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Applied Mathematical Modelling 34 (2010) 573–592

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Analysis of flow and thermal field in nanofluid using a single phasethermal dispersion model

Shailesh Kumar, Santosh Kumar Prasad, Jyotirmay Banerjee *

Department of Mechanical Engineering, Sardar Vallabhbhai National Institute of Technology, Surat 395007, India

a r t i c l e i n f o

Article history:Received 29 July 2008Received in revised form 15 May 2009Accepted 1 June 2009Available online 17 June 2009

Keywords:NanofluidThermal dispersionFinite volume methodBuoyancyGrashof numberPrandtl number

0307-904X/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.apm.2009.06.026

* Corresponding author. Tel.: +91 261 2201575; fE-mail address: [email protected] (J. Baner

a b s t r a c t

Flow and thermal field in nanofluid is analyzed using single phase thermal dispersionmodel proposed by Xuan and Roetzel [Y. Xuan, W. Roetzel, Conceptions for heat transfercorrelation of nanofluids, Int. J. Heat Mass Transfer 43 (2000) 3701–3707]. The non-dimen-sional form of the transport equations involving the thermal dispersion effect is solvednumerically using semi-explicit finite volume solver in a collocated grid. Heat transfer aug-mentation for copper–water nanofluid is estimated in a thermally driven two-dimensionalcavity. The thermo-physical properties of nanofluid are calculated involving contributionsdue to the base fluid and nanoparticles. The flow and heat transfer process in the cavity isanalyzed using different thermo-physical models for the nanofluid available in literature.The influence of controlling parameters on convective recirculation and heat transfer aug-mentation induced in buoyancy driven cavity is estimated in detail. The controlling param-eters considered for this study are Grashof number (103 < Gr < 105), solid volume fraction(0 < / < 0.2) and empirical shape factor (0.5 < n < 6). Simulations carried out with variousthermo-physical models of the nanofluid show significant influence on thermal boundarylayer thickness when the model incorporates the contribution of nanoparticles in the den-sity as well as viscosity of nanofluid. Simulations incorporating the thermal dispersionmodel show increment in local thermal conductivity at locations with maximum velocity.The suspended particles increase the surface area and the heat transfer capacity of thefluid. As solid volume fraction increases, the effect is more pronounced. The average Nus-selt number from the hot wall increases with the solid volume fraction. The boundary sur-face of nanoparticles and their chaotic movement greatly enhances the fluid heatconduction contribution. Considerable improvement in thermal conductivity is observedas a result of increase in the shape factor.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

Nanofluid is a suspension of nanoparticles in base fluid. Nanofluids have attracted enormous interest from researchersdue to their potential for high rate of heat exchange incurring either little or no penalty in pressure drop. The convectiveheat transfer characteristic of nanofluids depends on the thermo-physical properties of the base fluid and the ultra fine par-ticles, the flow pattern and flow structure, the volume fraction of the suspended particles, the dimensions and the shape ofthese particles. The utility of a particular nanofluid for a heat transfer application can be established by suitably modeling theconvective transport in the nanofluid.

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Nomenclature

Principal symbolsA AreaGr Grashof number ðgbf DTL3=m2

f ÞL Height and width of domain (Characteristic dimension)Nu Nusselt numberNu Average Nusselt numberP Pressure, N m�2

Pr Prandtl number (mf/af)Q Total heat transferS Surface area vector, m2

T Temperature, Kcp Specific heat at constant pressure, J kg�1 K�1

dp Diameter of the nanoparticle, nmV Control volume, m3

g Gravity acceleration, m2 s�1

kf Thermal conductivity of the fluid, W m�1 K�1

ks Thermal conductivity of the solid, W m�1 K�1

n Empirical shape factorT Time, secondu Velocity vector, m2 s�1

u, v Velocity component in coordinate directions, m s�1

x, y Distance in coordinate direction, m

Greek symbolsa Thermal diffusivity, m2 s�1

bf Fluid expansion coefficientbs Solid expansion coefficient/ Volume fraction of suspension particlesW Sphericityl Dynamic viscosity, N s m�2

t Kinematics viscosity, l/q, m2 s�1

q Density, kg m�3

h Non-dimensional temperature

Subscriptsf Base fluidnf Nanofluids Solidh Hot wall

Superscript* Non-dimensional terms

574 S. Kumar et al. / Applied Mathematical Modelling 34 (2010) 573–592

Several authors have tried to establish convective transport models for nanofluids. Nanofluid is a two-phase mixture inwhich the solid phase consists of nano-sized particles. In view of the nanoscale size of the particles, it may be questionablewhether the theory of conventional two-phase flow can be applied in describing the flow characteristics of nanofluid [1].Since the size of the particles is less than 100 nm, nanofluids behave like a fluid than a mixture [1–3]. Xuan and Roetzel[1] proposed homogeneous flow model where the convective transport equations of pure fluids are directly extended tonanofluids. This means that all traditional heat transfer correlations (e.g. Dittus–Boelter) could be used for nanofluids pro-vided the properties of pure fluids are replaced by those of nanofluids involving the volume fraction of the nanoparticles. Thehomogeneous flow models are however in conflict with the experimental observations of Maliga et al. [3], as they under pre-dict the heat transfer coefficient of nanofluids.

The main flow in nanofluid involves effect of gravity, Brownian force and friction force between the fluid and the ultrafineparticles, the phenomena of Brownian diffusion, sedimentation and dispersion. Thus although the nanoparticles areultrafine, the slip between the fluid and the particles may not be zero. The random and irregular movement of the particlesincreases the energy exchange rates in fluids. To account for this enhanced exchange rate Xuan and Roetzel [1] also proposeda single-phase thermal dispersion model. In this approach, the effect of nanoparticle/base fluid relative velocity is treated asa perturbation of the energy equation and an empirical dispersion coefficient is introduced to describe the heat transfer

S. Kumar et al. / Applied Mathematical Modelling 34 (2010) 573–592 575

enhancement. Khanafer et al. [4] developed a stream function vorticity based numerical algorithm based on the thermal dis-persion model. They found that the nanofluid heat transfer rate increases with increase in nanofluid volume fraction.

Buongiorno [5] proposed a mathematical model to capture the nanoparticle/base fluid slip by treating nanofluid as two-component mixture. The slip was considered due to Brownian diffusion and thermophoresis. He introduced diffusion param-eters due to both these mechanisms in the equation of conservation of mass and energy. Using an order of magnitudeanalysis he went on to predict that the effect of thermal dispersion is minimal in nanofluids. His analysis however didnot consider the influence of local velocity on the diffusion coefficients. Santra et al. [6] have tried to incorporate the influ-ence of rheology of the nanofluid by assuming the flow to be non-Newtonian.

It is worth mentioning that the work of Mansour et al. [7] in forced convection indicates that the assessment of the heattransfer enhancement potential of nanofluid is closely dependent on the way the nanofluid thermo-physical properties aremodeled. The presence of nanoparticles is expected to alter appreciably the thermal conductivity and viscosity of the basefluid. Several experimental and theoretical works [8–12] are reported in literature to predict suitable models for effectivethermal conductivity and viscosity of nanofluid. Maxwell [8] first published theoretical work for the effective thermal con-ductivity of two-phase mixtures containing powders with particle diameters in order of millimeters or even micrometers.Maxwell’s model predicted that the effective thermal conductivity of suspensions containing spherical particles increaseswith an increase in the volume fraction of the solid particles. Jhang and Choi [9] constructed correlation for effective thermalconductivity for dilute suspensions based on kinetic theory as well as nanoparticle boundary layer theory. Using an order ofmagnitude analysis they obtained expression for effective thermal conductivity of nanoparticles in terms of volume fractionof the nanoparticles. However comparison of the model predictions with the experimental data sets of Lee et al. [10] forCuO–water and Al2O3–water nanofluids shows significant differences as the volume fraction of nanoparticles were in-creased. Lee et al. [10] measured thermal conductivities of oxide nanoparticles by a transient hot-wire method. Das et al.[11] investigated the temperature dependence of thermal conductivity enhancement for Nanofluids. A temperature oscilla-tion technique was utilized for the measurement of thermal diffusivity and thermal conductivity was calculated from it. Thethermal conductivity prediction of Jhang and Choi shows good agreement with the experimental data of Das et al. [11] in thelower temperature range. The model prediction however fails when the temperature is higher than 40 �C. Kleinstreuer and Li[12] provided a comprehensive discussion on the various correlations proposed in literature for effective thermal conductiv-ity of nanofluids and compared the model predictions with experiments. Keblinski et al. [13] explored various possiblemechanisms for the anomalous increase in the thermal conductivity of nanofluid.

The correlation for effective viscosity of dilute suspensions containing spherical particles was first proposed by Brinkman[14]. Later a semi-empirical relation was proposed by Kreiger and Dougherty [15] for effective viscosity of the nanofluid as afunction of volume fraction, packing fraction and intrinsic viscosity for hard spherical particles. A decade later Nielson [16]simplified this expression in terms of volume fraction and the minimum packing fraction of the nanoparticles.. The recentexperimental results of Murshed et al. [17] for their viscosity measurements have shown close match with the models forviscosity proposed by Kreiger and Dougherty [15] and Nielson [16].

In the present work, the natural convection flow of copper–water nanofluid is analyzed to establish the sensitivity of ther-mo-physical models on heat transfer characteristics of the nanofluid, to estimate the effect of thermal dispersion due to slipbetween the nanoparticles and base fluid, and to evaluate the influence of control parameters on the heat transfer charac-teristics of nanofluid. Numerical formulation is developed using the single-phase model proposed by Xuan and Roetzel [1]that accounts for thermal dispersion of nanoparticles in the base fluid. The copper–water nanofluid is assumed to be New-tonian. Among a few experimental data available in literature [18–20], such a Newtonian behaviour was experimentally con-firmed for Al2O3/water suspensions as well as copper–water suspensions. The thermal dispersion coefficient is obtained as afunction of the flow pattern, properties of the base fluid and the nanoparticles, the dimensions and shape of the nanoparti-cles, and the volume fraction of the nanoparticles suspended in the mixture. The governing conservation equations for flowand heat transfer are solved using a semi-explicit finite volume formulation in collocated grid. The flow and heat transferbehavior of the nanofluid is analyzed for different thermo-physical models proposed in Literature. The density variationin the buoyancy force is calculated based on the Boussinesq approximation [21]. The heat transfer enhancement due to nano-particles is estimated by varying the volume fraction of the nanoparticle, maximum temperature difference in the domainand the shape of the nanoparticles.

2. Mathematical model

The basis of single phase model proposed by Xuan and Roetzel [1] is the fact that solid particles in nanofluid are ultra fine(<100 nm) and are easily fluidized. These particles can thus be approximately considered to behave like a fluid. Under theassumptions that there exist no motion slip between the discontinuous phase of the dispersed ultra fine particles and thecontinuous liquid and the local thermal equilibrium exists between the nanoparticles and the base fluid, the nanofluidcan be treated as a pure fluid [1,3]. All the equations of motion and energy for the pure fluid can thus directly be extendedto the nanofluids. However, while applying the existing dimensionless relations for pure fluids to the nanofluids, one needsto use the properties corresponding to the nanofluids.

Thermal properties involved in calculating the heat transfer rate of the nanofluid are heat capacity, thermal expansioncoefficient, viscosity, and thermal conductivity. These properties can be expressed in terms of the properties of the base fluid


and the nanoparticles. The classical models available in literature [4,8] are used in this work for the heat capacity (qcp)nf andthermal expansion coefficient parameters (qb)nf of the nanofluid and are expressed [4] as:

ðqcpÞnf ¼ ð1� /Þ � ðqcpÞf þ /� ðqcpÞs ð1aÞðqbÞnf ¼ /qsbs þ ð1� /Þqf bf : ð1bÞ

Brinkman [14] proposed the expression for evaluating the effective viscosity of a linearly viscous fluid of viscosity lf con-taining a dilute suspension of small rigid spherical particles of volume fraction /, as:

lnf ¼ lf � ð1þ 2:5� /Þ: ð2aÞ

This relation is restricted for low volume fraction (/ < 0.05). The equation was extended by Brinkman [14] for high vol-ume fraction (/ > 0.05) as:

lnf ¼lf

ð1� /Þ2:5: ð2bÞ

Each relation has its own application limitation. The direct and reliable access to obtaining the apparent viscosity of nano-fluid is by experiment. Recently experimental investigations of Murshed et al. [17] have found good agreement with thesemi-empirical model propose by Kreigger and Dougherty [15] for the effective viscosity of nanofluid. These models corre-late the effective viscosity as a function of particle volume fraction (/p), maximum packing fraction (/m) and intrinsic vis-cosity (g) of nanoparticles.

lnf

lf¼ 1�

/p

/m

� ��g/m

: ð2cÞ

For spherical particles, /m = 0.64 and g = 2.5. For a concentration of dispersed particles, Nielson [16] proposed a simplecorrelation expressed as:

lnf

lf¼ ð1þ 1:5/pÞe

/p1�/m : ð2dÞ

Apparent thermal conductivity is the most important parameter to indicate the enhancement potential of the nanopar-ticle liquid suspension. Research has shown that the thermal conductivity of the nanofluid is a function of thermal conduc-tivity of the base fluid and the nanoparticle material, the volume fraction, the surface area, and the shape of the nanoparticlessuspended in the liquid, and distribution of the dispersed particles. For solid particle suspensions in base fluid in which theratio of conductivity of two phases is larger than 100, Hamilton and Crosser [22] developed the following model:

knf

kf¼ ks þ ðn� 1Þ � kf � ðn� 1Þ � /� ðkf � ksÞ

ks þ ðn� 1Þ � kf þ /� ðkf � ksÞ: ð3Þ

Here n is empirical shape factor defined in terms of sphericity as:

n ¼ 3wðFor spherical particles w ¼ 1Þ: ð4aÞ

Here w is the sphericity, which is a measure of how spherical (round) an object is. It is a measure of compactness of a shape.Wadell [23] defined sphericity w, of a particle as the ratio of the surface area of a sphere (with the same volume as the givenparticle) to the surface area of the particle.

w ¼ p13 � ð6VpÞ

23

Ap: ð4bÞ

For particles other than a sphere, the shape factor n may vary from 0.5 to 6.0 or w varies from 6.0 to 0.5.Although the nanofluid behaves more like a fluid than the conventional solid liquid mixtures, the slip velocity between

the fluid and the particles may not be zero and the nano-sized particle will tend to move randomly in the fluid phase. Irreg-ular and random movement of the particles increases the energy exchange rates in the fluid. This thermal dispersion due tonanoparticles will tend to flatten the temperature distribution and make the temperature gradient between the fluid andwall steeper thereby resulting in augmentation in heat transfer rate between the fluid and the wall. The thermal dispersiondue to nanoparticle is accounted for in the present work via perturbation in the mean flow and thermal fields as discussed inthe following section.

2.1. The dispersion model

The energy equation for the nanofluids is expressed as:

oTotþr � ðuTÞ ¼ r � knf

ðqcpÞnfrT

!: ð5Þ


It is assumed that the irregular movement of the ultra fine particles induces small perturbations (T0and u0, respectively) toboth the local temperature and velocity of the nanofluid. Thus the intrinsic phase averages are given as

T ¼ T þ T 0; ð6aÞu ¼ uþ u0; ð6bÞ

where one has T ¼ 1V

RV TdV ;u ¼ 1

V

RV udV , 1

V

RV T 0dV ¼ 0.

Now, substituting the above values of T and u in Eq. (5) and rearranging,

oðT þ T 0Þot

þr � ½ðuþ u0ÞðT þ T 0Þ� ¼ r � knf

ðqcpÞnfrðT þ T 0Þ

!

oroTotþr � ðuTÞ ¼ r � ðknfrTÞ � 1

ðqcpÞnf� ðu0T 0Þ

!:

ð7Þ

The second term in R.H.S. of the Eq. (7) indicates the effect of the thermal dispersion resulting from chaotic movement of thenanoparticles in the flow. The heat flux induced by the thermal dispersion in nanofluid flow can be represented as:

qd ¼ ðu0T 0Þ ¼ �kd

ðqcpÞnfrTf ; ð8Þ

where kd is the coefficient of the dispersed thermal conductivity. It is expected that the thermal dispersion coefficient is afunction of the flow pattern, properties of the base fluid and the nanoparticles, the dimensions and shape of the nanoparti-cles, and the volume fraction of the nanoparticles suspended in the mixture. Khanafer et al. [4] proposed the dispersed ther-mal conductivity of nanofluid as:

kd ¼ CðqcpÞnf jujdp/: ð9Þ

Here C is an unknown constant which should be determined by matching experimental data and dp is diameter of thenanoparticles.

2.2. Governing equations for flow and heat transfer

The integral form of Navier–Stokes equations used in the present formulation for two dimensional, incompressible,homogenous single-phase model of the thermally driven nanofluid is expressed as follows:

Continuity equation
Zs
u � dS ¼ 0: ð10Þ

X-momentum equation
ZV
ouot

dV þZ

Suu � dS ¼ � 1

qnf

ZV

oPox

dV þlnf

qnf

ZSru � dS: ð11Þ

Y-momentum equation
ZV
ovot

dV þZ

Svu � dS ¼ � 1

qnf

ZV

oPoy

dV þlnf

qnf

ZSrv � dSþ

ðqbÞnf

qnf

ZV

gðT � TLÞdV ; ð12Þ

where the velocity vector is defined as: u = uî + vj.Energy equation with thermal dispersion effect
Z
V

oTot

dV þZ

STu � dS ¼

ZSr anf þ

kd

ðq � cpÞnf

" #T � dS: ð13Þ

Here the thermal diffusivity of the nanofluid is expressed as anf ¼knf

ðq�cpÞnf. In the absence of thermal dispersion effects the en-

ergy equation reduces to
ZV
oTot

dV þZ

STu � dS ¼ anf

ZSrT � dS: ð14Þ

2.3. Non-dimensionalization of governing equations

The above equations can be reduced to non-dimensional form using the following scales:

x� ¼ xL; y� ¼ y

L; u� ¼ uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

gbf DTLp ; v� ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

gbf DTLp ; t� ¼

tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigbf DTL

pL

; h ¼ T � TL

TH � TL;


p� ¼ Pqf ðgbf DTLÞ ; Gr ¼

gbf DTL3

m2f

; Pr ¼ mf

af; d�p ¼

dp

L;

where suffix f stand for the thermo-physical properties of base fluid while nf stands for thermo-physical properties of nano-fluid. The non-dimensional terms, Grashof number (Gr) and Prandtle number (Pr) here correspond to the properties of thebase fluid.

Non-dimensional form of continuity equation
ZS
u� � dS� ¼ 0 ð15Þ

Non-dimensional form of X-momentum equation
ZV
ouot

dV þZ

Suu � dS ¼ � 1

qnf

ZV

oPox

dV þlnf

qnf

ZSru � dS

ZV

ou�

ot�dV� þ

ZS

u�u� � dS� ¼ �qf

qnf

ZV

oP�

ox�dV� þ

lnf

qnf mf

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

f

gbf DTL3

vuut ZSru� � dS�

ZV

ou�

ot�dV� þ

ZS

u�u� � dS� ¼ �qf

qnf

ZV

oP�

ox�dV� þ

lnf

qnf mf� 1ffiffiffiffiffiffi

Grp

ZSru� � dS�:

ð16Þ

Non-dimensional form of Y-momentum equation
ZV
ovot

dV þZ

Svu � dS ¼ � 1

qnf

ZV

oPoy

dV þlnf

qnf

ZSrv � dSþ

ðqbÞnf

qnf

ZV

gðT � TLÞdV ;

ZV

ov�ot�

dV� þZ

Sv�u� � dS� ¼ �

qf

qnf

ZV

oP�

oy�dV� þ

lnf

qnf tf� 1ffiffiffiffiffiffi

Grp

ZSrv� � dS� þ

ðqbÞnf

qnf bf

ZV

hdV�:

ð17Þ

Non-dimensional form of energy equation
ZV
oTot

dV þZ

STu � dS ¼

ZSr anf þ

kd

ðq � cpÞnf

" #� TdS;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigbf DTL

pL

� L3 � DT �Z

V

ohot�

dV� þ L2 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigbf DTL

q� DT

Zs

hu� � dS� ¼ DT � L2

L

Zs

anf þkd

ðq � cpÞnf

" #rh � dS�;

ZV

ohot�

dV� þZ

shu� � dS� ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

gbf DTL3q Z

sanf þ

kd

ðq � cpÞnf

" #rh � dS�;

ZV

ohot�

dV� þZ

shu� � dS� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

f

gbf DTL3

vuut � 1mf� af

af

Zs

anf þkd

ðq � cpÞnf

" #rh � dS�;

ZV

ohot�

dV� þZ

shu� � dS� ¼ 1

prffiffiffiffiffiffiGrp

Zs

anf

afþ kd

ðq � cpÞnf � af

" #rh � dS�:

ð18Þ

Now, substituting, anf, (qcp)nf and kd from Eq. (9), in the Eq. (18),

anf ¼ knf =ðq � cpÞnf ; ðq � cpÞnf ¼ /qscpsþ ð1� /Þqf cpf

and kd ¼ C � /� ðq � cpÞnf � dp �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ v2

p:

The non-dimensional form of energy equation with the thermal dispersion effect will be in the following form
ZV
ohot�

dV� þZ

shu� � dS� ¼ 1


Zs

knf =ðq � cpÞnf

kf =ðq � cpÞfþ

C � /� ðq � cpÞnf � d�p �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigbf DTL

p� L�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu�2 þ v�2p

ðq � cpÞnf � af

" #rh � dS�;

ZV

ohot�

dV� þZ

shu� � dS� ¼ 1


Zs

knf =kf

ð1� /Þ þ / ðqcpÞsðqcpÞf

þC � /� tf � d�p �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigbf DTL

p� L�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu�2 þ v�2p

tf � af

24

35rh � dS�;

ZV

ohot

dV� þZ

shu� � dS� ¼ 1


Zs

knf =kf

ð1� /Þ þ / ðq�cpÞsðq�cpÞf

þ C � /� d�p � prffiffiffiffiffiffiGrp

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu�2 þ v�2

p24

35rh � dS�;

ZV

ohot

dV� þZ

shu� � dS� ¼ 1


Zs

k�nf þ k�dh i

rh � dS�;ZV

ohot

dV� þZ

shu� � dS� ¼ 1


Zs

k�eff

h irh � dS�:

ð19Þ


Here d�p is the non-dimensional ratio of the diameter of the nanoparticles to that of the characteristic dimension of the do-main (L). It can be observed that the effective thermal conductivity (keff) of the nanofluid is a function of velocity field, shapeand size of the particle, temperature, volume fraction of the nanoparticles and the thermal properties of both the nanopar-ticles and the base fluid. When the thermal dispersion effects are neglected the non-dimensional form of energy Eq. (19) willreduce to

Table 1Physica

Propert

cp (J/kgq (kg/mk (W/mb (K�1)

o

ot�

ZV

hdV� þZ

su� � hdS� ¼ 1

prffiffiffiffiffiffiGrp � anf

af

Zsrh � dS�: ð20Þ

2.4. Nusselt number calculation

The local variation of the Nusselt number of the nanofluid is calculated as:

Nu ¼ QQCond:fluid

¼ ðknf Þkf� oh

ox�: ð21Þ

The average Nusselt number ðNuÞ on the hot wall is calculated by integrating the local Nusselt number over the left wall

Nu� �

¼ 1H

Z 1

0Nudy: ð22Þ

3. Numerical method

The governing equations in non-dimensional form are solved using a semi-explicit finite volume solver. The momentumequations are discretized on non-staggered grids in an explicit manner with the exception of the pressure gradient terms,which are treated implicitly. The continuity equation is also enforced implicitly. Collocated storage of pressure and velocityvariables at the cell centers is used. The cell face mass fluxes are obtained by momentum interpolation.

4. Problem definition

Flow of incompressible copper–water nanofluid in a thermally driven square cavity (shown in Fig. 1) is considered in thepresent work. The physical properties of fluid and solid phase used in the present study are shown in Table 1. The boundarieshave no-slip tangential and zero normal velocity boundary condition. The left wall is hot and kept at constant temperature.The right wall is cold and kept at constant temperature. The bottom and top walls are insulated.

NANOFLUID

u=0, v=0,δ

u=0,

v=0,

u=0,

v=0,

u=0,v=0,

g

δθ/δy=0

θ/

θ h=1

δy=0

θ c=0

x

y

Fig. 1. Schematic diagram of buoyancy driven cavity with nanofluid inside the cavity.

l properties of liquid and solid phase used in present study.

y Fluid phase (water) Solid phase (copper)

K) 4179 3833) 997.1 8954K) 0.6 400

2.1 � 10�4 1.67 � 10�5


5. Code validation and grid independence study

The numerical formulation is validated against the available results obtained by Khanafer et al. [4]. Their formulation wasbased on stream-vorticity equations solved using the alternating direct implicit procedure. Fig. 2 shows the temperature dis-tribution along the horizontal centerline and the u-velocity profiles along the vertical centerline of the cavity for varyingGrashof number as 104, 105 and volume fraction varying as 0.0, 0.10 and 0.20. The left side of the figure shows the resultsobtained in the present calculations while the right side shows the results available in literature [4]. The comparison is alsoprovided when convection only in the base fluid is considered. In addition, the result of the present numerical formulations isalso compared with the experimental results of Krane and Jessee [24] and is shown in Fig. 3. The grid independence study iscarried out using three different uniform grids 41 � 41, 81 � 81 and 121 � 121. The u-velocity distribution along the verticalcenterline of the cavity is used as the grid sensitivity parameter. The variation in u-velocity for all the three grids is shown inFig. 3. The results obtained for grid sizes 81 � 81 and 121 � 121 show insignificant difference. Hence the grid size, 81 � 81 isused for all the results obtained in the present work. The results of all three grids shown in Fig. 4 for Grashof number 105,Pr = 6.2, d�p ¼ 10�6 and volume fraction 0.05 with empirical shape factor of the particles as 3.

6. Results and discussion

Simulations are carried out initially with four different thermo-physical models for the estimation of single phase effec-tive viscosity, density and the thermal expansion coefficient of the nanofluid. The influence of thermal gradient, thermal dis-persion, volume fraction and shape of nanoparticles on the heat transfer augmentation in the cavity is then estimated. Therange of controlling parameters considered for this study is: Grashof number (103 < Gr < 105), solid volume fraction (0 < /< 0.2) and empirical shape factor (0.5 < n < 6).

6.1. Influence of physical properties

The flow and thermal field in the cavity depends on the effective thermo-physical properties of the nanofluid. The ther-mo-physical properties of the nanofluid in the present single-phase model are calculated involving the contribution of thenanoparticles in the base fluid available in literature [4,8,14]. These expressions in terms of volume fraction of nanoparticlesare already discussed in Section 2. Four basic models of nanofluid based on its thermo-physical properties are proposed here.Model-I is a pure fluid model where the thermo-physical properties of the nanofluid is same as that of the base fluid other

u

y

-0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.080.0

0.2

0.4

0.6

0.8

1.0

Gr=105

=0.2,0.1,0

Gr=104=0,0.1,0.2

nanofluid

pure fluid

x0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Gr=104

Gr=105

=0.2,0.1,0

=0,0.1,0.2

nanofluidpure fluid

θ

Fig. 2. Comparison of u-velocity along the vertical centreline and temperature along the horizontal center line of the cavity obtained by the presentsimulations (left) with results available in literature (right) [4].


than the thermal conductivity which is taken as a function of volume fraction from Eq. (3). Thus nanoparticles do not haveany contribution to the overall viscosity, density and thermal expansion coefficient of the nanofluid in this model. Model-II isfor nanofluids where nanoparticles only contribute to conductivity and viscosity of the nanofluids as per (Eqs. (2a) and (2b))Brinkman’s model [14]. Model-III considers the contribution of nanoparticles to the conductivity, viscosity as well as densityof the nanofluids. Model-IV is a general model for nanofluid where all the thermo-physical properties of the nanofluid areinfluenced by the presence of nanoparticles. The thermo-physical properties and non-dimensional buoyancy and viscous

u

y

-0.2 -0.1 0.0 0.1 0.20.0

0.2

0.4

0.6

0.8

1.0

Experimental result [24]

Present numerical result

x

v0.0 0.2 0.4 0.6 0.8 1.0

-0.4

-0.2

0.0

0.2

0.4

Experimental result [24]

Present numerical result

x

Tem

pera

ture

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0Experimental result [24]Present numerical result

(a) (b)

(c)Fig. 3. Comparison of the temperature and velocity profile in a buoyancy driven cavity with the experimental results of Krane and Jessee [24]; (a) u-velocityvariation along vertical centerline; (b) v-velocity variation along horizontal centerline and (c) temperature variation along the horizontal centerline.

u

y

-0.04 -0.02 0.00 0.02 0.040.0

0.2

0.4

0.6

0.8

1.0

41×41

121×121

81×81

Fig. 4. Variation of u-velocity along the vertical center line for Gr = 105, Pr = 6.2, / = 0.05 and n = 3 obtained with three different grids.


terms for these models are tabulated in Table 2. The thermal conductivity of the nanofluid for all these models is calculatedusing Eq. (3). This equation accounts for the influence of thermal conductivity due to shape factor and volume fraction of thenanoparticles.

The flow and thermal field in the cavity and the heat transfer from the hot wall are established for all these models forGrashof number 105, Prandtl number 6.2, volume fraction 0.20, and shape factor 3. Fig. 5 shows the flow and thermal field inthe cavity obtained for simulations considering all the four models of the thermo-physical properties. The streamline patternfor Model-II and III shows small opposing re-circulation near the centre of the cavity. The strength of flow is observed to bereduced for Model-II as compared to the pure fluid because of the increase in viscosity of the fluid. As a result the hydrody-namic boundary layer thickness gets reduced while the thermal boundary layer thickness in Model-II remains unchanged.

Thermal boundary layer thickness is influenced in Model-III where the isotherms are observed to be mostly flat and den-sely packed even near the left bottom and right top of the cavity. This indicates maximum heat transfer from left and rightwalls. Since the thermal expansion coefficient for Model-IV is reduced the buoyant force reduces compared to viscous resis-tance resulting in reduced strength of the flow field compared to Model-III. Also, the streamline contours do not show anyopposing re-circulation near the centre of the cavity for Model-IV.

Nusselt number (Nu) represents the ratio of heat convected from the wall to the fluid, to that conducted up to the wall.Fig. 6a shows the variation of local Nusselt number from the hot wall for various models. The vertical isotherms near the hotwall are densely packed near the lower part of the hot wall and are observed to separate from the upper part of the wall. Thusthe Nusselt number which depends on gradient of temperature is the minimum at the top wall and increases towards thebottom part of the hot wall. The maximum value of local Nusselt number is observed near the lower part of the hot wall. Thestiffness of the curve is smaller for Models-II and IV while Model-III shows the maximum value of local Nusselt number andthe point of inflexion happens nearest to the bottom wall for this model.

To observe the influence of thermo-physical properties on heat transfer augmentation at varying volume fractions, thesimulations are carried out for all these models by varying the volume fraction from 0 to 0.20. Fig. 6b shows the comparisonof average Nusselt number obtained from the hot wall as function of volume fraction for all these models. The variation ofaverage Nusselt number along hot wall with volume fraction is observed to be linear for all the models. It is observed that forthe entire range of volume fraction the average Nusselt number obtained by Models-II and IV is lower than even for Model-Iwhere only the influence of conductivity due to nanoparticles are considered.

Table 3 shows the comparison of average Nusselt number from the hot wall for all the models for a volume fraction of0.20. Model-I estimates a higher average Nusselt number than Model-II due to a larger effective viscosity of nanofluid forModel-II. This is due to a thicker momentum boundary layer and an increase in the shear stress between the fluid layers.Model-III estimates the highest average Nusselt number due to a higher thermal expansion and density. Higher effectivedensity indicates higher momentum and consequently more heat transfer enhancement. It should be noted that Model-IVshows the lowest average Nusselt number due to the lower thermal expansion coefficient and reduced buoyant convection.An augmentation of 9.13% in heat transfer (average Nusselt number) is obtained with Model-III as compared to Model-I for avolume fraction of 0.20 and Grashof number of 105. Model-III provides maximum heat transfer augmentation and the heattransfer rates are significantly increased compared to pure fluid at high volume fraction.

6.1.1. Sensitivity to viscosity modelsSimulations are carried out to establish the influence of different nanofluid viscosity models proposed by Brinkman [14]

and Kreigger and Dougherty [15]. A further reduction in the average Nusselt number (K–D Model in Table 3) is observedwhen the nanofluid viscosity in Model-IV is replaced by the equation of viscosity proposed by Kreigger and Dougherty[15]. This fact is important in view of the work reported by Putra et al. [19] where they observed deterioration in the heattransfer characteristics of nanofluids in natural convection.

6.2. Effect of thermal dispersion

The random movement of ultrafine particles in the base fluid significantly affects heat transfer mechanism of nanofluid.The present single-phase model for the nanofluid assumes that the nanoparticles are completely fluidized and flows with thesame local velocity as that of the base fluid. However in reality there exist slip and random motion of the nanoparticles. It is

Table 2Different models of nanofluids based on thermo-physical properties.

Model Density Viscosity Thermal expansioncoefficient

Non-dimensionalviscous term

Non-dimensionalbuoyancy term

Physicalbasis

I qf lf bf 1=ffiffiffiffiffiffiGrp

1 Pure fluidII qf lnf ¼

lf

ð1�/Þ2:5bf

lnfffiffiffiffiGrp 1 qf � qs

III qnf = / � qs + (1 � /) � qf lnf ¼lf

ð1�/Þ2:5bf

lnf

qnf tf� 1ffiffiffiffi

Grp 1 qf – qs

IV qnf = / � qs + (1 � /) � qf lnf ¼lf

ð1�/Þ2:5bnf = k � bf

lnfqnf tf� 1ffiffiffiffi

Grp k ¼ ðq�bÞnf

qnf �bfGeneral model

Where (q � b)nf = / � qs � bs + (1 � /) � qf � bf.

-0.014+

(a)0.1

0.2

0.3

0.4

0.5

0.6

0.70.80.9

(b)

-0.013+

(c)0.1

0.2

0.3

0.4

0.5

0.6

0.70.80.9

(d)

+-0.015

(e) 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.80.9

(f)

++

-0.014

(g) 0.10.2

0.30.4

0.5

0.6

0.7

0.80.9

(h)Fig. 5. Streamline profile (left) and isotherms (right) obtained for Gr = 105, Pr = 6.2, / = 0.20, n = 3 for different models of nanofluids based on thermo-physical properties; (a, b) Model-I, (c, d) Model-II, (e and f) Model-III and (g, h) Model-IV.


y

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0Model1

Model2

Model3

Model4

Nu

(a)

Ave

rage

Nus

seltN

umbe

r

0.00 0.05 0.10 0.15 0.208

10

12

14

16

18

20

Model1

Model2

Model3

Model4

φ

(b)Fig. 6. Variation of (a) Nusselt number along the hot wall for / = 0.20 (b) averager Nusselt number ðNuÞ from hot wall for different volume fractions forGr = 105, Pr = 6.2, n = 3.


expected that these random motion of the nanoparticles significantly influence the heat transfer in the fluid. This heat trans-fer augmentation is modeled in the present work using the thermal dispersion coefficient (kd). The thermal dispersion coef-ficient incorporates the influence of random motion of the nanoparticles by assuming that the local thermal conductivity isaugmented due to local velocity. The dispersion coefficient (kd) thus depends on the size of solid particles with respect to thecharacteristic dimension of the cavity, the volume fraction of the nanoparticles, the local velocity and the thermal gradientdriving the flow in the cavity.

The influence of thermal dispersion on flow, thermal field and heat transfer augmentation in the cavity is discussed in thissection. Simulations are carried out to evaluate the flow and thermal field in the cavity by involving the thermal dispersioneffect. These are then compared with the flow and thermal field obtained when the thermal dispersion effect is neglected.The physical properties of the nanofluid are taken as that of Model-IV for simulations with and without dispersion. The localNusselt number variation and average Nusselt number on the hot wall are evaluated to compare the heat transferaugmentation.

Fig. 7 shows the comparison of flow and thermal field with and without dispersion effect for copper water nanofluid. Theother controlling parameters are kept constant at Grashof number of 105, shape factor of 3 and volume fraction of nanopar-ticles at 0.20. The streamline patterns and contours of isotherm for the case when thermal dispersion effect is considered, isdepicted by solid lines while the dashed lines show the contours of same magnitude when the dispersion effect is neglected.Grashof number is a measure of the buoyant driving force in relation to the viscous forces that resist the fluid motion. Buoy-ant fluid flow in the form of clockwise roll is initiated along the left hot wall of the cavity. The roll extends from the hot wallto the cold wall due to the temperature difference. The temperature of the hot wall is convected inside of the cavity due tothis buoyant flow. Densely packed isotherms near the hot wall indicates high heat transfer rate. The overlapped contours ofthe isotherms in Fig. 7b depict a shift of solid lines towards the hot wall compared to the dashed line. This signifies an in-crease in strength of convection roll when the thermal dispersion influences are considered in the simulations. This is clearlyillustrated in the streamline contours of Fig. 7a. The maximum value of stream function for the contours depicted by solidlines is more than those of dashed line. Significant difference in the flow pattern near the centre of the cavity is observedwhen the thermal dispersion effects are considered. This is characterized by formulation of small opposing vortices on bothsides of the cavity near the center.

Table 3Average Nusselt number from hot wall for various models of nanofluid based on thermo-physical properties (Gr = 105, Pr = 6.2, / = 0.20, n = 3).

Various models Average Nusselt number

Model-I 12.68Model-II 11.47Model-III 13.85Model-IV 11.03K–D Model [15] 10.30


To estimate the influence of flow velocity on the dispersed thermal conductivity in the domain, the contours of k�d (in Eq.(19)) is presented in Fig. 7c. It is observed that the dispersed thermal conductivity is more in the region near the walls wherethe magnitude of velocity is more compared to the center of the domain. Fig. 7d shows the influence of the size of the nano-particles on dispersed thermal conductivity. It is observed that maximum value of the dispersed thermal conductivity in-creases when the hydraulic diameter of the particles is increased. Table 4 shows the augmentation in average Nusseltnumber from hot wall when dispersion effects are considered in the simulation. It is observed that the influence of thermaldispersion increases as the volume fraction increases.

The next sections discuss the influence of controlling parameters on the heat transfer augmentation for nanofluids. Thethermal dispersion influence is considered through the dispersion coefficient in all the calculations henceforth.

6.3. Influence of Grashof number

Simulations are extended to observe the influence of Grashof number on the flow of nanofluid and thermal field in thecavity. The Grashof number (Gr) for these investigations is varied between 103

6 Gr 6 105. The volume fraction and shapefactor are kept constant as 0.20 and 3, respectively. The physical properties of Model-IV discussed in the Section 3.2 are used

-0.010

-0.009

-0.012 -0.011

(a)

0.9 0.80.7

0.60.5

0.4

0.20.3

0.1

(b)

0.192 0.162

0.161 0.144

0.072

0.072

0.024

0.360 0.34

6

(c)

0.022

0.124

0.124

0.316 0.297

0.336 0.336

0.672

0.66

4

(d)Fig. 7. (a) Effect of thermal dispersion on flow (streamline pattern on left) and (b) thermal field (isotherm on right). Solid lines are for simulation involvingthermal dispersion and dashed line are those neglecting thermal dispersion effect (c) contours for dispersed thermal conductivity (kd) (10�2) in the cavityfor d�p ¼ 10�6 (d) contours for dispersed thermal conductivity (kd) (10�2) in the cavity for d�p ¼ 2� 10�6.

Table 4Variation of average Nusselt number from the hot wall with and without thermal dispersion for Gr = 105, Pr = 6.2, n = 3.

Volume fractions Average Nusselt number with thermal dispersion Average Nusselt number without thermal dispersion

0.00 8.87 8.870.08 9.46 9.250.16 10.02 9.710.20 11.03 10.40


with (dp/L) of order 10�6. The thermal dispersion effect is considered in all these simulations. Grashof number accounts forthe effect of temperature difference, gravity, and kinematic viscosity of nanofluid, dimensions of cavity and thermal expan-sion coefficient of nanofluid. Increase in Grashof number increases the buoyancy driven term in y momentum equation andresults in larger velocity as shown in stream line profile from Fig. 8a, c and e. The central vortex developed for Gr = 1000 iscircular in nature and tends to become elliptic as the Grashof number is increased to 104. For further increase in Grashofnumber to 105, the central vortex is observed to break in two parts. The centre of vortex shifted towards the vertical wallsof the cavity. The temperature profile in the cavity flattens with increase in Gr. The isotherms are observed to be denselypacked near the hot wall indicating higher rate of heat transfer as the Grashof number increases.

-0.017

(a)

0.9

0.8

0.7

0.60.50.4

0.3

0.2

0.1

(b)

-0.018

(c)

0.8

0.70.6

0.50.4

0.3

0.2

0.1

0.9

(d)

-0.011

(e)

0.90.8

0.7

0.6

0.5

0.4

0.20.3

0.1

(f)

Fig. 8. Effect of Grashof number on flow (streamline pattern on left) and thermal field (isotherm on right) in the cavity for / = 0.20, Pr = 6.2, n = 3; (a, b)Gr = 1000, (c, d) Gr = 104, (e, f) Gr = 105.


Fig. 9a shows the variation of u-velocity along the vertical centerline of the cavity while Fig. 9b shows the v-velocity var-iation along the horizontal centerline of the cavity. It is observed that the velocity magnitudes are maximum near the wallsand are minimum near the centre of the cavity. The sinusoidal pattern of the vertical velocity component for lower Grashofnumber signifies the reversal of flow direction from the hot wall to the cold wall with a negligible velocity at the core. As theGrashof number increases the peak increases and shifts near the wall depicting high gradient of velocity near the wall andhigher convection flux. The non-dimensional temperature along the horizontal centre line of the cavity is shown in Fig. 9cand Nusselt number variation along the hot wall is shown in Fig. 8d for varying values of Grashof number. Sharp drop oftemperature near the hot wall is observed as the Grashof number increases, with point of inflexions occurring near the wallsat the center of the two broken vortices. The Nusselt number variation is observed to be maximum near the lower end of thehot wall and the peak value increases with increase in Grashof number. The augmentation in average Nusselt number alongthe hot wall with increase in Grashof number is shown in Table 5.

6.4. Influence of volume fraction of suspended particles

Simulations are also carried out to observe the influence of volume fraction of the nanoparticles on flow, thermal field andheat transfer augmentation in the cavity. These results are discussed in this section. The range of the volume fraction / usedin this study is varied from 0 to 0.20. For all these simulations the Grashof number and the shape factor of the nanoparticleare kept at 105 and 3, respectively. The thermo-physical properties of nanofluids are considered as that of Model-IV alongwith the dispersion effect via the dispersion coefficient.

u

y

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.080.0

0.2

0.4

0.6

0.8

1.0

(a) (b)

(e) (d)

x

v

0.0 0.2 0.4 0.6 0.8 1.0-0.20

-0.10

0.00

0.10

0.20

x

θ

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Gr=103

Gr=104

Gr=105

Nuh

y

0 5 10 15 20 250.0

0.2

0.4

0.6

0.8

1.0

Gr=103

Gr=104

Gr=105

Gr=103

Gr=104

Gr=105

Gr=103

Gr=104

Gr=105

Fig. 9. Effect of Grashof number on (a) u-velocity, (b) v-velocity profile (c) centerline temperature distribution (d) Nu from hot wall for / = 0.20, Pr = 6.2,n = 3.

Table 5Effect of Grashof number on average Nusselt number on hot wall for volume fraction 0.20, Pr = 6.2, n = 3.

Grashof number (Gr) Average Nusselt number ðNuÞhGr = 103 2.70Gr = 104 5.32Gr = 105 11.03

-0.009

-0.008

-0.008

-0.007

-0.005

(a)

0.90.8

0.60.7

0.5

0.4

0.3

0.10.2

(b)

-0.010-0.010

-0.008

-0.007

(c)

0.90.8

0.7

0.6

0.4

0.5

0.3

0.20.1

(d)

-0.011 -0.011

-0.010

-0.008

(e)

0.90.8

0.60.7

0.5

0.4

0.30.2

0.1

(f)

-0.011 -0.011

-0.010

-0.009

(g)

0.90.8

0.60.7

0.5

0.4

0.30.2

0.1

(h) Fig. 10. Effect of volume fraction on flow (streamline on the left) and thermal field (isotherms on the right) for Gr = 105, Pr = 6.2, n = 3. (a, b) / = 0.0,(c, d) / = 0.08, (e, f) / = 0.16 and (g, h) / = 0.20.



Fig. 10 shows the flow and thermal field in the cavity as the volume fraction increases. For the range of parameters con-sidered in Fig. 10, broken re-circulation vortices are observed at the centre of the cavity for pure fluid. These vortices tend tojoin in presence of nanoparticles and get elongated with increase in volume fraction of the nanopaticles. The maximum valueof stream function increases with volume fraction. This is due to high-energy transport through the flow associated with theirregular motion of ultra fine particles. As the volume fraction increases, the velocities along the vertical walls of cavity showa higher level of activity as predicted by thin hydrodynamic boundary layers. This is illustrated in the v velocity variation(Fig. 10b) along the horizontal central line of the cavity for various volume fractions. The isotherms are horizontal in the cen-tral region but near the left and right wall they become vertical and tend to increase the heat transport between fluid andwalls. The isotherms at the center of cavity are horizontal signifying stratification in the vertical direction. The isothermsbecome vertical only inside the thermal boundary layers at vertical walls. The isotherms get flattened with increase in vol-ume fraction due to dispersion.

The non-dimensional u-velocity variation along the horizontal centerline and the Nusselt number variation from the hotwall of the cavity for varying volume fraction are shown in Fig. 11. As volume fraction increases, the u velocity component ofnanofluid increases as a result of an increase in the energy transport through the fluid. The peak of Nusselt number is also

u

y

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.060.0

0.2

0.4

0.6

0.8

1.00.0

0.08

0.16

0.20

(a) (b) x

v

0.0 0.2 0.4 0.6 0.8 1.0-0.10

-0.05

0.00

0.05

0.10

φ=0.0,0.08,0.16,0.20

Nuh

y

0 5 10 15 20 250.0

0.2

0.4

0.6

0.8

1.00.0

0.08

0.16

0.20

(c) Fig. 11. Variation of (a) u-velocity along vertical centerline (b) v-velocity along horizontal centerline and (c) Nusselt number along the hot wall of the cavityfor varying volume fraction.

Table 6Effect of volume fraction on average Nusselt number from hot wall Gr= 105, Pr = 6.2, n = 3.

Volume fraction of solid particles (%) Average Nusselt number ðNuÞh0 8.878 9.4616 10.0220 11.03


observed to increase with increase in volume fraction. Table 6 shows the value of average Nusselt number for various volumefractions. The numerical results indicate that the heat transfer feature of a nanofluid increases remarkably with the volumefraction of nanoparticles. As the volume fraction increases, irregular and random movements of particles increases energyexchange rates in the fluid with the penalty on wall shear stress and consequently enhances the thermal dispersion inthe flow.

-0.009 -0.008

-0.007

-0.006

(a)

0.9 0.8

0.7

0.6

0.5

0.40.3

0.20.1

(b)

-0.012

-0.010

-0.008

-0.011

(c)

0.8

0.6

0.5

0.4

0.3

0.10.2

0.9

(d)

-0.014 -0.014-0.013

-0.011

(e)

0.90.8

0.7

0.6

0.5

0.40.3

0.2

0.1

(f) Fig. 12. Effect of shape factor on flow (streamlines on the left) and thermal field (isotherms on the right) for Gr = 105, Pr = 6.2, / = 0.20; (a, b) n = 0.5 (c, d)n = 3, (e, f) n = 6.


6.5. Influence of shape factor

The surface area of contact between the nanoparticle and base fluid contribute significantly to the heat transfer betweenparticles and base fluid. Simulations are carried out to observe the heat transfer augmentation by varying the shape factor ofnanoparticles. The shape factor values considered here are n = 0.5, 3 and 6. For all these simulations, the Grashof number(Gr), Prandtl number (Pr), and volume fraction of nanoparticles are kept constant as 105, 6.2 and 0.20, respectively.

The streamline profile and isotherms for various values of shape factors are shown in Fig. 12. The streamline intensityincreases with the shape factor. For n = 0.5 the central vortices are observed to be broken. As the shape factor increases, thesebroken vortices tend to rejoin with an increase in strength of vortex. The increment in intensity of streamline indicates high-er velocity of fluid inside the cavity. Due to the presence of thin hydrodynamic layer near the hot and cold wall, the stream-line and isotherms are densely packed near these zones of cavity.

The u-velocity variation along the vertical centerline and v-velocity variation in horizontal centerline shown in Fig. 13shows increase in peak values near the walls as the shape factor increases. This is because of higher surface to volume ratioand hence greater heat transport and higher velocity variation.

The local Nusselt number variation along the hot wall for various shape factor ‘n’ is presented in Fig. 13c. Results showthat the heat transfer increases from the top of the hot wall towards the lower portion of the wall. The maximum value of

u

y

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.060.0

0.2

0.4

0.6

0.8

1.0n=0.5

n=3

n=6

(a) (b)

(a) (b)

x

v

0.0 0.2 0.4 0.6 0.8 1.0-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15n=0.5

n=3

n=6

x

θ

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

n=0.5,3,6

Nuh

y

0 5 10 15 20 250.0

0.2

0.4

0.6

0.8

1.0n=0.5

n=3

n=6

Fig. 13. Variation of (a) u-velocity along vertical centerline (b) v-velocity along horizontal centerline (c) temperature along horizontal centerline and (d)Nusselt number along the hot wall of the cavity for varying shape factor; Gr = 105, Pr = 6.2, / = 0.20.

Table 7Effect of shape factor on average Nusselt number from hot wall; Gr = 105, Pr = 6.2, / = 20%.

Shape factor (n) Average Nusselt number ðNuÞh0.5 7.783 11.036 13.11


local Nusselt number increases as ‘shape factor’ increases. In some cases the size of particles may produce reduction in veloc-ity. This is almost negligible for small volume fraction and small shape factor. The peak values of the local Nusselt numberalong the hot wall (Fig. 13d) also increases as the shape factor is increased. Table 7 presents the average Nusselt numberalong hot wall for different values of shape factor.

7. Conclusions

Numerical simulations are carried out for analysis of flow and thermal field in a thermally driven cavity using nanofluid.The contribution of random motion of the nano-sized particles in the base fluid is accounted for using a thermal dispersionmodel. The model accounts for the enhancement of local thermal conductivity of the nanofluid due to motion of the particles.It is observed that the dispersed thermal conductivity is more in the region near the walls where the magnitude of velocity ismore compared to the center of the domain.. It is also observed that maximum value of the dispersed thermal conductivityincreases when the hydraulic diameter of the particles is increased. The flow and thermal field in the cavity are analyzedusing four different models for estimating effective viscosity, density and thermal expansion coefficient of the nanofluid.The general model (Model-IV) shows the lowest average Nusselt number due to the lower thermal expansion coefficientand reduced buoyant convection. A further reduction in the average Nusselt number is observed when the nanofluid viscos-ity in Model-IV is replaced by the equation of viscosity proposed by Kreigger and Dougherty [15]. The augmentation in heattransfer due to nanofluid is estimated numerically by varying the Grashof number, volume fraction and shape of nanopar-ticles. The influence of shape of the nanoparticle on heat transfer augmentation is simulated using an empirical shape factorparameter which depicts the ratio of surface area to volume of the nanoparticles.

References

[1] Y. Xuan, W. Roetzel, Conceptions for heat transfer correlation of nanofluids, Int. J. Heat Mass Transfer 43 (2000) 3701–3707.[2] S. Lee, S.U.S. Choi, S. Li, J.A. Eastman, Measuring thermal conductivity of fluids containing oxide nanoparticles, ASME Trans., J. Heat Transfer 121 (2)

(1999) 280–289.[3] S.E.B. Maliga, S.M. Palm, C.T. Nguyen, G. Roy, N. Galanis, Heat transfer enhancement using nanofluid in forced convection flow, Int. J. Heat Fluid Flow 26

(2005) 530–546.[4] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass

Transfer 46 (2003) 3639–3663.[5] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250.[6] A.K. Santra, S. Sen, N. Chakraborty, Study of heat transfer augmentation in a differentially heated square cavity using copper–water nanofluid, Int. J.

Therm. Sci. 47 (2008) 1113–1122.[7] R. Ben Mansour, N. Galanis, C.T. Nguyen, Effect of uncertainities in physical properties in forced convection heat transfer with nanofluids, Appl. Therm.

Eng. 27 (2007) 240–249.[8] J.C. Maxwell, A Treatise on Electricity and Magnetism, second ed., Oxford University Press, Cambridge, 1904.[9] S.P. Jhang, S.U.S. Choi, Effect of various parameters on nanofluid thermal conductivity, ASME J. Heat Transfer 129 (2006) 617–623.

[10] S. Lee, S.U.S. Choi, S. Li, J.A. Eastman, Measuring thermal conductivity of fluids containing oxide nanoparticles, ASME Trans., J. Heat Transfer 121 (2)(1999) 280–289.

[11] S.K. Das, N. Putra, P. Thiesen, W. Roetzel, Temperature dependence of thermal conductivity enhancement for nanofluids, ASME J. Heat Transfer 125(2003) 567–574.

[12] C. Kleinstreuer, Jie Li, Discussion: Effect of various parameters on nanofluid thermal conductivity, vol. 130/025501/1-3, 2008 (ASME J. Heat Trans., vol.129, 2006, pp. 617–623).

[13] P. Keblinski, S.R. Philpot, S.U.S. Choi, J.A. Eastman, Mechanisms of heat flow in suspensions of nano-sized particles (nanofluids), Int. J. Heat MassTransfer 45 (2002) 855–863.

[14] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952) 571–581.[15] I.M. Krieger, T.J. Dougherty, A mechanism for non-Newtonian flow in suspensions of rigid sphere, Trans. Soc. Rheol. 3 (1959) 137–152.[16] L.E. Nielson, Generalised equation for the elastic moduli of composite materials, J. Appl. Phys. 41 (1970) 4626–4627.[17] S.M.S. Murshed, K.C. Leong, C. Yang, Investigations of thermal conductivity and viscosity of nanofluids, Int. J. Therm. Sci. 47 (2008) 560–568.[18] S.Z. Heris, S. Gh. Etemad, M.N. Esfahany, Experimental investigation of oxide nanofluids laminar flow convective heat transfer, Int. Commun. Heat Mass

Transfer 33 (2006) 529–535.[19] N. Putra, W. Roetzel, S.K. Das, Natural convection of nanofluids, Heat Mass Transfer 46 (2003) 775–784.[20] R. Prasher, D. Song, J. Wang, Measurements of nanofluid viscosity and its implication for thermal applications, Int. J. Heat Fluid Flow 26 (2005) 855–

864.[21] Jyotirmay Banerjee, K. Muralidhar, Role of internal radiation during Czochralski Growth of YAG and Nd: YAG crystals, Int. J. Therm. Sci. 45 (2006) 151–

167.[22] R.L. Hamilton, O.K. Crosser, Thermal conductivity of heterogeneous two-component system, I EC Fundam. (1962) 182–191.[23] H. Wadell, Volume, shape and roundness of quartz particles, J. Geol. 43 (1935) 250–280.[24] R.J. Krane, J. Jessee, Some detailed field measurements for a natural convection flow in a vertical square enclosure, in: Proceedings of the First ASME–

JSME Thermal Engineering Joint Conference, vol. 1, 1983, pp. 323–329.