International Journal of Heat and Mass Transfer 55 (2012) 1475–1485

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International Journal of Heat and Mass Transfer

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Unconfined laminar nanofluid flow and heat transfer around a square cylinder

Vahid Etminan-Farooji a, Ehsan Ebrahimnia-Bajestan a,⇑, Hamid Niazmand a, Somchai Wongwises b,c

a Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iranb Fluid Mechanics, Thermal Engineering and Multiphase Flow Research Lab. (FUTURE), Department of Mechanical Engineering, King Mongkut’s University of TechnologyThonburi, Bangmod, Bangkok, Thailandc The Academy of Science, The Royal Institute of Thailand, SanamSueaPa, Dusit, Bangkok 10300, Thailand

a r t i c l e i n f o

Article history:Received 13 April 2011Received in revised form 10 October 2011Accepted 10 October 2011Available online 12 December 2011

Keywords:NanofluidSquare cylinderPeclet numberThermophysical propertiesUnconfined flowHeat transfer enhancement

0017-9310/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2011.10.030

⇑ Corresponding author. Tel.: +98 915 300 6795; faE-mail address: ehsan.ebrahimnia@gmail.com (E. E

a b s t r a c t

The momentum and forced convection heat transfer for a laminar and steady free stream flow of nano-fluids past an isolated square cylinder have been studied numerically. Different nanofluids consisting ofAl2O3 and CuO with base fluids of water and a 60:40 (by mass) ethylene glycol and water mixture wereselected to evaluate their superiority over conventional fluids. Recent correlations for the thermal con-ductivity and viscosity of nanofluids, which are functions of particle volumetric concentration as wellas temperature, have been employed in this paper. The simulations have been conducted for Pe = 25,50, 100 and 200, with nanoparticle diameters of 30 and 100 nm and particle volumetric concentrationsranging from 0% to 4%. The results of heat transfer characteristics of nanofluid flow over a square cylindershowed marked improvement comparing with the base fluids. This improvement is more evident in flowswith higher Peclet numbers and higher particle volume concentration, while the particle diameterimposes an adverse effect on the heat transfer characteristics. In addition, it was shown that for any givenparticle diameter there is an optimum value of particle concentration that results in the highest heattransfer coefficient.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Due to the immense theoretical and practical importance ofcross flow past bluff bodies, this field of study has received sub-stantial attention from researchers. Common examples of thiskind of flow include flow in tubular and pin-type heat exchang-ers, the use of thin cylinders as measuring probes as well as sen-sors, and in the continuous thermal treatment of food particles(such as slices and chips of carrots and potatoes) in viscous flu-ids (high Prandtl number). Over the years, a considerableamount of information dealing with different aspects of the heattransfer and flow over bluff bodies has been added to the liter-ature, a great majority of which relates to the flow over a circu-lar cylinder [1–5]. In contrast to much theoretical, experimentaland numerical data on the flow around circular cylinders over awide range of Reynolds numbers, there are a limited number ofsimilar studies and information available on flow around squarebodies. However, in recent years there have been some efforts toexpand the knowledge of flow over square cylinders in the liter-ature. Turki et al. [6] presented a numerical study to analyse theunsteady flow field and heat transfer characteristics in a

ll rights reserved.

x: +98 511 876 3304.brahimnia-Bajestan).

horizontal channel with a built-in heated square cylinder. Thestudy was carried out for two channel blockage ratios (B = 1/4and 1/8), and different Reynolds and Richardson numbers rang-ing from 62 to 200 and from 0 to 0.1 respectively at Pr = 0.71.They showed that the overall heat transfer of a square cylinderwas slightly affected by the blockage ratio and presented corre-lations for B = 1/4 and B = 1/8.

Sharma and Eswaran [7] have carried out a detailed study onheat and fluid flow over square cylinders. In a study concerningthe unconfined two dimensional laminar regime for Reynoldsnumbers between 1 and 160, they showed that the transition tounsteadiness occurs between Re = 40 and 50. The results indi-cated that the heat transfer characteristics in the steady flowregime (Re 6 40) and unsteady periodic 2-D flow regime(50 6 Re 6 160) are markedly different. The time-averaged rear-surface Nusselt number is more strongly dependent on Reynoldsnumbers in the periodic flow regime, in contrast to the otherfaces of the cylinder. Finally, heat transfer correlations for bothconstant temperature and constant-heat-flux boundary condi-tions were proposed.

Dihman et al. [8] researched the flow and heat transfer charac-teristics of an isolated square cylinder in a crossflow placed sym-metrically in a channel for the range of conditions of 1 6 Re 6 45,0.7 6 Pr 6 4000, Pe 6 4000 and B = 1/8, 1/6 and 1/4. It was shown

Nomenclature

B blockage ratio, B = L/Hcd drag coefficientcp specific heat, J/kg Kd diameter, mh heat transfer coefficient, W/m2 KH height of the computational domain, mk thermal conductivity, W/m KL side of the square cylinder, mn normal direction to the surface, mNu Nusselt number, Nu = hL/kP pressure, PaPe Peclet number, Pe = Re � PrPr Prandtl number, Pr = lcp/kRe Reynolds number, Re = qu1L/lT temperature, KV velocity, m/sx streamwise coordinate, x = x0/LXd downstream face distance of the cylinder from the

outlet, m

Xu upstream face distance of the cylinder from the inlet, my transverse coordinate, y = y0/L

Greek symbolsd size of the CV clustered around the cylinder, mD size of the CV far away from the cylinder, ml dynamic viscosity, kg/m sU particle volumetric concentration, %q density, kg/m3

j Boltzmann constant, 1.381 � 10�23 J/K

Subscripts1 inlet conditionbf base fluidnf nanofluidnp nanoparticlew wall

1476 V. Etminan-Farooji et al. / International Journal of Heat and Mass Transfer 55 (2012) 1475–1485

that under the same conditions of B, Re and Pr, the use of the con-stant heat flux boundary condition yields slightly higher values ofthe Nusselt number than constant temperature case. The resultsindicated that for a fixed block ratio, the increase in Prandtl and/or Reynolds number ends in the enhancement of the average Nus-selt number. Finally, heat transfer correlations were obtained forthe constant temperature and constant heat flux cases of the solidsquare cylinder in the previously mentioned range of physicalparameters. In a recent study [9] on the effects of Reynolds andPrandtl numbers on the heat transfer from a square cylinder, anumerical study has been carried out in a 2D unsteady crossflowfor the range of conditions 60 6 Re 6 160 and 0.7 6 Pr 6 50 (themaximum value of Peclet number being 4000). The results showedthat the overall mean Nusselt number increases with the Reynoldsand Prandtl numbers. It was also shown that the Nusselt numberprofiles for the two boundary conditions were qualitatively similar,though the constant heat flux case had a numerically higher meanNu, for identical conditions. The front surface of the cylinder exhib-ited the highest value of the surface average Nusselt number; how-ever the value for the rear face exceeded the values at the top orbottom surfaces at high Reynolds and Prandtl numbers.

One of the main difficulties with industrial fluids such as water,mineral oils and ethylene glycol is their poor heat transfer proper-ties that impose a limitation in improving the heat transfer aug-mentation and compactness of the heat exchangers. There areseveral methods to improve the heat transfer characteristics, suchas employing the flow inserts and nanofluids. Recently, much re-search [10,11] has been done on the suspension of nanoparticlesin conventional heat transfer fluids named nanofluid, due to theiranomalous thermal conductivity enhancement. Applying thisnew generation of heat transfer fluids eliminates the limitationof low thermal conductivity of conventional heat transfer fluids.Several effective parameters on the thermal conductivity of nano-fluids have been presented, such as nanofluid temperature [12,13],nanoparticle size [12], concentration [12,13], shape [14] and mate-rials [13]. Furthermore, some researchers believe that the Brown-ian motion of nanoparticles inside the fluid is a key mechanismof heat transfer in nanofluids [15]. A number of models arepresented to consider these different effective parameters on theconductive heat transfer of nanofluids [13,15,16] and other ther-mophysical properties such as viscosity [17,18].

Ding et al. [19] conducted experiments on the laminar convec-tive heat transfer of a Carbone nanotube-water mixture flowingthrough a horizontal tube. They reported a maximum enhance-ment of 3.5 times the heat transfer coefficient of the nanofluid rel-ative to pure water. They showed that the effect of shape andconcentration of nanoparticles is significant in determining theconvective heat transfer coefficient. An experimental investigationon the convective heat transfer characteristics in the developingregion of tube flow was carried out by using water/Al2O3 nanofl-uids by Anoop et al. [20]. Their results indicated that nanofluidswith a smaller average particle size result in higher heat transfercoefficient.

The effect of types of nanoparticle on the laminar convectiveheat transfer of nanofluids was studied by Rea et al. [21]. Theyshowed that water/Al2O3 nanofluid increases the heat transfercoefficient more than water/ZrO2 nanofluid. Zeinali et al. [22]investigated the heat transfer performance of water/Al2O3 andwater/CuO nanofluids flowing through a circular tube under a lam-inar flow regime. The results of their experiments indicated thatthe water/Al2O3 nanofluids cause a greater enhancement in heattransfer coefficient compared with CuO/water. They suggested thatit may result from the large particle size of CuO nanoparticles com-pared with Al2O3. Ebrahimnia-Bajestan et al. [23] carried out anumerical study on the effects of different parameters on the con-vective heat transfer of nanofluids inside a straight circular pipe.Their results indicated that increasing particle concentration, as-pect ratio of nanoparticles and flow Reynolds number enhancethe heat transfer while increasing nanoparticle diameter reducesthe heat transfer coefficient. Furthermore, the results demon-strated that the heat transfer characteristics of nanofluids are sig-nificantly affected by Brownian motion of particles as well as typesof base fluid and nanoparticles.

Due to the outstanding potential of nanofluids in enhancingheat transfer many researchers have studied their effects and char-acteristics, but most of these studies are oriented toward simplegeometries. To the best of the authors’ knowledge, no researchon nanofluids flow and heat transfer over bluff bodies has beendocumented in the literature. In this study, the unconfined flowsof two conventional fluids as well as various types of nanofluidsover a square cylinder are numerically simulated. The convectiveheat transfer coefficients of these flows are determined and the

Xu Xd

L Hu∞

T∞

Square Cylinder Tw

Slip Boundary

Slip Boundary

xy

Fig. 1. Schematics of the flow and the computational domain.

V. Etminan-Farooji et al. / International Journal of Heat and Mass Transfer 55 (2012) 1475–1485 1477

effect of using nanofluids in enhancing the heat transfer from thecylinder to the flow is carefully studied. The conventional fluids(also base fluids) were chosen to be water and a 60% ethylene gly-col and 40% water mixture by mass (EG:W) which are widely usedas the heat transfer fluid in heat exchangers, building heating andautomobile radiators. The material of the nanoparticles were se-lected as Al2O3 and CuO with two different diameters of dnp = 30and 100 nm. The values of the nanoparticle volumetric concentra-tion u were also chosen to be 2% and 4%. All the simulations wereconducted at four different Peclet numbers of 25, 50, 100 and 200.

2. Mathematical modelling

2.1. Geometrical configuration

The system of interest in this case is the unconfined flow arounda two-dimensional square cylinder with side L, which is also thenon-dimensionalising length scale. In this system, the square cylin-der (block) is placed symmetrically between the upper and lowerboundaries and is maintained at a constant temperature Tw. Theblock is exposed to a free stream with uniform velocity u1 andtemperature T1 (Fig. 1). In addition, the free slip condition is usedfor artificial confining boundaries. Sohankar et al. [24] have shownthat for avoiding the misleading effects of the presence of theupper and lower boundaries on the flow characteristics near thecylinder, the blockage ratio B which is defined as L/H should notbe more than 5. Therefore, H/L = 20 has been used in the presentwork as the channel blockage ratio. Moreover, the non-dimen-sional upstream distance between the channel inlet and the frontsurface of the cylinder, Xu/L, is taken as 8 and the non-dimensionaldownstream distance between the rear surface of the cylinder andchannel outlet, Xd/L, is taken as 15. More details on the procedureof choosing these values are presented in Section 2.4.

2.2. Governing equations

The unconfined flow of water and the mixture of ethylene glycoland water (60:40 EG:W)can be considered to be incompressible. Itis also reasonable to neglect viscous dissipation. Under these con-ditions, the following conservation equations were solved numer-ically in a two-dimensional flow field using the computational fluiddynamics software Fluent6.3 [25]:

– Continuity

ðr � VÞ ¼ 0; ð1Þ

– Momentum

qnf ðr � VÞV ¼ �rP þr:ðlnfrVÞ; ð2Þ

– Energy

qnf cP;nf ðV � rÞT ¼ r:ðknfrTÞ: ð3Þ

2.3. Numerical method

The system of governing Eqs. (1)–(3) was solved by the controlvolume method using Fluent [25]. Discretization of convectionterms, diffusion terms, and other quantities resulting from the gov-erning equations was carried out using the first-order upwindscheme. The pressure–velocity coupling was modelled with theSIMPLE algorithm. Fluent uses a point implicit linear equation inconjunction with an algebraic multi-grid (AMG) method to solvethe resultant block system of equations for all dependent variablesin each cell. For all the simulations performed in this study, thesolutions were only considered to be converged when the residuals

were lower than 10�9. The resulting data were then post-processedto obtain the streamlines, temperature field and convective heattransfer coefficients of the block walls.

2.4. Choice of numerical parameters

It has been shown that in numerical simulations of flow aroundbluff bodies, the numerical parameters such as blockage ratio B,upstream computational domain Xu, downstream computationaldomain Xd and grid size, M � N, are of great importance and shouldbe chosen carefully [24,26]. Since there is no available informationon nanofluids flow around a square cylinder, the values of afore-mentioned parameters used when studying the air flow over bluffbodies were used as initial guesses. In order to study the confiningupper and lower boundaries, three blockage ratios of 1/15, 1/20and 1/25 were tested for nanofluid EG:W/Al2O3 4% (dnp = 30 nm)at the lowest Peclet number (Pe = 25). A monotonic behaviourwas discovered in the variation of cd and Nu with blockage ratio,and the changes in the values of cd and Nu were 1.68% and0.47%, respectively as the blockage ratio reduced from 1/20 to 1/25. Therefore, B = 1/20 was found to be adequate to simulate nano-fluid unconfined flow.

In order to explore the influence of the upstream distance be-tween the inlet and the front surface of the cylinder, four valuesof Xu/L = 5, 8, 10 and 15 were studied while keeping Xd/L constantat 40. Once again the nanofluid EG:W/Al2O3 4% (dnp = 30 nm) wassimulated at Pe = 25 and results showed that relative percentagechanges in the values of cd and Nu were about 0.042% and 0.012%for Xu/L = 8 with respect to Xu/L = 15, respectively. Thus Xu/L = 8was found to be an appropriate value, which is consistent with ear-lier research [7,8,27].

An extensive study on the influence of the downstream distanceof the cylinder from the outlet Xd is carried out in [24] and also in anumber of other studies including [8,28]. Based on the values usedin these papers, four values of Xd/L = 10, 15, 20 and 30 were testedfor water/CuO 4% (dnp = 100 nm) at the highest Peclet number(Pe = 200), while keeping Xu/L constant at 8. This experiment re-vealed that the resulting gain in the accuracy of cd and Nu wereas small as 0.019% and 0.01%, respectively, at the expense of a con-siderable increase in CPU time when increasing the Xd/L from 15 to30. As a result, Xd/L = 15 is used for all the calculations in this study.

2.5. Grid structure

The grid structure approach adopted in [7,8] was found to beadequate for simulating the unconfined flow of interest. First, auniform fine grid with constant cell size d was distributed aroundthe cylinder over a distance of 1.5 units to adequately capturewake-wall interactions. A much coarser mesh with constant cellsize D was made outside the region around the cylinder thatextended 4 units upstream, downstream, and to the sides. An

Table 1Details of the grids used at Pe = 200 for grid independence study.

d D Xu Xd B Size (M � N)

Grid 1 0.015 0.5 8 15 1/20 163 � 67Grid 2 0.01 0.25 8 15 1/20 274 � 129Grid 3 0.008 0.2 8 15 1/20 342 � 162Grid 4 0.006 0.15 8 15 1/20 458 � 210

1478 V. Etminan-Farooji et al. / International Journal of Heat and Mass Transfer 55 (2012) 1475–1485

algebraic expression was used to generate the grid between thesetwo regions.

2.6. Grid independence study

A grid independence study was carried out using the differentnon-uniform grids listed in Table 1. Four values of d and D weretested to study the effects of near-region and far-field grid resolu-tion for Pe = 200 and a blockage ratio of B = 1/20. The fluid waschosen to be water/CuO4% (dnp = 100 nm) and the upstream dis-tance Xu/L and downstream distance Xd/L were kept constant at8 and 15, respectively. The results of grids 1 and 2 demonstrateconsiderable differences while the relative changes in the valueof cd and Nu for grid 3 relative to grid 4 is only 0.31% and0.74%, respectively. The excessive computation time taken tosolve grid 4 justifies the small errors involved in using grid 3.Therefore, grid 3 is used in this study for unconfined (B = 1/20)flow. The grid structure was generated using Gambit [29] and isshown in Fig. 2. It shows the non-uniform grid for the whole com-

Y

0 5 100

5

10

X

Y

7 7.5 8 8.50

0.5

1

1.5

2

Fig. 2. Non-uniform compu

putational domain (Fig. 2(a)) and a magnified view near the block(Fig. 2(b)).

2.7. Nanofluid modelling

The thermophysical properties of nanofluids such as density,specific heat, thermal conductivity and viscosity are calculatedbased on relationships presented in the literature. These formulasare listed below:

– DensityThe density of the nanofluids is calculated based on the equa-tion proposed by Pak and Cho [30]:

X

tational

qnf ¼ ð1� /Þqbf þ /qnp: ð4Þ

– Specific heatThe specific heat of nanofluids is determined using the equationgiven by Xuan and Roetzel [31] that assumes thermal equilib-rium between the base fluid and the nanoparticles:

Cp;nf ¼ð1� /ÞðqcpÞbf þ /ðqcpÞnp

ð1� /Þqbf þ /qnp: ð5Þ

– Thermal conductivityKoo and Kleinstreuer [15] presented their thermal conductivitymodel as a two-term function which takes into account theeffect of particle size dnp, particle volumetric concentration /,temperature T and the properties of the base fluid as well asthe effect of the Brownian motion of nanoparticles. Later, Vajjhaand Das [32] revised this relation using a broader set of data:

15 20

9 9.5 10

grid structure.

Table 2Curve-fit relations proposed by Vajjha and Das [32].

Type of particles b Concentration

Al2O3 8:4407ð100/Þ�1:07304 1% 6 / 6 10%

CuO 9:881ð100/Þ�0:9446 1% 6 / 6 6%

Table 3Physical properties of nanoparticles.

Material q (kg/m3) Cp (J/kg�K) k (W/m K)

Al2O3 3970 765 40CuO2 6350 535.6 76.5

Table 4Physica

Nano

Wate

Wate

V. Etminan-Farooji et al. / International Journal of Heat and Mass Transfer 55 (2012) 1475–1485 1479

knf ¼knp þ 2kbf � 2ðkbf � knpÞ/knp þ 2kbf þ ðkbf � knpÞ/

kbf

þ 5� 104b/qcp;bf �ffiffiffiffiffiffiffiffiffiffiffiffiffiffijT

qnpdnp

sf ðT;/Þ; ð6aÞ

where

f ðT;/Þ ¼ 2:8217� 10�2/þ 3:917� 10�3� � T

T0

þ �3:0669� 10�2/� 3:91123� 10�3� �

: ð6bÞ

In the above equation, T0 is set at 273 K and also the expressionsof b for Al2O3 and CuO nanoparticles are given in Table 2. Theserelations are valid for 298 K 6 T 6 363 K.

– ViscosityIn a recent paper, Masoumi et al. [17] developed a theoreticalmodel for the prediction of the effective viscosity of nanofluidsbased on Brownian motion. They showed that their presentedmodel could accurately predict the effective viscosity of differ-ent nanofluids. Their model is:

lnf ¼ lbf þqnpVBd2

np

72Cd; ð7aÞ

in which:

VB ¼1

dnp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi18jT

pqnpdnp

s; ð7bÞ

d ¼ffiffiffiffiffiffip

6/3

rdnp; ð7cÞ

C ¼ l�1bf c1dnp � 109 þ c2

� �/þ c3dnp � 109 þ c4

� �h i; ð7dÞ

where:

c1 ¼ �0:000001133;c2 ¼ �0:000002771;c3 ¼ 0:00000009;c4 ¼ �0:000000393:

l properties of nanofluids with the base fluid of water at T = 300 K.

fluid dnp (nm) / ð%Þ q (kg/m3)

r/Al2O3 30 2 1056.04 1115.5

100 2 1056.04 1115.5

r/CuO 30 2 1103.64 1210.7

100 2 1103.64 1210.7

The thermophysical properties equations summarised in theprevious sections require the values of the properties of the basefluid as well as nanoparticles. The properties of nanoparticles aretaken to be constant in the present operating range of 300 K toabout 320 K (Table 3).The thermophysical properties of water and EG:W were ob-tained from the ASHRAE Handbook [33] were curve fitted as afunction of the temperature with the following equations.

– Thermophysical properties of water

cp (J

392369392369

376341376341

qbf ¼ �0:0036T2 þ 1:9159T þ 748:19; ð8Þcp;nf ¼ �0:0001T3 þ 0:1155T2 � 41:296T þ 9017:8; ð9Þkbf ¼ �8� 10�6T2 þ 0:0062T � 0:5388; ð10Þlbf ¼ 0:00002414� 10ð247:8=ðT�140ÞÞ; ð11Þ

– Thermophysical properties of EG:W

qbf ¼ �0:002475T2 þ 0:9998T þ 1002:5; ð12ÞCp;bf ¼ 4:248T þ 1882:4; ð13Þkbf ¼ �3:196� 10�6T2 þ 0:0025T � 0:1054; ð14Þ

lbf ¼ 0:001� exp 3135:6� 1T� 8:9367

� �: ð15Þ

Then they were substituted into the density, specific heat, thermalconductivity and viscosity equations, Eqs. (4)–(7), to evaluate theproperties of nanofluids at different temperatures, concentrationsand nanoparticle diameters. This was done by developing variousUser Defined Functions (UDFs) in C language which were then inter-preted and incorporated into the main solver. Hence, in this studythe properties of nanofluids are temperature dependent. Tables 4and 5 show the physical properties of nanofluids based on waterand EG:W at T = 300 K, respectively.

2.8. Validation of results

The results were validated using the available data in the liter-ature which are mostly oriented to air (Pr = 0.7). Separate runswere necessary to determine the results at specific Reynolds num-bers of 5, 20 and 40. Table 6 compares the drag coefficient cd andoverall Nusselt number of the cylinder Nu obtained in the presentresearch with those from references [7,26]. It is clear that the re-sults are in very good agreement with previous studies.

3. Results and discussion

Computations were carried out in a complete laminar and stea-dy regime. The Peclet numbers were chosen to be 25, 50, 100 and200 so the corresponding Reynolds numbers would not exceed 45(for Re > 50 the flow becomes unsteady). A total number of 72 sim-ulations were carried out for four nanofluids of water/Al2O3, water/

/kg K) k (W/m K) l (kg/m s) Pr

4.3 0.6857 9.942E�04 5.6894.8 0.7215 1.188E�03 6.0864.3 0.6679 8.715E�04 5.1214.8 0.7043 8.909E�04 4.674

1.7 0.6870 1.032E�03 5.6506.3 0.7272 1.277E�03 6.0011.7 0.6689 8.767E�04 4.9306.3 0.7082 9.012E�04 4.348

Table 6Comparing the values of Nu and CD with the data in the literature (air, Pr = 0.7).

Re = 5 Re = 20 Re = 40

Nu cd Nu cd Nu cd

Present study 1.27 4.83 2.07 2.43 2.72 1.83Paliwali et al. [26] 1.22 – 2.07 – 2.71 1.98Sharma and Eswaran [7] 1.24 4.86 2.05 2.35 2.71 1.75

8 9 10 11 12-2

-1

0

1

2

8 9 10 11 12-2

-1

0

1

2

Fig. 3. Streamlines of water/CuO 4% (dnp =

Table 5Physical properties of nanofluids with the base fluid of EG:water (60:40) at T = 300 K.

Nanofluid dnp (nm) u (%) q (kg/m3) cp (J/kgK) k (W/mK) l (kg/ms) Pr

EG:Water/Al2O3 30 2 1137.5 2989.8 0.4143 5.310E�03 38.3204 1195.3 2839.0 0.4356 6.347E�03 41.374

100 2 1137.5 2989.8 0.3997 4.655E�03 34.8194 1195.3 2839.0 0.4215 4.758E�03 32.055

EG:Water/CuO 30 2 1185.1 2875.9 0.4149 5.511E�03 38.1934 1290.5 2640.9 0.4394 6.823E�03 41.003

100 2 1185.1 2875.9 0.4002 4.682E�03 33.6494 1290.5 2640.9 0.4238 4.813E�03 29.991

1480 V. Etminan-Farooji et al. / International Journal of Heat and Mass Transfer 55 (2012) 1475–1485

CuO, EG:W/Al2O3, and EG:W/CuO using values of dnp = 30 and 100as well as / ¼ 0, 2% and 4%. Twelve extra simulations were alsoconducted to determine the required values for presenting Fig. 10.

Fig. 3 shows the streamlines plots in the vicinity of the squarecylinder at Pe = 25, 50, 100 and 200 for water/CuO4% (dnp = 30 nm).It is clear from Fig. 3(a)–(d) that the flow is attached to the blockuntil it reaches the trailing edge of the upper surface. At this point,the flow separates from the block and forms a recirculation regionbehind the cylinder. At Pe = 25 (Fig. 3(a)), this region is small but byincreasing the Peclet number the recirculation region develops andat Pe = 200 (Fig. 3(d)) extends to less than 3 times L. In the case of

8 9 10 11 12-2

-1

0

1

2

8 9 10 11 12-2

-1

0

1

2

30 nm) at different Peclet numbers.

00.11

00.11 00.11

00.33

00.33

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0 9

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22

Fig. 4. Temperature contours of water (upper half) and water/CuO 4% (dnp = 30 nm, lower half) at different Peclet numbers.

V. Etminan-Farooji et al. / International Journal of Heat and Mass Transfer 55 (2012) 1475–1485 1481

water/CuO 4% (dnp = 30 nm), the Peclet numbers of 25, 50, 100 and200 correspond to Reynolds numbers of 4.2, 8.3, 16.7 and 33.3.

Fig. 4 depicts the non-dimensional temperature contours (tem-perature is non-dimensionalised as h = (T � T1)/(Tw � T1)) aroundthe block for different Peclet numbers. The top half and the lowerhalf show the isotherms of water and water/CuO4% (dnp = 30 nm),respectively. Clearly, the contours of both fluids have the same pat-tern but the nanofluids show a higher capability of transferring theheat from the cylinder, particularly at higher Peclet numbers. It isevident from Fig. 4 that temperature contours are much densernear the front surface of the block. This indicates the higher tem-perature gradient and, subsequently, greater heat transfer coeffi-cient. At the trailing edge of the upper surface of the cylinder,where the flow separates from the block, the temperature contoursstart to disperse. As the Pe increases and the recirculation regionbehind the cylinder grows, this dispersion reduces and the iso-therms become closer to the rear side of the block.

Fig. 5 shows the local heat transfer coefficient along the cylindersurfaces for water/CuO 4% (dnp = 30 nm) at different Peclet num-bers. The heat transfer coefficient around the block is defined as:

h ¼ knf oT=onTw � T1

: ð16Þ

As expected, the h values increase with increasing the Peclet num-ber. Generally, the front surface has the highest heat transfer coef-ficient which is predictable according to Fig. 4. It is also clear that

heat transfer coefficient curve reaches a peak at the corners of theblock which is due to the high temperature gradients at thesepoints. The averaged values of h over the rear surface of the cylinderis lower than the front and upper surfaces as the temperature con-tours are less crowded behind the block.

Fig. 6 presents the average heat transfer coefficient around theblock at different Peclet numbers, nanoparticles volume fractionsand diameters for the cases of Al2O3/water and CuO/water nanofl-uids. The results indicate that the average heat transfer coefficientof nanofluid rises with increasing Peclet number as well as the par-ticle volume fraction and decreases with an increase in particlediameter.

Since the denominator of Eq. (16) is constant, the effect of nano-fluids on the heat transfer coefficient can be explained based on thevariation in thermal conductivity and wall temperature gradient.

In this paper the inlet velocity is adjusted to achieve a constantPeclet number (Pe ¼ qu1LCp

k ) for different cases of nanofluids.According to Table 4, at higher concentration levels, the values ofthermal conductivity and density increase and the heat capacityreduces. However, the overall interaction of these parameters issuch that the inlet velocity should increase to achieve a constantPeclet number which may lead to a higher velocity gradient andas a result, depending on the Prandtl behaviour, the temperaturegradient around the block changes. On the other hand, increasingthe particle concentration boosts the viscosity which may tend todecrease the velocity gradient near the block. Therefore, these

0

5

10

15

20

25

0 0.5 1 1.5 2

h(W

/m2 -

K)

h(W

/m2 -

K)

h(W

/m2 -

K)

h(W

/m2 -

K)

Distance along the cylinder surfaces (m)

Water

Water/CuO 4% (30nm)

0

5

10

15

20

25

0 0.5 1 1.5 2

Distance along the cylinder surfaces (m)

Water

Water/CuO 4% (30nm)

0

5

10

15

20

25

0 0.5 1 1.5 2

Distance along the cylinder surfaces (m)

Water

Water/CuO 4% (30nm)

0

5

10

15

20

25

0 0.5 1 1.5 2

Distance along the cylinder surfaces (m)

Water

Water/CuO 4% (30nm)

Fig. 5. Local heat transfer coefficient of water and water/CuO 4% (dnp = 30 nm) along the cylinder surfaces at different Peclet numbers.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

200150100500

_ h(W

/m2 -K

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

_ h(W

/m2 -K

)

Pe200150100500

Pe

Water

Water/Al2O3 2% (30nm)

Water/Al2O3 4% (30nm)

Water/Al2O3 2% (100nm)

Water/Al2O3 4% (100nm)

Water

Water/CuO 2% (30nm)

Water/CuO 4% (30nm)

Water/CuO 2% (100nm)

Water/CuO 4% (100nm)

Fig. 6. Average heat transfer coefficient of the cylinder as a function of Peclet number for water and different nanofluids with base fluid of water.

1482 V. Etminan-Farooji et al. / International Journal of Heat and Mass Transfer 55 (2012) 1475–1485

opposing behaviours during particle concentration enhancementshould be considered to realise the resultant heat transfer charac-teristics of nanofluids.

Fig. 6 shows that, at any given Pe, the heat transfer coefficient isenhanced as the particle concentration increases. According to Eq.(16), this manner proves that for the nanofluids shown in this fig-ure, the upward trend of thermal conductivity dominates the heat

transfer coefficient behaviour. It can also be seen that the effect ofparticle concentration becomes more significant at higher Pecletnumbers.

Similarly, for a given particle concentration the heat transfercoefficient increases at a higher Pe. That is due to the fact thatfor constant thermophysical properties and a fixed geometry theonly way to increase the Peclet number is to increase the inlet

_ h(W

/m2 -K

)

_ h(W

/m2 -K

)

200150100500

Pe

200150100500

Pe

0.0

0.5

1.0

1.5

2.0

2.5

EG:Water

EG:W/Al2O3 2% (30nm)

EG:W/Al2O3 4% (30nm)

EG:W/Al2O3 2% (100nm)

EG:W/Al2O3 4% (100nm)

0.0

0.5

1.0

1.5

2.0

2.5

EG:Water

EG:W/CuO 2% (30nm)

EG:W/CuO 4% (30nm)

EG:W/CuO 2% (100nm)

EG:W/CuO 4% (100nm)

Fig. 7. Average heat transfer coefficient of the cylinder as a function of Peclet number for ethylene glycol:water (EG:W) and different nanofluids with base fluid of EG:W.

_ h(W

/m2 -K

)

200150100500

Pe

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Water/Al2O3 2% (30nm)

Water/Al2O3 4% (30nm)

Water/CuO 2% (30nm)

Water/CuO 4% (30nm)

Fig. 8. Average heat transfer coefficient of the cylinder as a function of Pecletnumber for different water based nanofluids.

V. Etminan-Farooji et al. / International Journal of Heat and Mass Transfer 55 (2012) 1475–1485 1483

velocity which leads to a higher velocity and temperature gradientnear the block.

On the other hand, the particle diameter has an inverse effect onheat transfer characteristics, which is obvious in Fig. 6, especiallyat lower particle concentrations.

Similarly, Fig. 7 presents the effects of particle concentrationand diameter on averaged heat transfer coefficients at differentvalues of Peclet number. In this figure the base fluid is a 60:40 mix-ture of ethylene glycol and water (EG:W).

Comparison of Figs. 6 and 7 indicates that the effect of nanopar-ticle diameter on the heat transfer coefficient is more significant inthe cases of nanofluids with the base fluid of EG:W. Since the Prandtlnumber of EG:W is noticeably higher than water at the same Pe

200150100500

Pe

1.11

1.13

1.15

1.17

1.19

1.21

1.23

1.25

Qnf.

./ Q

bf

Water/Al2O3 2% (30nm)

Water/Al2O3 4% (30nm)

Water/Al2O3 2% (100nm)

Water/Al2O3 4% (100nm)

Fig. 9. Heat transfer coefficient ratio

number (see Tables 4 and 5), the heat transfer coefficient of waterbased nanofluids is higher than that of EG:W based nanofluids.

The effect of nanoparticle material on heat transfer coefficientof nanofluids is shown in Fig. 8. The results indicate that the heattransfer coefficient of water/CuO nanofluid is a little higher thanwater/Al2O3 at higher particle concentrations and Peclet numberdue to the more noticeable difference in the thermophysical prop-erties of CuO as compared with Al2O3.

Fig. 9 presents the ratio of overall heat transfer rate of nanofl-uids to the corresponding base fluids (at the same Peclet number)for different particle concentrations and diameters. The resultsshow that the ratio of heat transfer rate increases with increasingparticle concentration / and decreases with particle diameterdnp. As Fig. 9 shows, the effect of particle concentration in enhanc-ing the heat transfer is more significant than that of particle diam-eter and the effect of particle diameter is more considerable inlower particle concentrations. The enhancement of heat transferrate is nearly constant at different Peclet number flows, especiallyat higher Peclet numbers. Comparison of Fig. 9(a) and (b) demon-strates that the effect of nanofluids on the heat transfer rateenhancement is considerably greater for the case of EG:W basednanofluids than water based nanofluids.

Fig. 10 shows the effect of particle diameter on the heat transfercoefficient for different particle concentrations at Pe = 100. The re-sults indicate that the effect of particle diameter is substantial atlower particle concentrations. As aforementioned, in order tomaintain a constant Peclet number the inlet velocity should be in-creased with respect to the particle concentration and conse-quently the heat transfer coefficient enhances. However, in thecase of the particle diameter of 15 nm, there is an optimum particle

200150100500

Pe

Qnf.

./Q

bf

1.11

1.13

1.15

1.17

1.19

1.21

1.23

1.25

EG:W/Al2O3 2% (30nm)

EG:W/Al2O3 4% (30nm)

EG:W/Al2O3 2% (100nm)

EG:W/Al2O3 4% (100nm)

of nanofluids to the base fluids.

2.4

2.5

2.6

2.7

2.8

2.9

3.0

3.1

0% 1% 2% 3% 4%

Φ

Water/Al2O3 (15nm)

Water/Al2O3 (30nm)

Water/Al2O3 (60nm)

Water/Al2O3 (100nm)

_ h(W

/m2 -K

)

Fig. 10. Average heat transfer coefficient of the cylinder as a function of volumetricratio for different Al2O3 nanoparticle diameters at Pe = 100.

1484 V. Etminan-Farooji et al. / International Journal of Heat and Mass Transfer 55 (2012) 1475–1485

concentration with the largest heat transfer coefficient and forhigher particle concentrations the heat transfer decreases.

In the explanation of Fig. 6, two opposing behaviours of the wallvelocity gradient were observed during particle concentrationenhancement which should be considered to realise the resultantheat transfer characteristics of nanofluids. For a given Pe, increas-ing the particle concentration enhances the thermal conductivityof the nanofluid, but based on Eq. (7a) and Table 5 the viscosity in-crease is more dominant at small particle diameters which cangovern the wall velocity gradient behaviour. Higher viscositycauses a lower velocity gradient near the block which results in alower temperature gradient. Therefore, in the nominator of Eq.(16) the thermal conductivity increases with particle diameterand the temperature gradient decreases, which denotes that thereis a certain particle concentration with better heat transfer charac-teristics as shown in Fig. 10 for dp = 15 nm. Considering Fig. 10, thisbehaviour is expected to be observed at higher particle concentra-tions for other particle diameters.

4. Conclusion

The effects of using nanofluids instead of conventional fluids inthe unconfined flow and heat transfer over a square cylinder wereinvestigated numerically. Heat transfer characteristics of Al2O3 andCuO nanofluids were studied by varying effective parameters suchas particle volumetric concentrations /, nanoparticle diameters dnp

and base fluids. For example, at a Peclet number of 200, usingnanofluids water/CuO 4% (dp = 30) and water/Al2O3 4% (dp = 30) in-creases the heat transfer by 21% and 19.66% relative to water. Forthe same value of Pe, EG:W/CuO 4% (dp = 30) and EG:W/Al2O3 4%(dp = 30) augmented heat transfer by 25.1% and 23.6% relative toEG:W, respectively.

It was also concluded that heat transfer enhancement rises withthe Peclet number as well as particle concentration, but falls withincreasing nanoparticle diameter dnp. The results show that theeffect of particle concentration on the heat transfer is moresignificant than the particle diameter.

Finally, studying the variation of heat transfer coefficient hversus / revealed that for any given particle diameter there is anoptimum value of nanoparticle concentration that results the max-imum heat transfer coefficient. Therefore, extreme care should betaken in choosing the values of / and dnp.

Acknowledgements

The second author spent one year for this research work atFluid Mechanics, Thermal Engineering and Multiphase FlowResearch Lab. (FUTURE), Department of Mechanical Engineering,

King Mongkut’s University of Technology Thonburi, whose guid-ance and assistance are gratefully acknowledged. The fourthauthor would like to thank the Thailand Research Fund and the Na-tional Research University Project for the support.

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