The effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking/stretching sheet

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International Journal of Heat and Fluid Flow

journal homepage: www.elsevier .com/locate / i jhf f

Effects of variable viscosity and thermal conductivity of Al2O3–water nanofluidon heat transfer enhancement in natural convection

Eiyad Abu-Nada *

Department of Mechanical Engineering, Hashemite University, Zarqa 13115, Jordan

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 November 2008Received in revised form 9 February 2009Accepted 10 February 2009Available online xxxx

Keywords:NanofluidViscosityThermal conductivityNatural convectionAnnulus

0142-727X/$ - see front matter � 2009 Elsevier Inc. Adoi:10.1016/j.ijheatfluidflow.2009.02.003

* Tel.: +962 390 3333; fax: +962 382 6613.E-mail address: [email protected]

Please cite this article in press as: Abu-NaHeat Fluid Flow (2009), doi:10.1016/j.ijhe

Heat transfer enhancement in horizontal annuli using variable properties of Al2O3–water nanofluid isinvestigated. Different viscosity and thermal conductivity models are used to evaluate heat transferenhancement in the annulus. The base case uses the Chon et al. expression for conductivity and the Ngu-yen et al. experimental data for viscosity which take into account the dependence of these properties ontemperature and nanoparticle volume fraction. It was observed that for Ra P 104, the average Nusseltnumber was reduced by increasing the volume fraction of nanoparticles. However, for Ra = 103, the aver-age Nusselt number increased by increasing the volume fraction of nanoparticles. For Ra P 104, the Nus-selt number was deteriorated every where around the cylinder surface especially at high expansion ratio.However, this reduction is only restricted to certain regions around the cylinder surface at Ra = 103. ForRa P 104, the difference in Nusselt number between the Maxwell Garnett and Chon et al. model predic-tion is small. But, there was a deviation in prediction at Ra = 103 and this deviation becomes more signif-icant at high volume fraction of nanoparticles. The Nguyen et al. data and Brinkman model givescompletely different predictions for Ra P 104 where the difference in prediction of Nusselt numberreached 30%. However, this difference was less than 10% at Ra = 103.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

Natural convection heat transfer is an important phenomenonin engineering systems due to its wide applications in electroniccooling, heat exchangers, and thermal systems. Enhancement ofheat transfer in such systems is very essential from the industrialand energy saving perspectives. The low thermal conductivity ofconventional heat transfer fluids, such as water, is considered a pri-mary limitation in enhancing the performance and the compact-ness of such thermal systems. An innovative technique forimprovement of heat transfer using nano-scale particle dispersedin a base fluid, known as nanofluid (Choi, 1995), has been studiedextensively in recent years (Daungthongsuk and Wongwises, 2007;Trisaksri and Wongwises, 2007) mainly for forced convectionapplications. However, natural convection heat transfer researchusing nanofluids has received very little attention and there is stilla debate on the effect of nanoparticles on heat transfer enhance-ment in natural convection applications.

Examples of these controversial results are the results reportedby Khanafer et al. (2003) who studied Cu–water nanofluids in atwo dimensional rectangular enclosure. They reported an increasein heat transfer with the increase in percentage of the suspended

ll rights reserved.

da, E. Effects of variable visatfluidflow.2009.02.003

nanoparticles at any given Grashof number. Oztop and Abu-Nada(2008) showed similar results, where an enhancement in heattransfers was registered by the additions of nanoparticles. How-ever, contrary experimental findings were reported by Putraet al. (2003) using Al2O3 and CuO water nanofluids. They reportedthat the natural convection heat transfer coefficient was lowerthan that of clear flow. Additionally, another experimental work,in natural convection, by Wen and Ding (2006) reported deteriora-tion in heat transfer by the addition of nanoparticles. Most re-cently, Abu-Nada et al. (2008) showed that the enhancement ofheat transfer in natural convection depends mainly on Rayleighnumber and for certain Rayleigh numbers, Ra = 104, the heat trans-fer was not sensitive to nanoparticles concentration whereas athigher values of Rayleigh number an enhancement in heat transferwas taking place. Therefore, there is still a controversy on the effectof nanofluids on heat transfer in natural convection and thenumerical simulations seem to over estimate the enhancement ofheat transfer in natural convection.

In fact, convective heat transfer is affected by the thermophys-ical properties of the nanofluid such as viscosity and thermal con-ductivity. A recent nanofluid heat transfer study on forcedconvection conducted by Ben Mansour et al. (2007) revealed thatfor forced convection different expressions for the thermophysicalproperties of nanofluids lead to totally different predictions for theperformance of system. All of the previous mentioned numerical

cosity and thermal conductivity of Al2O3–water nanofluid ... Int. J.

Nomenclature

cp specific heat at constant pressure (J/kg K)D diameter of inner cylinder (m)g gravitational acceleration (m/s2)h local heat transfer coefficient (W/m2 K)k thermal conductivity (W/m K)L gap between inner and outer cylinder, i.e., L = ro � ri (m)Nu Nusselt numberPr Prandtl numberqw heat transfer at the inner cylinder surface (W/m2)Ra Rayleigh numberRe Reynolds number~r radial coordinate measured from the inner cylinder sur-

face (m)r nondimensional radial distance~T dimensional temperature (K)T nondimensional temperatureu dimensional tangential velocity (m/s)U nondimensional tangential velocityv dimensional radial velocity (m/s)V nondimensional radial velocity

Greek symbolsa thermal diffusivity (m2/s)

b thermal expansion coefficient (1/K)h angle measured from the lower symmetry planel dynamic viscosity (N s/m2)m kinematic viscosity (m2/s)q density (kg/m3)/ nanoparticle volume fractionW nondimensional stream functionw dimensional stream function (m2/s)X nondimensional vorticityx dimensional vorticity (1/s)

Subscriptsavg averageC coldf fluidH hotnf nanofluidp particleo reference value at cold condition

TC

g

r

θ

ro

ri

TH

θ = 0º

θ = 90º

θ = 180º

L

Fig. 1. Sketch of problem geometry.

2 E. Abu-Nada / International Journal of Heat and Fluid Flow xxx (2009) xxx–xxx

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results conducted on natural convection used the Brinkman modelfor the viscosity. This model is shown to underestimate the effec-tive viscosity of the nanofluid (Nguyen et al., 2007; Polidori et al.,2007). The Brinkman model does not consider the effect of nano-fluid temperature or nanoparticles size. Besides, the Brinkmanmodel was derived for larger particles size compared to the sizeof the nanoparticles. On the other hand, for the thermal conductiv-ity, most of numerical simulations reported in literature used theMaxwell Garnett (MG) model. This model does not consider mainmechanisms for heat transfer in nanofluids such as Brownian mo-tion and does not consider nanoparticle size or temperature depen-dence. Therefore, numerical simulations need more robust modelsfor viscosity and thermal conductivity that take into account tem-perature dependence and nanoparticle size. Recently, Nguyen et al.(2007) and Angue Minsta et al. (2009) studied the effect nanopar-ticles concentration, nanoparticles size on nanofluids viscosity un-der a wide range of temperatures experimentally. They showedthat viscosity drops sharply with temperature especially for highconcentration of nanoparticles. Moreover, the effect of tempera-ture, nanoparticle size, and nanoparticles volume fraction on ther-mal conductivity was studied experimentally by Chon et al. (2005)and they showed that nanofluid thermal conductivity is also af-fected by temperature, volume fraction of nanoparticles, and nano-particle size. Thus, such physics cannot be neglected and thedependence of nanofluid properties on temperature, volume frac-tion of nanoparticles, must be taken into account in order to pre-dict the correct role of nanoparticles on heat transferenhancement.

Therefore, the scope of the current research is to implementmore appropriate models for nanofluids properties and study theeffect of these models on heat transfer in natural convection. Theexperimental results reported by Nguyen et al. (2007) will be usedto derive a correlation for nanofluids viscosity as a function of tem-perature and nanoparticles concentration. Also, the Chon et al.model (2005) will be used for the thermal conductivity. The cur-rent research will evaluate the role of both viscosity and thermalconductivity, derived from experimental data, on heat transfer innatural convection. Besides, these models will be compared to fre-quent used models in literature namely the Brinkman model for

Please cite this article in press as: Abu-Nada, E. Effects of variable visHeat Fluid Flow (2009), doi:10.1016/j.ijheatfluidflow.2009.02.003

viscosity and the MG model for thermal conductivity. Theenhancement in heat transfer will be evaluated under wide rangeof temperatures and wide range of volume fraction ofnanoparticles.

2. Governing equations and problem formulation

Fig. 1 shows a schematic diagram of differentially heated annu-lus. The fluid in the annulus is a water-based nanofluid containingAl2O3 nanoparticle. The nanofluid is assumed incompressible andthe flow is assumed to be laminar. It is assumed that water and

cosity and thermal conductivity of Al2O3–water nanofluid ... Int. J.

Table 1Thermophysical properties of fluid and Al2O3 nanoparticles.

Physical properties Fluid phase (water) Al2O3

cp (J/kgK) 4179 765q (kg/m3) 997.1 3970k (W/mK) 0.613 25b � 10�5 (1/K) 21 0.85Dp (nm) 0.384 47

Table 2Grid independence tests.

Grid size Air Nanofluid

Nuavg Relative error* Nuavg Relative error*

21 � 21 2.3925802 – 6.925662 –31 � 31 2.4544397 0.025203 6.808558 �0.017241 � 41 2.4698942 0.006257 6.769043 �0.0058451 � 51 2.4701051 8.54 � 10�05 6.772485 0.00050861 � 61 2.4701027 �9.7 � 10�07 6.772491 8.86 � 10�07

71 � 71 2.4701036 3.64 � 10�07 6.772487 �5.9 � 10�07

81 � 81 2.4701031 �2 � 10�07 6.772492 7.38 � 10�07

101 � 101 2.4701028 �1.2 � 10�07 6.772489 �4.4 � 10�07

* Relative error = (Nu large grid � Nusmall grid)/Nularge grid.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0r

T

180º

90º

0º

a

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nanoparticles are in thermal equilibrium and no slip occurs be-tween them. The thermophysical properties of the nanofluid are gi-ven in Table 1. The inner cylinder is maintained at a constanttemperature (TH) higher than the outer cylinder (TC). The densityof the nanofluid is approximated by the Boussinesq model. The vis-cosity as well as the thermal conductivity of the nanofluid is con-sidered variable properties that vary with temperature and volumefraction of nanoparticles.

The governing equations for the laminar, steady state naturalconvection in terms of the stream function-vorticity formulationare given as:

Vorticity

@

@~rx@W@h

� �� @

@hx@W@~r

� �¼ 1

qnf

@

@~rlnf ~r

@x@~r

� �þ @

@hlnf

1~r@x@h

� �� �

þðuqsbs þ ð1�uÞqf bf Þ

qnf

� g@~T@~r

~r sin hþ @~T@h

cos h

!¼ 0 ð1Þ

Energy

@

@~r~T@W@h

� �� @

@h~T@W@~r

� �¼ @

@~ranf ~r

@~T@~r

!þ @

@hanf

1~r@~T@h

!ð2Þ

Kinematics

@

@~r~r@W@~r

� �þ @

@h1~r@W@h

� �¼ �~rx ð3Þ

The radial and tangential velocities are given by the following rela-tions, respectively,

v ¼ 1~r@w@h

; ð4Þ

u ¼ � @w@~r: ð5Þ

Fig. 2. Comparison between viscosities calculated using Eq. (13) and the Nguyenet al. data (2007).

Please cite this article in press as: Abu-Nada, E. Effects of variable visHeat Fluid Flow (2009), doi:10.1016/j.ijheatfluidflow.2009.02.003

The thermal diffusivity, in Eq. (2), is given as:

anf ¼knf

ðqcpÞnfð6Þ

The effective density of the nanofluid is given as

qnf ¼ ð1�uÞqf þuqp ð7Þ

The heat capacitance of the nanofluid is expressed as:

ðqcpÞnf ¼ ð1�uÞðqcpÞf þuðqcpÞp ð8Þ

The effective thermal conductivity of the nanofluid is calculated bythe Chon et al. model (2005):

knf

kbf¼ 1þ 64:7u0:7640 df

dp

� �0:3690 kf

kp

� �0:7476

Pr0:9955 Re1:2321 ð9Þ

0

0.1

0.2

0.3

0.4

0.5

0.6

0 20 40 60 80 100 120 140 160 180

Nu

/(R

a D)0

.25

Angle (θ)

Present Work

Kuhen and Goldstein, 1980 (experimental)

Wang et al., 1990 (numerical)

b

Fig. 3. (a) Comparison of present work (solid lines) and Kuhen and Goldsteinexperimental results (1976), experimental data points: h: 0�, e: 90�, s: 180�(Ra = 4.7 � 104, Pr = 0.706, and L/D = 0.8). (b) Comparison between present workand other published data for the distribution of Nusselt number around the cylindersurface for pure natural convection case (RaD = 105, Pr = 0.7).

cosity and thermal conductivity of Al2O3–water nanofluid ... Int. J.

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ARTICLE IN PRESS

The results using Eq. (9) will be compared to the MG model givenby:

knf

kf¼ kp þ ðn� 1Þkf � ðn� 1Þðkf � kpÞu

kp þ ðn� 1Þkf þ ðkf � kpÞuð10Þ

The Pr and Re in Eq. (9) are given as, respectively (Chon et al., 2005):

Pr ¼lf

qf afð11Þ

Re ¼qf kbT

3pl2f lf

ð12Þ

where f stands for the base fluid which is water in the current studyand kb is the Boltzmann constant, 1.3807 � 10�23 J/K and lf is themean path of base fluid particles given as 0.17 nm (Chon et al.,2005). This model considers the effect of nanoparticle size and tem-perature on nanofluids thermal conductivity with a wide range oftemperature between 21 and 70 �C. This model was further testedexperimentally by Angue Minsta et al. (2009) for Al2O3 and CuOnanoparticles and found to predict the thermal conductivity ofnanofluid accurately up to a volume fraction of 9% for both CuOand Al2O3 nanoparticles. Therefore, the current study adopted theChon et al. model to predict the thermal conductivity of Al2O3–water nanofluid.

The correlation for dynamic viscosity of Al2O3–water nanofluidis derived using the available experimental data of Nguyen et al.(2007). Actually, no explicit correlation is given in Nguyen et al.(2007) that give the viscosity of Al2O3–water nanofluid as a func-tion of the temperature and the volume fraction of nanoparticlessimultaneously. Therefore, in the recent work, a two-dimensionalregression is further performed on the raw experimental data re-ported in Nguyen et al. (2007) to develop such a correlation. Thisderived correlation gives the viscosity of nanofluid as a functionof temperature and volume fraction of nanoparticles. The R2 of

0

2

4

6

8

10

12

0 30 60 90 120 150 180

θ

Nu ϕ = 0%

ϕ = 1%ϕ = 3%ϕ = 4%ϕ = 5%ϕ = 7%ϕ = 9%

( (a)

0

0.5

1

1.5

2

2.5

3

3.5

0 30 60

Nu

ϕ = 0%ϕ = 1%ϕ = 3%ϕ = 4%ϕ = 5%ϕ = 7%ϕ = 9%

(c)

Fig. 4. Nusselt number distribution around inner cylinder surface using various volume

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the regression is 99.8% and a maximum error is 5%. The correlation,obtained from the two-dimensional regression, is given as:

lAl2O3¼ �0:155� 19:582

~Tþ 0:794uþ 2094:47

~T2� :192u2

� 8:11u~T� 27463:863

~T3þ :0127u3 þ 1:6044

u2

~T

þ 2:1754u~T2

ð13Þ

The viscosity in Eq. (13) is expressed in centi poise. Fig. 2 pre-sents a plot for the viscosity of Al2O3–water nanofluid as a functionof temperature and concentration of nanoparticles calculated byusing Eq. (13). Also, the figure plots the measured data from Ngu-yen et al. experiment (2007). It is very clear that the current regres-sion is in good agreement with the measurements of Nguyen et al.(2007).

The results, using Eq. (13), will be compared to the Brinkmanmodel given by:

lnf ¼lf

ð1�uÞ2:5ð14Þ

The following dimensionless groups are introduced:

X ¼ xL2

afo

; W ¼ wafo

; V ¼ vLafo

U ¼ uLafo

; T ¼~T � TC

TH � TC; and r ¼

~r � ri

L

k ¼ knf

kfo;a ¼ anf

afo; l ¼

lnf

lfo

ð15Þ

where the subscript ‘‘o” stands for the reference temperature whichis taken as 22 �C in the current study. The temperature differencebetween the hot and the cold surfaces is kept constant at 30 �C.

0

1

2

3

4

5

6

7

0 30 60 90 120 150 180

θ

Nu

ϕ = 0%ϕ = 1%ϕ = 3%ϕ = 4%ϕ = 5%ϕ = 7%ϕ = 9%

)b

90 120 150 180

θ

fractions of Al2O3 nanoparticles, L/D = 0.8: (a) Ra = 105, (b) Ra = 104, and (c) Ra = 103.

cosity and thermal conductivity of Al2O3–water nanofluid ... Int. J.

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By using the dimensionless parameters, the governing equa-tions are written as:

@

@rX@W@h

� �� @

@hX@W@r

� �

¼ Prð1�uÞ þu qp

qf

@

@rl r þ ri

L

� � @X@r

� �þ @

@hl 1

r þ riL

� � @X@h

! !

þ RaPr1

ð1�uÞu

qf

qpþ 1

bs

bfþ 1

uð1�uÞ

qf

qpþ 1

24

35 @T

@rr þ ri

L

� �sin hþ @T

@hcos h

� �

ð16Þ

0.1

0.2

0.3

0.4

0.5

0.60.7

0.80.9

a)(

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.80.9

(c)(

0.1

0.2

0.3

0.4

0.50.6

0.7

0.80.9

(e)

Fig. 5. Temperature isotherms for L/D = 0.8: (a) Ra = 105 / = 9%, (b) Ra = 105, / = 4%, (c

Please cite this article in press as: Abu-Nada, E. Effects of variable visHeat Fluid Flow (2009), doi:10.1016/j.ijheatfluidflow.2009.02.003

@

@rT@W@h

� �� @

@hT@W@r

� �

¼ 1

ð1�uÞ þu ðqcpÞpðqcpÞfo

@

@rk r þ ri

L

� � @X@r

� �þ @

@hk

1r þ ri

L

� � @X@h

! !

ð17Þ

@

@rr@w@r

� �þ @

@hÞ ¼ �rx ð18Þ

where

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.80.9

b)(

0.1

0.2

0.3

0.4

0.5

0.6

0.70.80.9

d)

0.1

0.2

0.3

0.4

0.5

0.60.7

0.80.9

(f)

) Ra = 105, / = 1%, (d) Ra = 103, / = 9%, (e) Ra = 103, / = 4%, and (f) Ra = 103, / = 1%.

cosity and thermal conductivity of Al2O3–water nanofluid ... Int. J.

6 E. Abu-Nada / International Journal of Heat and Fluid Flow xxx (2009) xxx–xxx

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Ra ¼ gbðTH � TCÞL3

v foafo

ð19Þ

Pr ¼ mfo

afoð20Þ

The ratio between the gap length between the inner and outercylinders divided by the inner cylinder diameter, i.e., L/D, is definedas the aspect ratio. The effective thermal expansion coefficient ofthe nanofluid appears on the right hand side of Eq. (17) which isgiven as (Khanafer et al., 2003)

bnf

bf¼ 1

ð1�uÞu

qf

qpþ 1

bp

bfþ 1

uð1�uÞ

qf

qpþ 1

24

35 ð21Þ

The dimensionless radial and tangential velocities are given as,respectively:

V ¼ 1r þ ri

L

� � @w@h

;

U ¼ � @w@r:

ð22Þ

Due to the symmetry, only half of the cylinder will be solved(i.e., 0 6 h 6 180). Therefore, the dimensionless boundary condi-tions are as follows:

Fig. 6. Tangential velocity for Ra = 105, L/D = 0.8 h = 90�: (a) Ra = 105 and (b)Ra = 103.

Please cite this article in press as: Abu-Nada, E. Effects of variable visHeat Fluid Flow (2009), doi:10.1016/j.ijheatfluidflow.2009.02.003

On the inner cylinder surface; W ¼ 0

x ¼ � @2W@r2 ; and T ¼ 1

On the outer cylinder surface; W ¼ 0

x ¼ � @2W@r2 ; and T ¼ 0

Symmetry lines : W ¼ 0; X ¼ 0; and@T@h¼ 0

ð23Þ

3. Numerical implementation

Eqs. (17)–(19) and the variable properties, given by Eqs. (9) and(13), with the boundary conditions given in Eq. (24) are solvedusing the finite volume approach (Patankar, 1980; Versteeg andMalalasekera, 1995). The diffusion term in the vorticity and energyequations is approximated by a second-order central differencescheme which gives a stable solution and a second-order upwinddifferencing scheme is adopted for the convective terms. For fulldetails of numerical implementation, the reader is referred toAbu-Nada et al. (2008).

After solving W, X, and T, further useful quantities are obtained.For example, the Nusselt number can be expressed as:

Nu ¼ hðDÞkf

ð24Þ

The heat transfer coefficient is expressed as

h ¼ qw

TH � TCð25Þ

The thermal conductivity is expressed as

knf ¼ �qw

@~T=@~rð26Þ

By substituting Eqs. (26), (27), (9) into Eq. (25), and using thedimensionless quantities, the Nusselt number on the inner cylinderis written as:

Nu ¼ � knf

kf

� �@T@r

ð27Þ

where (knf/kf) is calculated using Eq. 9. The average Nusselt numberis calculated as:

Nuavg ¼1p

Z h¼p

h¼0NuðhÞdh ð28Þ

The integration of Eq. (27) is carried out by using the 1/3rdSimpson’s rule of integration. A normalized Nusselt number is de-fined as the ratio of Nusselt number at any volume fraction ofnanoparticles to that of pure water and is given as:

Nu�avgðuÞ ¼NuðuÞ

Nuðu ¼ 0Þ ð29Þ

The Nusselt number is used as an indicator of heat transferenhancement where an increase in Nusselt number correspondsto enhancement in heat transfer.

4. Grid testing and code validation

An extensive mesh testing procedure was conducted to guaran-tee a grid independent solution. Two cases of Ra = 0.53 � 104 andRa = 105 using Pr = 0.7 are tested for grid independence with nonanoparticles in the flow field. The present code was tested for gridindependence by calculating the average Nusselt number aroundthe inner cylinder surface. It was found that a grid size of61 � 61 guarantees a grid independent solution for both cases.

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The Nusselt number for the grid independent solution is comparedwith the results of Guj and Stella (1995) and Shu et al. (2000) forconcentric horizontal annulus and Ra = 0.53 � 104. The calculatedaverage Nusselt number by the current code gives a value of2.47010 which falls between the results obtained by Guj and Stella(Nuavg = 2.4220) and the results of Shu et al. (Nuavg = 2.5560). Fur-thermore, a grid independence test was carried out for the Al2O3–water nanofluid using / = 9%, Ra = 105, and L/D = 0.8. It was foundthat the same grid size 61 � 61 guarantees a grid independentsolution. Details of grid independence solution are given in Table 2.

Due the lack of experimental data for natural convection in anannulus with the presence of nanoparticles, the present numericalsolution is validated by the experimental results of Kuhen andGoldstein (1976) using Ra = 4.57 � 104 and Pr = 0.7. The compari-sons for three temperature profiles at three different angles areshown in Fig. 3a. Furthermore, another validation test was carriedfor pure natural convection of air around a heated horizontal cylin-der in free air for Ra = 105 and Pr = 0.70 with the experiment of Ku-hen and Goldstein (1980) and the numerical work of Wang et al.(1990) as shown in Fig. 3b. It is clear that present results are ingood agreement with other published data.

5. Results and discussion

The range of Rayleigh number, volume fraction of nanoparticles,and expansion ratio, are Ra = 103–105, 0 6 / 6 9%, and 0.2 6 L/D 6 0.8, respectively. Fig. 4 presents Nusselt number distributionaround the inner cylinder surface using various volume fractions

-2-6

-10

-14-18

-22

a)(

(c)

Fig. 7. Streamlines for Ra = 105, L/D = 0.8:

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of Al2O3 nanoparticles for L/D = 0.8. For the case of Ra = 105 andRa = 104 the increase in the volume fraction of nanoparticles causesa reduction in Nusselt number every where around the inner cylin-der surface. However, for Ra = 103 there is a reduction in Nusseltnumber for h < 90 and an enhancement in heat transfer is takingplace for h > 90. The difference between / = 1% and the pure fluidcase is negligible whereas the effect of nanoparticles become moreevident at higher concentrations. Also, it is clear that for high vol-ume fraction of nanoparticles, / = 9%, and for the case for Ra = 103,the Nusselt number variation around the inner cylinder surface be-comes less pronounced compared to lower volume fraction ofnanoparticles. Actually, for Ra = 103, the more addition of nanopar-ticles reduces the difference between the maximum and minimumNusselt number around the cylinder surface. For example, for /= 9% the maximum Nusselt number is 2.9 and the minimum is2.3. This is best illustrated by looking at Fig. 5 where the case of/ = 9%, and for the case for Ra = 103 the temperature is almost uni-form around the inner cylinder surface. Also, Fig. 5 illustrates howthe thickness of thermal boundary layer is influenced by the addi-tion of nanoparticles. Also, it is very interesting to note how theplume region is affected by increasing the volume fraction fornanoparticles where for the case of 105, as shown in Fig. 5a, theplume region spreads wider and for the case of Ra = 103, as shownin Fig. 5d, the plume region disappears completely. This behavior isrelated to the increased viscosity at high volume fraction of nano-particles, see Fig. 2, where high / causes the fluid to become moreviscous and this causes the velocity to decrease accordingly, seeFig. 6, which reduces convection. Fig. 6 shows how the maximum

-5-10

-15

-20

-25

-30

b)(

-5-10

-15-20

-25

-30

-35

(a) / = 9%, (b) / = 4%, and (c) / = 1%.

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velocity drops to almost one half when the volume fraction ofnanopartilces is increased from zero (clear fluid) to 9% for the caseof Ra = 105. Also, Fig. 6 shows that this reduction becomes more se-vere at Ra = 103 where the maximum velocity drops to almost onefourth of the maximum velocity when the volume fraction ofnanopartilces is increases from zero to 9%. Besides, Fig. 6 revealshow the velocity gradients at the inner cylinder wall are affectedby presence of nanopartilces. This velocity gradient change will af-fect the convection around the inner cylinder surface and accord-ingly will influence the Nusselt number around the innercylinder. The reduction in velocities and convection will causethe fading of the plume at Ra = 103 and the spreading of the plumeregion for the case of Ra = 105. The increase of thermal boundarylayer thickness is responsible for the reduction in temperature gra-dients at the inner surface which causes a reduction in Nusseltnumber accordingly, see Eq. (29). It is observed from Fig. 5, forRa = 105, that the addition of nanoparticles causes an increase inthermal boundary layer thickness which explains the reduction

0

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a)(

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ϕ = 0%ϕ = 1%ϕ = 3%ϕ = 4%ϕ = 5%ϕ = 7%ϕ = 9%

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Nu ϕ = 0%

ϕ = 1%ϕ = 3%ϕ = 4%ϕ = 5%ϕ = 7%ϕ = 9%

(e)(

Fig. 8. Nusselt number distribution around inner cylinder surface using various volumRa = 104, L/D = 0.4, (d) Ra = 104, L/D = 0.2, (e) Ra = 103, L/D = 0.4, and (f) Ra = 103, L/D = 0.

Please cite this article in press as: Abu-Nada, E. Effects of variable visHeat Fluid Flow (2009), doi:10.1016/j.ijheatfluidflow.2009.02.003

in the value of Nusselt number every where around the inner cyl-inder surface. However, for the case of Ra = 103, the addition ofnanoparticles causes the thermal boundary layer thickness to in-crease for h < 90; however, this thickness decreases for h > 90 be-cause of the plume disappearance. This explains the behaviorobserved in Fig. 4c for the Nusselt number distribution aroundthe inner surface. Fig. 7 shows the streamlines for the case ofRa = 105. It is clear that by increasing the volume fraction of nano-particles the maximum strength of streamlines is reduced due tothe increased viscosity of the nanofluids as mentioned earlier.

Fig. 8 studies the effect of addition of nanoparticles on Nusseltnumber by using different aspect ratios for the annulus. The figureshows that for the case of Ra = 105, the behavior encountered usingsmaller aspect ratio is similar to that of high aspect ratio. However,for Ra = 103 different behavior is observed using smaller aspect ra-tios. For the case of Ra = 103 the angle at which Nusselt numberswitches from a reduction to increasing is reduced to 60� forL/D = 0.4 and to an almost complete enhancement all over the

0

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)b(

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)f

e fractions of Al2O3 nanoparticles: (a) Ra = 105, L/D = 0.4, (b) Ra = 105, L/D = 0.2, (c)2.

cosity and thermal conductivity of Al2O3–water nanofluid ... Int. J.

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cylinder surface for L/D = 0.2 compared to the 90� encountered at L/D = 0.8; see Fig. 4. Thus, the region around the inner cylinder sur-face where an enhancement in heat transfer is taking place in-creases. This causes an enhancement in the total Nusselt numberaround the inner cylinder surface as shown in Fig. 9.

Fig. 9 shows the average Nusselt number as well as normalizedaverage Nusselt number around the inner cylinder surface. It is ob-served that, for all expansion ratios, and for the case of Ra = 104 andRa = 105 a decrease in Nusselt number is taking place for volumefraction of nanoparticles greater than 5%. However, such a decreaseis not observed for volume fraction less than 5% where a fluctua-tion in Nusselt number is detected. In general, the influence ofnanoparticles has two opposing effects on Nusselt number: afavorable effect that is due to the presence of high thermal conduc-tivity of nanoparticles and an undesirable effect due to the high le-vel of viscosity experienced at high volume fraction of

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)f(e)(

Fig. 9. (a) Average Nusselt number L/D = 0.8, (b) normalized Nusselt number L/D = 0.8,average Nusselt number L/D = 0.2, (f) normalized Nusselt number L/D = 0.2.

Please cite this article in press as: Abu-Nada, E. Effects of variable visHeat Fluid Flow (2009), doi:10.1016/j.ijheatfluidflow.2009.02.003

nanoparticles. The heat transfer in natural convection at high Ray-leigh number is dominated by convection and at low Rayleighnumbers is dominated by conduction. So, for Ra = 105 and 104,the heat transfer is dominated by convection and the presence ofnanoparticles will cause the nanofluid to become more viscouswhich will reduce convection currents and accordingly tempera-ture gradient at the cylinder surface and Nusselt number. This willbe accompanied by some enhancement in heat transfer due to thehigh thermal conductivity of nanoparticles. However, suchenhancement is small compared to the deterioration caused by vis-cosity because the convection currents in the annulus will be re-duced, see Fig. 7. In fact, the role of Brownian motion becomesless pronounced because the thermal conductivity of nanofluid isinversely proportional to the viscosity squared as shown in Eq.(12). Therefore, for high Rayleigh number the more addition ofnanoparticles, for / > 5, will have an adverse effect on Nusselt

0.7

0.75

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)

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(c) average Nusselt number L/D = 0.4, (d) normalized Nusselt number L/D = 0.4, (e)

cosity and thermal conductivity of Al2O3–water nanofluid ... Int. J.

0.8

0.85

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MG & BrinkmanChon et al. & BrinkmanMG + Nguyen Exp. DataChon et al. & Nguyen et al.

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MG & BrinkmanChon et al. & BrinkmanMG + Nguyen Exp. DataChon et al. & Nguyen et al.

(b)(a)

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1.15

1.2

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uav

g*

MG & BrinkmanChon et al. & BrinkmanMG + Nguyen Exp. DataChon et al. & Nguyen et al.

(c)

Fig. 10. Effect of the conductivity and viscosity models on Nusselt number, L/D = 0.8: (a) Ra = 105, (b) Ra = 104, and (c) Ra = 103.

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Fig. 11. Isotherms for Ra = 105, L/D = 0.8, / = 9%: (a) MG and Brinkman model, (b) Chon et al. model (2005) and Brinkman model, (c) MG model and Nguyen et al. data (2007),and (d) Chon et al. model (2005) and Nguyen et al. data (2007).

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Please cite this article in press as: Abu-Nada, E. Effects of variable viscosity and thermal conductivity of Al2O3–water nanofluid ... Int. J.Heat Fluid Flow (2009), doi:10.1016/j.ijheatfluidflow.2009.02.003

-5-10

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Fig. 12. Streamlines for Ra = 105, L/D = 0.8, / = 9%: (a) MG and Brinkman model, (b) Chon et al. model (2005) and Brinkman model, (c) MG model and Nguyen et al. data(2007), and (d) Chon et al. model (2005) and Nguyen et al. data (2005).

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number. However, for / 6 5, the role of viscosity is less pro-nounced and the adverse effect of viscosity is sort of balanced bythe favorable effect of thermal conductivity which explains thefluctuation in Nusselt number for / 6 5. Also, it is worth to men-tion that the case of Ra = 104 experience more deterioration in Nus-selt number compared Ra = 105. Actually, for Ra = 104 the inertiaforces are smaller than that of Ra = 105 which will cause the ad-verse effect of nanoparticles to become more severe at Ra = 104

which will cause more reduction in Nusselt number compared toRa = 105.

On the other hand, for low Rayleigh numbers i.e., Ra = 103, theheat transfer is dominated by conduction and by adding morenanoparticles the conduction is enhanced due to the high thermalconductivity of nanoparticles and accordingly the heat transfer isenhanced. The more addition of nanoprtilces will cause the strongconvection currents that are mainly experienced in the plume re-gions to diminish which will cause a disappearance of the plumeregion, see Fig. 5d–f. This will cause an increase in temperaturegradient all over the cylinder surface and accordingly an increasein Nusselt number.

An interesting comparison between various models used forthermal conductivity and viscosity on the average Nusselt numberis shown in Fig. 10. This figure shows results obtained using fourdifferent approaches. The first approach is using MG model forthe thermal conductivity and Brinkman model for the viscosity ofnanofluids. This combination is used by most researchers in litera-ture. The second approach is using the Chon et al. model (2005) forthermal conductivity and the Brinkman model for the viscosity of

Please cite this article in press as: Abu-Nada, E. Effects of variable visHeat Fluid Flow (2009), doi:10.1016/j.ijheatfluidflow.2009.02.003

the nanofluid. The third approach is using the MG model for thethermal conductivity and the experimental data of Nguyen et al.(2007) for viscosity. The fourth approach, the one that is used inthe current study as the base case, is using the Chon et al. model(2005) for thermal conductivity and Nguyen et al. model for viscos-ity of nanofluids. From Fig. 10a and b, it is clear that difference be-tween average Nusselt number calculated using the Chon et al.model and the MG model is relatively small. However, the differ-ence in Nusselt number when using the Nguyen et al. data andBrinkman model is much more significant. This tells that the effectof thermal conductivity models is less significant than viscositymodels at high Rayleigh number. So, the prediction of Nusseltnumber using the Nguyen data is completely different from usingthe Brinkman model. According to the Brinkman model prediction,there is an over estimation in the enhancement in Nusselt numberby increasing the volume fraction of nanoparticles. However, theNguyen et al. data shows deterioration in Nusselt number by add-ing nanoparticles. Actually, the Brinkman model is used for dis-persed particles in solution with particle size much higher thanthe nano-scale particles and its applicability to nanoparticles isquestionable. On the other hand for low Rayleigh number, i.e.,Ra = 103 when Nguen et al. data is used the Nusselt number isnot sensitive to volume fraction less than 4%; however, for highervolume fractions an enhancement in heat transfer is observedwhich is opposite to the behavior registered at high Rayleigh num-ber (using the Nguyen et al. data). It is interesting to note that forthis Rayleigh number all of the four approaches predict anenhancement in heat transfer. Also, the difference in Nusselt pre-

cosity and thermal conductivity of Al2O3–water nanofluid ... Int. J.

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diction between the Brinkman model and the Nguyen et al. data issmall compared to high Rayleigh numbers. For example, the max-imum difference, at / = 9%, is approximately 10% compared to 30%at Ra = 105.

Also, Fig. 10 shows that the deviation between the Chon et al.model and MG model becomes more pronounced for Ra = 103

especially at high volume fractions of nanoparticles. This tells thatsuch difference becomes more appreciable at high volume fractionof nanoparticles and this difference cannot be neglected. In generalthe MG model over predicts the enhancement in heat transfercompared to the Chon et al. model at high volume fractions ofnanoparticles (/ > 5%). Fig. 10 is very useful to know when theMG model is appropriate for natural convections applications andwhen its applicability becomes less accurate. Besides, it tells thatfor higher Rayleigh numbers the Brinkman prediction is far fromNguyen et al. measured data prediction and this difference cannotbe neglected which limits the applicability of Brinkman model forthese Rayleigh numbers. However, this difference is less pro-nounced at lower Rayleigh number and the Brinkman model couldbe used with approximately 10% deviation in prediction from themeasured data. Figs. 11 and 12 show the isotherms and thestreamlines for the four mentioned approaches, respectively. It isvery clear how different models give different temperature andstreamlines distribution in the annulus.

6. Conclusions

For Ra P 104 and / > 5%, the average Nusselt number is reducedby increasing the volume fraction of nanoparticles. However, theinfluence of nanoparticles is less pronounced at low volume frac-tion where a fluctuation in Nusselt number is noticed for / 6 5%.For Ra = 103, the average Nusselt number is enhanced by increasingthe volume fraction of nanoparticles. Generally speaking, forRa P 104, the Nusselt number is deteriorated every where aroundthe cylinder surface especially at high expansion ratio. However,this reduction is only limited to certain regions around the cylindersurface at Ra = 103. For Ra P 104, the difference in MG and Chonet al. model prediction is small. However, there is a deviation atRa = 103 and this deviation becomes more significant at high vol-ume fraction of nanoparticles. The Nguyen data and Brinkmanmodel gives completely different predictions for Ra P 104 wherethe difference in prediction of Nusselt number could more than30%. However, this difference reduces to less than 10% at Ra = 103.

Please cite this article in press as: Abu-Nada, E. Effects of variable visHeat Fluid Flow (2009), doi:10.1016/j.ijheatfluidflow.2009.02.003

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