International Journal of Heat and Mass Transfer 92 (2016) 76–82

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Natural convection in a trapezoidal enclosure filled with carbonnanotube–EG–water nanofluid

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.08.0360017-9310/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding authors.E-mail addresses: M.hemmatesfe@semnan.ac.ir (M. Hemmat Esfe),

wmyan@ntut.edu.tw (W.-M. Yan).

Mohammad Hemmat Esfe a,⇑, Ali Akbar Abbasian Arani b, Wei-Mon Yan c,⇑, Hamidreza Ehteramb,Alireza Aghaie b, Masoud Afrand a

aDepartment of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, IranbDepartment of Mechanical Engineering, University of Kashan, Kashan, IrancDepartment of Energy and Refrigerating Air-Conditioning Engineering, National Taipei University of Technology, Taipei 10608, Taiwan, ROC

a r t i c l e i n f o

Article history:Received 29 June 2015Received in revised form 13 August 2015Accepted 14 August 2015

Keywords:NanofluidNatural convectionCarbon nanotubeTrapezoidal enclosureEG–water

a b s t r a c t

In the present study, natural convection heat transfer inside a trapezoidal enclosure filled with carbonnanotube–EG–water nanofluid using variable properties has been numerically investigated. The bottomand top walls of trapezoidal enclosure are kept at constant temperatures Th and Tc, respectively, while theside walls of the cavity are thermally insulated. Using the finite volume method and the SIMPLER algo-rithm, the governing equations have been discretized. Simulations have been carried out for differentaspect ratios, Rayleigh numbers of 103–106 as well as solid volume fractions of 0.0015, 0.03, and 0.045.The results show that at low Rayleigh number ðRa 6 104Þ, the average Nusselt number NuAvg decreaseswith increasing the inclination angle (aspect ratio) at all solid volume fractions. While for Ra = 106, theNuAvg increases and then decreases with inclination angle (aspect ratio) with maximum NuAvg occurringat c ¼ 30�. Moreover, NuAvg increases with increasing Rayleigh number at fixed inclination angle (aspectratio) and solid volume fraction.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Natural convection heat transfer in square enclosures hasreceived considerable interest due to wide range of application inengineering. A few important applications involving this type ofheat transfer are air conditioned system in building, furnace andhome heating, electronic equipment cooling, drying foods, doublepane windows, etc. The main advantages of natural convectioncooling systems are their simplicity, low noise, and minimum cost.Increasing of heat transfer performance in this type of systems isan essential topic. Primary limitation in enhancing of heat transferperformance is low thermal conductivity of conventional fluidssuch as water and oils. Due to small sizes and very large specificsurface areas of the nanoparticles, nanofluids have superior prop-erties like high thermal conductivity. In addition metallic nanopar-ticles have larger thermal conductivity than conventional fluids(700–3000 times).

There are a large number of research works about using ofnanofluid that report significant heat transfer enhancement in an

enclosure, Khanafer et al. [1], Jou and Tzeng [2], Oztop andAbu-Nada [3], Ogut [4], Das and Ohal [5], Hemmat Esfe et al.[6–8], Garoosi et al. [9] and Saedodin et al. [10]. Contradictoryopinion was presented by Putra et al. [11]. They reported a system-atic deterioration base on their experimental investigation. Inanother work that conducted by Santra et al. [11], they reportedsame opinion as Putra et al. based on their numerical simulation.One can address the possible determining factors for the heattransfer enhancement/reduction in nanofluids includes the varia-tions of the size, shape, and distribution of nanoparticles anduncertainties in the thermophysical properties of nanofluids.

There are complex interactions between nanofluid with thewalls of the cavity. This complexity may increases with a changeof the geometry or orientation of the cavity. In fact study of naturalconvection fluid flow and heat transfer in a trapezoidal geometry ismore difficult than that of square or rectangular enclosures due tothe presence of sloping walls. There are a large number of investi-gations on natural convection fluid flows and heat transfers intrapezoidal cavities have been published [11–25]. As an examplea numerical work was performed by Basak et al. [26] for a uni-formly and non-uniformly heated bottom wall. Another works ontrapezoidal enclosure were done by Varol et al. [23–25]. Naturalconvection heat transfer in a nanofluid-filled trapezoidal enclosure

Nomenclature

cp specific heat, J/kg KGr Grashof numberAR aspect ratiog gravitational acceleration, m/s2

h heat transfer coefficient, W/m2 Kk thermal conductivity, W/m KH height of cavity, mL bottom length of enclosure, mL1 top length of enclosure, mn normal directionNu Nusselt numberp pressure, N/m2

P dimensionless pressurePr Prandtl numberRa Rayleigh numberT temperature, Ku, v velocity components, m/sU, V dimensionless velocity componentsx, y Cartesian coordinates, mX, Y dimensionless Cartesian coordinates

Greek symbolsa thermal diffusivityb thermal expansion coefficientu solid volume fractionq densityl dynamic viscosityh dimensionless temperaturec inclination angle

SubscriptAvg averagec cold wallf fluidf,0 fluid without nano particleh hot wallnf nanofluidp nanoparticles

Fig. 1. Schematic of the present study.

M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 92 (2016) 76–82 77

was analyzed by Saleh et al. [27]. They found that sloping wall andCu nanoparticles with high concentration can increase the rate ofheat transfer.

Buoyancy induced fluid flow and heat transfer in inclined trape-zoidal cavity was analyzed by Lee [28]. He analyzes the behavior ofthe flow and heat transfer characteristics at different Rayleigh andPrandtl numbers in his numerical study. In this numerical study itwas shown that for Ra = 104 and Pr = 0.1, the heat transfer was astrong function of the orientation angle of the cavity. Kumar andKumar [18] conducted a numerical analysis on the natural convec-tion heat transfer in a trapezoidal cavity filled with a porous med-ium. They showed that the inclination of the side wall was animportant role on the fluid flow and temperature distribution inthe cavity. Moukalled and Acharya [29] studied the conjugate nat-ural convection heat transfer in a trapezoidal cavity with a dividerattached onto inclined wall. Moukalled and Darwish [30] per-formed a numerical analysis of natural convection in a partitionedtrapezoidal cavity. They showed that the presence of bafflesdecreased heat transfer as high as 70%. Other similar studies onnatural convection in trapezoidal cavities were done by Peric[31], Van Der Eyden et al. [32], Boussaid et al. [33], Kumar [23],Papanicolaou and Belessiotis [34], Hammami et al. [35] andNatarajan et al. [36]. Recently Nasrin and Parvin [37] conducted anumerical research in order to investigate the transportmechanism of natural convection in a trapezoidal cavity filled withCu–water nanofluid. They found that both aspect ratio and Prandtlnumber affect the fluid flow and heat transfer in the cavity. In addi-tion they developed a correlation for the average Nusselt numberas a function of the Prandtl number as well as the cavity aspectratio.

Despite a number of research studies on trapezoidal cavityreported in the literature, there is a serious lack of informationregarding the problem of fluid flow and heat transfer enhancementin trapezoidal enclosures filled with nanofluids. To the best of ourknowledge, a little investigation of natural convection heat transferof the nanofluids in a trapezoidal enclosure has been undertakenyet. In addition, only limited numbers of publications are availableon the convective heat transfer enhancement, using CNT nanoflu-ids, with inconsistent results. Therefore, the aim of the presentpaper is to analyze numerically the problem of natural convectionin a trapezoidal enclosure filled with CNT–EG–water nanofluids

using variable properties based on experimental study. In the fol-lowing sections a complete analysis on the flow fields, temperaturedistributions and the rate of heat transfer are presented for differ-ent Rayleigh number and cavity aspect ratio (ratio of base lengthand height) graphically by presenting the streamlines, isotherms,and local and mean Nusselt numbers. The obtained results in thispaper is hoped to be a useful guide for the trapezoidal enclosurefilled with CNT–EG–water nanofluid to increase the rate of heattransfer.

2. Analysis

A schematic of the present study is depicted in Fig. 1. The sidewalls of the cavity are insulated and the bottom and top wallsare kept at constant temperatures of Th and Tc, respectively. Thelength of the bottom and top walls of cavity are assumed to be con-stant and equal to L1 and L, respectively. The aspect ratio, AR, andthe inclination angle of the side wall, c, are defined as to beAR = 2H/(L � L1) and k ¼ tan�1ðARÞ, respectively. Five inclinedangles c of 15�, 30�, 45�, 60�, and 75� are simulated in this work.

Table 1Comparison of the present results for average Nusselt number with those of Oztopand Abu-Nada [39].

Ra u Present study NuAvg Oztop and Abu-Nada [39]

103 0.00 1.008 1.0040.50 1.119 1.1220.10 1.253 1.251

104 0.00 1.996 2.0100.50 2.097 2.1220.10 2.188 2.203

105 0.00 3.993 3.9830.50 4.234 4.2710.10 4.459 4.440

Table 2Grid independence study (u = 0.0015, Ra = 104 and c ¼ 45�).

NuAve Number of nodes

1.283 35 � 711.357 45 � 911.386 55 � 111.392 65 � 1311.393 71 � 151

78 M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 92 (2016) 76–82

The cavity is filled with CNT–EG–water nanofluid which isassumed incompressible and Newtonian. With the followingdimensionless variables,

ðX;YÞ ¼ ðx; yÞL

; ðV ;UÞ ¼ ðv ;uÞLaf ;0

; h ¼ T � Tc

Th � Tc; P ¼ pL2

qnfa2f ;0

;

l� ¼ lnf

lf ;0; k� ¼ knf

kf ;0ð1Þ

Pr ¼ tf ;0af ;0

; Ra ¼ bf ;0gðTh � TcÞL3af ;0tf ;0

the present problem can be described in the following dimension-less governing equations.

@U@X

þ @V@Y

¼ 0 ð2Þ

U@U@X

þ V@U@Y

¼ � @P@X

þ lf ;0

qnfaf ;0

@

@Xl� @U

@X

� �þ @

@Yl� @U

@Y

� �� �ð3Þ

U@V@X

þ V@V@Y

¼ � @P@Y

þ lf ;0

qnfaf ;0

@

@Xl� @U

@X

� �þ @

@Yl� @U

@Y

� �� �

þ ðqbÞnfqnfbf ;0

Ra Prh ð4Þ

U@h@X

þ V@h@Y

¼ lf ;0

ðqcpÞnfaf ;0

@

@Xk�

@h@X

� �þ @

@Yk�

@h@Y

� �� �ð5Þ

The corresponding boundary conditions are described asfollows:

U ¼ V ¼ 0; h ¼ 1 On bottom wallU ¼ V ¼ 0; h ¼ 0 On top wallU ¼ V ¼ 0; @h

@n ¼ 0 The side wallsð6Þ

The density, heat capacity, thermal expansion coefficient, andthermal diffusivity of the nanofluid are evaluated from the follow-ing equations, respectively:

qnf ¼ uqp þ ð1�uÞqf ð7Þ

ðqcpÞnf ¼ uðqcpÞp þ ð1�uÞðqcpÞf ð8Þ

ðqbÞnf ¼ uðqbÞp þ ð1�uÞðqbÞf ð9Þ

anf ¼ knfðqcpÞnf

ð10Þ

In this work, a mixture with 70 vol.% water and 30 vol.% ethy-lene glycol (EG) as the base fluid is used. In addition, sodium dode-cyl benzene sulphonate as a surfactant, with 0.1% volume fraction,and multi-walled carbon nanotubes (MWCNT), with average diam-eter of 30–50 nm, length of 10–20 lm, and specific surface area of60 m2/g and purity of 95% are used to produce the nanofluids. Tocalculate the thermal conductivity and viscosity of CNT nanofluidas a function of temperature, the correlations from Ref. [38] areused.

The Nusselt number based on the hot wall length of the trape-zoidal cavity is calculated from the following relation:

Nu ¼ � knfkf

@h@n

����wall

ð11Þ

The average Nusselt number on the hot wall is evaluated asfollows:

NuAvg ¼ 1L

Z L

0Nu dX ð12Þ

3. Numerical method

To solve the governing equations, the finite volume method andthe SIMPLER algorithm are employed. Also, the diffusion term isapproximated by a second order central difference scheme.Furthermore, to approximate the convective terms, a hybridscheme is used. To validate the numerical code, a special case isconsidered and simulated by the present code. Consequently, theobtained results are compared with the results available in the lit-erature. Natural convection fluid flow and heat transfer within asquare cavity filled with Cu–water nanofluid are numerically sim-ulated. Table 1 illustrates comparisons between the present resultsand those of Oztop and Abu-Nada [39] for different Rayleigh num-bers and solid volume fractions. As observed from Table 1, there isa very good agreement between the Nusselt numbers obtained bythe present study and those of Oztop and Abu-Nada [39].

A grid independence study was also carried out for natural con-vection heat transfer in the trapezoidal cavity to obtain a suitablegrid for the numerical simulations. The trapezoidal cavity is filledwith nanofluid containing volume fraction of 0.0015, and the cal-culations are accomplished for Ra = 104 and h = 45�. Four differentuniform grids are employed for the numerical simulations. Theresults for average Nusselt number on the hot wall are presentedin Table 2. As it can be understood from Table 2, a 65 � 131 gridensures the grid independence simulation.

4. Results and discussion

In this work, results of numerical simulation of natural convec-tion fluid flow and heat transfer of CNT–EG–water nanofluid insidea trapezoidal enclosure are presented. The simulations have beenperformed for Ra = 103, 104, 105, and 106, c = 15�, 30�, 45�, 60�,and 75� as well as solid volume fractions of 0.0015, 0.03, and0.045. The streamlines and isotherms for different Rayleigh num-bers, inclination angles (aspect ratios), and solid volume fractionsare illustrated in Figs. 2–5.

(b) Ra=103, φ=0.0045(a) Ra=103, φ=0.0015

γ=15

γ=30

=45

=60

=75

Fig. 2. Streamlines (left) and isotherms (right) for Ra = 103 and different inclination angles (aspect ratios) at (a) u = 0.0015, (b) u = 0.0045.

(b) Ra=104 , φ=0.0045(a) Ra=104 , φ=0.0015

γ=15

γ=30

γ=45

γ =60

γ=75

Fig. 3. Streamlines (left) and isotherms (right) for Ra = 104 and different inclination angles (aspect ratios) at (a) u = 0.0015, (b) u = 0.0045.

M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 92 (2016) 76–82 79

As seen from Fig. 2, at Ra = 103, conduction heat transfer domi-nates at all values of inclination angles (aspect ratios) and solidvolume fractions, and weak eddies are formed along the side wallsof trapeze. Moreover, at Ra = 103, eddies become stronger andmore elongated with increasing c up to 60� at all volume fractions.At c = 75�, eddies strength is relatively weak than that of c = 60�.However its value is more than c = 30� and c = 45�. As depictedin Fig. 3, at Ra = 104, the curvature of the isotherms indicates theinfluence of natural convection heat transfer in the cavity. Thecurvature of the isotherms increases as the inclination angle(aspect ratio) increases, however it decreases with increasing the

solid volume fraction. At high values of solid volume fraction, theviscosity increases. Therefore, with decreasing natural convection,the conduction heat transfer becomes more effective. As a result,the curvature of the isotherms is reduced. As observed fromFig. 2, the inclination angle (aspect ratio) of the cavity has animportant role on the eddy strength.

As shown in Fig. 4, the curvature of the isotherms dramaticallyincreases at Ra = 105 than Ra = 104. This indicates that natural con-vection heat transfer is more predominant than the conductionheat transfer. Moreover, the secondary eddies are formed besidethe primary eddies at c = 30� and volume fraction of 0.0015. As

(b) Ra=105 , φ=0.0045(a) Ra=105 , φ=0.0015

γ=15

γ=30

γ =45

γ=60

γ=75

Fig. 4. Streamlines (left) and isotherms (right) for Ra = 105 and different inclination angles (aspect ratios) at (a) u = 0.0015, (b) u = 0.0045.

(b) Ra=106 , φ=0.0045(a) Ra=106 , φ=0.0015

γ=15

γ=30

γ=

45

γ=60

γ=

75

Fig. 5. Streamlines (left) and isotherms (right) for Ra = 106 and different inclination angles (aspect ratios) at (a) u = 0.0015, (b) u = 0.0045.

80 M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 92 (2016) 76–82

displayed in Fig. 5, for Ra = 106, natural convection heat transferbecomes completely dominant over the conduction heat transferat all considered k except for c ¼ 15�. Also, for this Rayleigh num-ber at c = 30� and all volume fractions, the secondary eddies areformed at the adjacent of the primary eddies. At c = 75�, someeddies with counter rotation direction are formed. The maximumvalue of the absolute streamlines strength for all Rayleigh numbersexcept for Ra = 103 is obtained at c = 75� and volume fraction of0.0015. At Ra = 104, 105, and 106 as well as all solid volume frac-tions, with increasing the inclination angle (aspect ratio), the value

of the absolute streamlines increases, and eddies become moreelongated. For all Rayleigh numbers, with increasing the solid vol-ume fraction, the eddy strength decreases at all aspect ratios. Atc = 15�, the conduction heat transfer dominates to natural convec-tion heat transfer even at high Rayleigh numbers, since the heightof the enclosure is so small that the convection heat transfer is verylow.

The variations of average Nusselt number NuAvg with respect tothe inclination angles (aspect ratios) for different Rayleigh num-bers at three solid volume fractions, 0.0015, 0.003, and 0.0045,

Fig. 6. Variations of average Nusselt number with respect to the inclination angles(aspect ratios) for different Rayleigh numbers at (a) u = 0.0015, (b) u = 0.003, (c)u = 0.0045.

M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 92 (2016) 76–82 81

are presented in Fig. 6. As it can be understood from Fig. 6, for lowRayleigh number ðRa 6 104Þ, the average Nusselt numberdecreases with increasing c. As expected with increasing c, the dis-tance between the hot and cold walls increases. In addition, sincethe value of Rayleigh number is low, the flow strength is not suffi-cient to generate stronger eddies and consequently higher naturalconvection heat transfer. At Ra = 106, average Nusselt number has amaximum value at k ¼ 30�. In fact, for this Rayleigh number, atc ¼ 15�, although the flow strength is high, there is not enoughspace in the cavity for natural convection heat transfer. Hencethe maximum average Nusselt number does not occur at c = 15�.

With increasing the volume fraction, the viscosity increases morethan thermal conductivity. Therefore, for all Rayleigh numbers,maximum average Nusselt number is corresponding to the mini-mum volume fraction, i.e. u = 0.0015, at all inclination angles(aspect ratios).

5. Conclusions

The problem of natural convection fluid flow and heat transferin a trapezoidal enclosure filled with CNT–EG–water nanofluidusing variable properties were numerically studied. The effects ofpertinent parameter such as aspect ratios, Rayleigh number, andvolume fraction of nanoparticles on streamlines, isotherms andNusselt number were investigated. Based on the obtained results,the following conclusions can be drawn:

� At low Rayleigh numbers, namely, 103 and 104, conduction heattransfer is dominant factor in this work. While, as Rayleighnumber increases, buoyancy-induced convection heat transferbecomes dominant.

� At low Rayleigh number ðRa 6 104Þ, the average Nusselt num-ber NuAvg decreases with increasing the inclination angle(aspect ratio) at all solid volume fractions.

� For Ra = 106, the NuAvg increases and then decreases with incli-nation angle (aspect ratio) with maximum NuAvg occurring atc ¼ 30�.

Acknowledgment

The financial support by the Ministry of Science andTechnology, R.O.C., through the contract MOST 103-2221-E-027-107-MY2 is also highly appreciated.

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