journal forHeat and Fluid Flow, Vol. 21, pp. 58-64.
Further reading
Abu-Nada, E., Masoud, Z. and Hijazi, A. (2008), “Natural convection heat transfer enhancementin horizontal concentric annuli using nanofluids”, International Communications in Heatand Mass Transfer, Vol. 35, pp. 657-65.
Corresponding authorVassilios C. Loukopoulos can be contacted at: [email protected]
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