Lattice Boltzmann simulation of natural convection heat transfer of SWCNT-nanofluid in an open enclosure

Ain Shams Engineering Journal (2015) xxx, xxx–xxx

Ain Shams University

Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com

MECHANICAL ENGINEERING

Lattice Boltzmann simulation of natural convection heat transfer of SWCNT-nanofluid in an open enclosure Mohammad Jafari a,b,*, Mousa Farhadi c, Sanaz Akbarzade d, Mahboubeh Ebrahimi b a

Faculty of Mechanical Engineering, Lorestan University, P.O. Box 465, Khoramabad, Iran Mechanical Engineering Department, Babol University of Technology, Babol, Iran c Faculty of Mechanical Engineering, Babol University of Technology, Babol, Iran d Mechanical Engineering Department, Semnan University, Semnan, Iran b

Received 29 June 2014; revised 24 November 2014; accepted 20 December 2014

KEYWORDS Lattice Boltzmann Method; Single Walled Carbon Nanotube (SWCNT); Effective thermal conductivity; Effective viscosity; Natural convection; Open-ended cavity

Abstract The effects of Single Walled Carbon Nanotube and Copper nanoparticles on natural convection heat transfer in an open cavity are investigated numerically. The problem is studied for different volume fractions of nanoparticles (0–1%) and aspect ratio of the cavity (1–4) when Rayleigh number varies from 103 to 105. The volume fraction of added nanoparticles to Water is lower than 1% to make a dilute suspension. Although, results show that adding nanoparticles to the base fluid enhances the heat transfer, make a comparison between SWCNT and Cu-nanoparticles shows that the SWCNT-nanoparticle has better performance to enhance the convection rate. It is found that the aspect ratio of the cavity plays an important role on natural convection. An increase of this parameter leads to heat transfer reduction in the target problem. It is concluded that the Carbon Nanotubes can be applied as a passive way to enhance heat transfer in convection problems.  2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

* Corresponding author at: Faculty of Mechanical Engineering, Lorestan University, P.O. Box 465, Khoramabad, Iran. Tel.: +98 9166677044; fax: +98 6614200022. E-mail addresses: [email protected], [email protected] (M. Jafari), [email protected] (M. Farhadi), sanaz.akbarzadh@gmail. com (S. Akbarzade), [email protected] (M. Ebrahimi). Peer review under responsibility of Ain Shams University.

Production and hosting by Elsevier

1. Introduction The enhancement of fluids heat transfer is an interesting topic for different kinds of industrial and engineering applications. Heat Exchangers, boilers and condensers are some examples of important industrial equipments which have a close relation with the heat transfer subject. There are varied methods to enhance the convection rate in different conditions. Generally, these methods are classified in three sections named Active, Passive and Component methods. The passive methods have

http://dx.doi.org/10.1016/j.asej.2014.12.012 2090-4479  2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Jafari M et al., Lattice Boltzmann simulation of natural convection heat transfer of SWCNT-nanofluid in an open enclosure, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.012

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M. Jafari et al.

Nomenclature c discrete lattice velocity Cu copper cs speed of sound in Lattice scale cp specific heat at constant pressure (kj kg1 k1) F external force in direction of lattice velocity f, g distribution function gy acceleration due to gravity (m s2) H height of cavity (m) k thermal conductivity (W m1 k1) Ma mach number Nu Nusselt number Pr Prandtl number (m/a) Ra Rayleigh number (gb(Th  Tc)L3/(mÆa)) SWCNT Single Walled Carbon Nanotube T dimensional temperature (K) W width of cavity (m) X, Y dimensionless coordinate

b h Dt q s u w w

thermal expansion coefficient (1Æk1) dimensionless temperature lattice time step lattice fluid density lattice relaxation time solid volume fraction  dimensionless stream-function ðw ¼ w=Wu 0Þ weighting factor

Subscripts ave average c cold eq equilibrium f fluid h hot nf nanofluid s solid particle ð Þ dimensional parameters

Greek symbols a discrete lattice direction

no need for external forces for making enhancement in heat transfer rate. When these methods are adopted in heat exchangers can prove that the overall thermal performance improves significantly. Liu and Sakr presented a paper [1] which reviews experimental and numerical works taken by researchers on the passive techniques such as twisted tape, wire coil, and swirl flow generators. The authors concluded that twisted tape inserts perform better in laminar flow than turbulent flow. However, the other several passive techniques such as ribs, conical nozzle, and conical ring are generally more efficient in the turbulent flow than in the laminar flow. A well-known passive way to enhance the rate of heat transfer of conventional fluids such as water, oil and ethylene glycol which has low thermal conductivity [2,3] is adding nanoscale conductive particles. The added particles can be metals [4], nonmetals [5] or Carbon nanotubes [6,7]. Because of the high thermal conductivity of these particles, they can improve conductivity of the suspensions systematically. Nowadays, nanoparticle added fluids are known as nanofluid that first time Choi [8] named this kind of fluid suspensions. The great potential of nanofluids for heat transfer enhancement in highly suited practical applications may create an opportunity to develop compact and effective heat transfer equipments for many industrial applications, including transportation, nuclear reactors, electronics, biomedicine, and food. At this way, in recent years, many studies conducted to study heat transfer of nanofluids numerically and experimentally [9–15]. Khanafer et al. [9] presented a heat transfer enhancement by adding nanoparticles to fluid in a two dimensional enclosures at natural convection regime for the different Grashof numbers. They presented an increase of Nuave about 7.5%, 12%, 15.5% and 20%, respectively, by adding a volume fraction of added Cu nanoparticles to the Water equal to 4%, 6%, 8% and 10% when Gr = 104. A useful review paper presented by Hussein et al. [10] shows that, according to the

majority of experimental and numerical studies, suspensions of solid nanoparticles significantly enhance heat transfer and the heat transfer coefficient of nanofluids is found to be larger than that of its base fluid at the same physical condition. In 1991, Iijima [16] discovered Carbon nanotubes as an allotrope of Carbon which is made of long-chained molecules of Carbon with Carbon atoms arranged in a hexagonal complex to form a tubular structure. Single Walled (SWCNT), Double Walled (DWCNT) and Multi Walled (MWCNT) are different classes of Carbon nanotubes depending on the number of tubules of Carbon in their structure. In last decade Carbon nanotubes have been mentioned as attractive topic by many researches which is generally due to their special properties at physical view such mechanical and thermal properties. At this point of view, the Carbon nanotubes have extraordinary thermal properties such as thermal conductivity about twice as high as diamond [17] or thermal stability up to 2800 C in vacuum. The higher thermal conductivity of Carbon nanotubes relative to other nanoparticles led to that the nanofluids containing cylindrical Carbon nanotubes are expected to have better heat transfer properties compared with the other nanofluids with spherical nanoparticles [18,19]. Gavili et al. [20] simulated the mixed convection in a two-sided lid-driven differentially heated square cavity for nanofluids containing Carbon nanotubes for nanofluids. Natural convection in a square cavity and its fluid flow is a classical benchmark in heat transfer problems. Open cavities are a kind of 2-D cavity which has an open side. These kinds of cavities have special physics for both flow and temperature fields in open side because of outgoing of flow from this side. Many studies have been done on analysis of buoyant flows and their heat transfer in open cavities [21–28]. Javam and Armfield [21] investigated stability of stratified natural convection flow in open cavities. Mohammad et al. [22] presented natural convection in an open ended cavity and slots they analyzed the

Please cite this article in press as: Jafari M et al., Lattice Boltzmann simulation of natural convection heat transfer of SWCNT-nanofluid in an open enclosure, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.012

Lattice Boltzmann simulation of SWCNT-nanofluid in an open enclosure effect of aspect ratio of cavity on heat transfer rate. They also presented a good procedure for simulating open boundaries in Lattice Boltzmann Method. Their technique demonstrated the ability of the LBM to simulate Natural convection in open cavities. In a numerical study via Lattice Boltzmann Method, natural convection in an open enclosure in the presence of water/Cu nanofluid has been investigated by Kefayati et al. [23]. Calculations were performed for different Rayleigh numbers (104–106) and volume fractions of nanoparticles (1–5%). Their results show that the average Nusselt number increases with augmentation of Rayleigh number and the volume fraction of nanoparticles. They reported an enhancement of natural convection heat transfer about 3.98%, 7.30%, 11.62%, 14.27% and 18.42% by adding a u = 1%, 2%, 3%, 4% and 5% of Cu nanoparticles at Ra = 104. Also when Rayleigh number is equal to 105, and other conditions are same, these values are about 2.97%, 5.73%, 8.71%, 11.26%, 13.60%. The convection heat transfer of different nanoparticlebased nanofluids in a cavity has been mentioned in current years [29–32]. Fattahi et al. [29] applied Lattice Boltzmann Method to study natural convection heat transfer of Al2O3 and Cu nanofluid in a cavity. Their results show that increasing the solid volume fraction led to a heat transfer enhancement at any Rayleigh number and also heat transfer increases with increase in Rayleigh number for a particular volume fraction. Nemati et al. [30] applied LBM to investigate the effect of Cu, Al2O3 and CuO nanoparticles on mixed convection in a lid-driven cavity. Their results show that adding nanoparticles increase the rate of mixed convection heat transfer of the base fluid for all tested Reynolds numbers. AbuNada and Chamkhac [31] conducted to study of mixed convection heat transfer of Water–Al2O3 nanofluid in an inclined cavity. Heat transfer enhancement due to increase of nanoparticles volume fraction at different Richardson and Grashof numbers was presented in their results. Their results show that Nuave on hot wall enhances about 3.3%, 8.4%, 11.9% and 17.2% respectively for u = 2%, 5%, 7% and 10% at Ri = 0.2 and Gr = 102 when the inclination angle of the cavity is equal to zero and also these values are 3%, 8.5%, 12% and 17% for Ri = 2. The investigation of the heat transfer enhancement by adding the Single Walled Carbon Nanotubes (SWCNTs) to the base fluid in an open-ended cavity is the main aim of the present study. Also a comparison between SWCNTs and Cu-nanoparticles is one of the major tasks of the study. In this simulation the effective conductivity and viscosity were calculated based on the new theoretical models. The presented model by Masoumi et al. [33] is applied for effective viscosity of the Cu and SWCNT-nanofluid. To simulate thermal conductivity of SWCNT-nanofluid a new theoretical model presented by Sabbaghzadeh and Ebrahimi [34] is used. This new thermal conductivity model captures the effects of different phenomena of energy transport in nanofluids. At molecular point of view, some of the most important incidents in energy transport are: Collision among the base fluid molecules, thermal diffusion of nanoparticles, thermal diffusion in the nanolayer in the fluid, thermal interaction of dynamic complex nanoparticles with the base fluid molecules and finally collision between nanoparticles caused by Brownian motion. On the other hand, the study applies the Patel model [35] to evaluate of the effective thermal conductivity of Cu-nanofluid.

3

By considering the extraordinary thermal conductivity and low density of Carbon nanotubes in comparison with other nanoparticles and also using a good expansion of effective thermal conductivity of nanofluids which covers molecular phenomena of energy transport of nanofluids, it is predicted that a great heat transfer enhancement can be achieved in this study. This enhancement can be assumed as a very important result for many industrial applications which are concern about the convection rate. Therefore the importance of new studies about the ways of heat transfer enhancement is still obvious. The progress of using the Lattice Boltzmann Method (LBM) as a numerical technique to simulate the heat transfer and fluid flow has been obvious in the last decade [36–41]. The Lattice Boltzmann Method has well-known advantages such easy implementation, possibility of parallel coding and simulating of complex geometries and fluid dynamic problems such as melting, fuel cell, porous media, and nanofluids. To the best of author knowledge, the effects of Carbon nanotubes on flow and thermal fields of natural convection are an unknown perspective which is not understand heretofore. Also, a lack of using Lattice Boltzmann method (LBM) as a suitable numerical method to study of convection heat transfer of nanofluids containing Carbon nanotubes is sensible. Therefore, in this study the used numerical method is LBM with coupled double population approach for flow and temperature fields. Also the Bounce Back method is applied in LBM to model boundary condition in solid boundaries which supply second order accuracy in both flow and temperature fields. The effect of the volume fraction of the Cu and SWCNT-particles on average Nusselt number, streamlines and temperature contours is investigated for various values of Rayleigh number. Also the aspect ratio of the cavity is studied as an important geometric parameter of the 2D cavities in different conditions to exhibit the role of this parameter on both heat transfer rate and fluid flow of the base fluid and the nanofluid. 2. Lattice Boltzmann Method for flow and thermal fields The LBM utilizes two distribution functions, for the flow and temperature fields. It uses modeling of movement of fluid particles to define macroscopic parameters of fluid flow. The basic form of LBM applies uniform Cartesian cells to discrete problem domain. Each cell of the grid has a constant number of distribution functions, which represent the number of fluid

Figure 1

Discrete velocity vectors for the D2Q9 model of LBM.

Please cite this article in press as: Jafari M et al., Lattice Boltzmann simulation of natural convection heat transfer of SWCNT-nanofluid in an open enclosure, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.012

4 particles movement in these separated directions. The distribution functions are obtained by solving the Lattice Boltzmann Equation (LBE), which is a special form of the Kinetic Boltzmann Equation. Lattice Boltzmann Method can be operated on a number of different lattices, both cubic and triangular, and with or without rest particles in the discrete distribution function. A well known system of classifying the different methods by lattice is the ‘‘DnQm’’ scheme. Here ‘‘Dn’’ stands for ‘‘n dimensions’’ while ‘‘Qm’’ stands for ‘‘m speeds’’. The D2Q9 model (see Fig. 1) is used in the current study which is a two-dimensional Lattice Boltzmann model on a square grid, with rest particles present. For a two dimensional model, a particle is restricted to stream in a possible of 9 directions, including the one staying at rest. The basic form of the Lattice Boltzmann Equation with an external force by introducing BGK approximations can be written as follows for both the flow and the temperature fields [36]:  Dt  eq fa ðx þ ca Dt; t þ DtÞ ¼ fa ðx; tÞ þ f ðx; tÞ  fa ðx; tÞ sm a þ Dtca Fa Dt ga ðx þ ca Dt; t þ DtÞ ¼ ga ðx; tÞ þ ½geq ðx; tÞ  ga ðx; tÞ st a

ð1Þ

M. Jafari et al. 8 a¼0 > < ð0;0Þ ea ¼ cðcosha þ sin ha Þ; ha ¼ ða  1Þp=2 a ¼ 1;2;3; 4 > : pffiffiffi 2cðcosha þ sinha Þ; ha ¼ ða  5Þp=2 þ p=4 a ¼ 5;6;7; 8 ð6Þ 8 > < 4=9 a ¼ 0 wa ¼ 1=9 a ¼ 1; 2; 3; 4 > : 1=36 a ¼ 5; 6; 7; 8

In order to incorporate external force in collision part of Lattice Boltzmann model (Eq. (1)), radiation heat transfer and viscous dissipation are neglected at the numerical simulation. Therefore, to capture buoyancy force effects in the flow field, the Boussinesq approximation is applied. Thus, to model buoyancy force in Eq. (1), the external force needs to be assumed as below in the needed direction: Fa ¼ 3wa ga bðT  Tref Þ

ð2Þ

Characteristic for both natural pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi velocity (Vnatural ¼ bgy DTH) and force (Vforce = vRe/W) regimes must be small in comparison with the fluid speed of sound. By considering D2Q9 model for applied lattice scheme for both flow and temperature fields, equilibrium distribution functions eq for flow field (feq a ) and temperature field (ga ) are calculated as follows in different a directions:   9 3 !2 ! ! 2 ð4Þ feq a ¼ wa q 1 þ 3ðea  u Þ þ ðea  u Þ  u 2 2   3 !2 geq ¼ w qRT u 0 0 2   3 3 9 3! ! ! 2 eq ga ¼ wa qRT þ ðea  u Þ þ ðea  u Þ  u 2 for a ¼ 1  4 2 2 4 2   2 9 3 ! ! !2 for a ¼ 5  8 ð5Þ ¼ w qRT 3 þ 6ðe  u Þ þ ðe  u Þ  u geq a a a a 2 2 The D2Q9 lattice is a Cartesian 2D lattice with nine velocity directions. For this model, the values of ea and wa for various a directions will be, respectively:

ð8Þ

Finally, macroscopic variables can be calculated as follows: X Flow density : q ¼ fa ; Momentum : qui ¼

where fa(x, t) is a distribution function on the mesoscopic level. Here c = Dx/Dt is a streaming speed. Also, Dx and Dt are the lattice cell size and the lattice time step size, respectively. At these equations (Eqs. (2) and (3)), ca is the discrete lattice velocity in direction a and Fa is the external force term in the direction of discrete velocity. Also, wa is a weighting factor and q is the lattice fluid density. sm and st are the dimensionless collision-relaxation times for the flow and temperature fields, respectively. They defined as follows [42]: pffiffiffiffiffiffiffiffi Ma  W 3Pr pffiffiffiffiffiffi sm ¼ 0:5 þ c2 Dt Ra 3v ð3Þ st ¼ 0:5 þ Pr c2 Dt

ð7Þ

X fa cia ;

a

Temperature : T ¼

X ga

a

ð9Þ

a

3. Nanofluids modeling In the present study, nanofluid is assumed as a single phase fluid. The added nanoparticles to the base fluid changes thermal conductivity, viscosity and other basic characteristic of the base fluid, thus innovated Carbon-based nanofluid has special characteristic as a combination of the pure fluid and Carbon nanotubes. Thermal diffusivity of nanofluid is as follows: anf ¼

knf ðqcp Þnf

ð10Þ

The density, heat capacitance and thermal expansion of nanofluid can be defined as [43]: qnf ¼ ð1  uÞqf þ uqs ðqcp Þnf ¼ ð1  uÞðqcp Þf þ uðqcp Þs ; bnf ¼ ð1  uÞbf þ ubs

ð11Þ

3.1. Effective viscosity of nanofluid The effects of nanoparticles on the viscosity of nanofluids are introduced by the so-called apparent viscosity which is presented by lapp. The numerical procedure to simulate the nanofluid viscosity is same as those presented by Masoumi et al. [33]. According to this analytical model of nanofluid viscosity, the effective viscosity can be defined as follows: lnf ¼ lbf þ lapp ¼ lbf þ

qnp VBrownian dnp 72Cd

ð12Þ

As presented, Masoumi et al. [33] considered Brownian motion as an important parameter that creates a relative velocity between the nanoparticle and the base fluid in nanofluids [33]. VB is the Brownian velocity and the corresponding Reynolds number is defined as follows [44]:

Please cite this article in press as: Jafari M et al., Lattice Boltzmann simulation of natural convection heat transfer of SWCNT-nanofluid in an open enclosure, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.012

Lattice Boltzmann simulation of SWCNT-nanofluid in an open enclosure

ReBrownian

1 ¼ vbf

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 18Kb T pqnp dnp

ð13Þ

ð14Þ

While, Kb is the Boltzmann Constant (=1.38 · 1023 J K1). Also, d is the approximately distance between the centers of particles and it will be obtained from [33]: rffiffiffiffiffiffi p dnp d¼ 3 ð15Þ 6u Now the correction factor, C, must be determined. C ¼ l1 bf ½ðc1 dnp þ c2 Þu þ ðc3 dnp þ c4 Þ

ð16Þ

Using the experimental data associated with the nanofluid viscosity c1, c2, c3 and c4 can be determined. For Cu-nanoparticles the values of these constants are assumed as follows [33]. c1 ¼ 0:000 001 133; c2 ¼ 0:000 002 771; c3 ¼ 0:000 000 09; c4 ¼ 0:000 000 393

ð17Þ

Also, according to the experimental data about the viscosity of Carbon nanotube based nanofluids presented by Chen et al. [45] the values of c-constants are estimated as follows: c1 ¼ 0:000 000 896; c2 ¼ 0:000 002 054; c3 ¼ 0:000 000 738; c4 ¼ 0:000 000 306

ð18Þ

u* is the mean velocity of the complex nanoparticles and L is the specific length of the cylindrical shape that can be calculated as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3 prnp lM L¼ ð23Þ u where l is the cylinder length, from the molecular theory of heat, u is determined as follows [46]: rffiffiffiffiffiffiffiffiffiffiffi 3Kb T u ¼ ð24Þ mn As cited, Kb is the Boltzmann Constant, mn is the mass of the complex nanoparticle which can be calculated as follows: " # qbf 2 2 ð25Þ mn ¼ prnp lqnp ðM  1Þ þ 1 qnp Sabbaghzadeh and Ebrahimi [34] presented more details of thermal conductivity calculation in the nanofluids based on Carbon nanotube existence in convective fluids in their published research paper. The effective thermal conductivity of Cu–Water nanofluid is calculated using the presented model by Patel et al. [35] which is as follows: knf ks As As ¼1þ þ cks Pe kf kf Af kf A f

The applied effective thermal conductivity of Carbon nanotubes-based nanofluid is same as those proposed by Sabbaghzadeh and Ebrahimi [34]. This Theoretical model contains five phenomena of energy transport in nanofluids: collision among the base fluid molecules; thermal diffusion of nanoparticles, thermal diffusion in the nanolayer in the fluid; thermal interaction of dynamic complex nanoparticles (nanoparticles and nanolayer) with the base fluid molecules and finally collision between nanoparticles due to Brownian motion. Their theoretical model can be expressed as [34]:

ð19Þ

Pe ¼

The relation is valid for 101 < Rebf < 105, The Reynolds number, Rebf is defined as follows: Rebf ¼

u L vbf

2Kb T plf d2s

ð27Þ

4.1. Problem description In the present study, the effects of Single Walled Carbon Nanotubes (SWCNT) and Cu-nanoparticles on natural convection heat transfer in a 2D deep open cavity are investigated using the Lattice Boltzmann Method based on double population approach and bounce back method. The horizontal walls of the cavity are assumed to be insulated while the left wall is

Adiabatic Wall

ð20Þ

The term of Knl in Eq. (19) indicates the thermal conductivity of nanolayer which has a value equal to:   K Kbf rnp M Knpbf  1 lnðMÞ   Knl ¼ ð21Þ K dnp ln M Knpbf

us ¼

4. Main problem

SWCNT-Based Nanofluid

Hot Wall ( T=Th)

rnl þ rnp rnp

us ds ; af

where us is the Brownian motion term of nanoparticle velocity and Kb is the Boltzmann constant.

where j, D and M are respectively thermal Kapitza resistance, diameter of nanoparticles complex and ratio of radius complex nanoparticle to nanoparticle which is defined as follows: M¼

ð26Þ

where As ds u ¼ ; Af df 1  u

3.2. Thermal conductivity of nanofluid

Keff ¼ Kbf ½1  uM2  þ uðjKnp þ Knl ðM2  1ÞÞ þ uKf M2  df   0:35 þ 0:56Re0:52 Pr0:3 bf bf D

5

H

g

Cold Open End ( T=Tc )

VBrownian

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 18Kb T ¼ dnp pqnp dnp

W

Y X

ð22Þ Figure 2

Adiabatic Wall

Schematic geometry of the problem.

Please cite this article in press as: Jafari M et al., Lattice Boltzmann simulation of natural convection heat transfer of SWCNT-nanofluid in an open enclosure, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.012

6

M. Jafari et al.

maintained at a uniform temperature (Th) differentially higher than the open end side temperature (Tc). The considered physical system is presented in Fig. 2. The effect of SWCNT/Cunanoparticles on both fluid flow and heat transfer is investigated for different values of Rayleigh number at the range of 103 6 Ra 6 105 while the Prandtl number of based fluid (Water) at reference temperature (22 C) is 6.57. It is assumed that the both nanotubes and nanoparticles are in constant form with uniform diameter which are at thermal equilibrium state with base fluid and also the nanotubes and nanoparticles have same velocity with the base fluid thus there is no slip velocity between added nanotubes/nanoparticles and molecules of the base fluid. Thermo-physical properties of SWCNT-particles, Cu-nanoparticles and Water at reference temperature are tabulated in Table 1. The dimensionless quantities are as follows: x¼ ¼

x ; H

y¼

T  Tc ; Th  Tc

y ; H

u¼

Pr ¼

u ; u0

vqCp ; K

v¼

v ; u0

Ra ¼

h

gbDhW3 am

ð28Þ

The boundary conditions on the solid walls are in the following forms: 8 > < u ¼ v ¼ 0; h ¼ 1; for x ¼ 0; 0 6 y 6 1 @h u ¼ v ¼ 0; @x ¼ 0; for 0 < x < W; y ¼ 0 ð29Þ > : @h ¼ 0; for 0 < x < W; y ¼ 1 u ¼ v ¼ 0; @x The boundary conditions on the east open side (x = W, 0 6 y 6 1) are in the following forms: ( @u @v @h ¼ 0; @x ¼ 0; @x ¼ 0 if u > 0 @x ð30Þ @u @v ¼ 0; ¼ 0; h ¼ 0 if u < 0 @x @x The boundary conditions in the LBM can be implemented through the distribution function. For both flow and thermal fields, the distribution functions out of the domain are known from the streaming process. The unknown distribution functions are those toward the domain. Bounce back boundary condition is used on the straight solid boundaries (no-slip velocity wall) which have adiabatic thermal conditions, the unknown distribution function in needed direction are calculated as follows: fa ¼ f~a ðflow fieldÞ;

ga ¼ g~a ðthermal fieldÞ

ð31Þ

Which a indicates needed direction and (–) shows opposite direction. The () symbol presents the known distribution function which obtained after the streaming and collision steps. At the west wall, for the flow field the Bounce Back is applied (fa ¼ f~a ) when the thermal field uses the following boundary condition for constant temperature boundary conditions at this wall: Table 1 Thermo-physical properties of the base fluid and nanoparticles [20,35]. Property cp (J/Kg K) q (kg/m3) k (W/m K) b · 105 (K1) dp (nm)

Water 4179 997.1 0.613 21 0.384

Cu 383 8954 400 0.167 100

8 > < g1 ¼ ðx1 þ x3 Þhw  g3 g5 ¼ ðx5 þ x7 Þhw  g7 > : g8 ¼ ðx8 þ x6 Þhw  g6

At the outlet, the used boundary condition is same as those presented by Mohamad et al. [22]. At this side, the temperature boundary condition is applied according to the axial velocity component: ga ðnÞ ¼ 13 ½4ga ðn  1Þ  ga ðn  2Þ 8 > < g3 ðnÞ ¼ ðx1 þ x3 Þhc  g1 ðnÞ g7 ðnÞ ¼ ðx5 þ x7 Þhc  g5 ðnÞ > : g6 ðnÞ ¼ ðx8 þ x6 Þhc  g8 ðnÞ

if u > 0 if u < 0

ð33Þ

where n is the lattice on the boundary, (n  1) and (n  2) are the lattices inside the enclosure adjacent to the open-side. The local Nusselt number is defined along the hot bottom wall as follows:    knf @h  Nu ¼ H ð34Þ kf @x where H is the height of the cavity. By integrating the local Nusselt number along the Y direction, the average Nusselt number is calculated as follows: Z H 1 Nuave ¼ Nudy ð35Þ H 0 4.2. Code validation, solution convergence and mesh independent treatment The numerical code is validated by the results presented by Mohammad et al. [22] for natural convection in an open cavity at Ra = 104, Pr = 0.71 and A = 1 (Fig. 3). Also for more consistency, Table 2 shows the comparison of the values of average Nusselt number of this study and those presented by Mohamad et al. [22] which shows a good agreement. The agreement of the result with the pervious results confirms the accuracy of numerical code to simulate the flow and temperature fields of convection heat transfer. A grid independency treatment is used to select an adequate number of nodes in the mesh grid. This treatment is presented in Fig. 4 for a special case study. It is visible that a grid with 100 nodes in each direction can satisfy the grid independency treatment when aspect ratio of the cavity is equal to unity. Therefore, in used numerical code to model various aspect ratio of the cavity, 100 lattice nodes in Y direction is used and number of nodes in X direction is 100–400 for aspect ratio of 1 to 4, respectively. To get a steady state result in the numerical solution, the results are compared at different time steps. This algorithm was done for all case studies to get convergent results by applying an error function as presented in the following equation in the used numerical code:

SWCNT 709.0 2200 2000 1.5 10

ð32Þ

Err ¼

ðWni;j  Wn1 i;j Þ 6 105 number of mesh nodes

ð36Þ

where Wni;j stands for a variable of flow or temperature fields such as temperature or velocity component in each node of the grid at nth iteration. Steady state solution was declared when the relative change in the variables of flow and tempera-

Please cite this article in press as: Jafari M et al., Lattice Boltzmann simulation of natural convection heat transfer of SWCNT-nanofluid in an open enclosure, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.012

Lattice Boltzmann simulation of SWCNT-nanofluid in an open enclosure

Average Nusselt number along the hot wall.

7.331 6.936 6.551 6.155

0.10 0.26 0.43 0.28

16 15 14

ture fields between two consecutive iterations at through solution domain is satisfied by the presented criterion. The Schematic of applied numerical convergence treatment to get convergent condition is presented in Fig. 5 for an arbitrary point in problem domain in a special case study. The values of CPU-time of Calculations are tabulated in Table 3 at different geometrical and physical conditions. The presented data in this table explain how aspect ratio and Rayleigh number can change value of numerical code’s run-time. 5. Result and discussion At pervious sections, the numerical simulation of natural convection heat transfer in an open ended enclosure has been

Nuave

13 12

0.4 %

Deviation%

2.8 %

Mohammad [22]

8.6 %

Present study Ra = 105, Pr = 0.71 A=1 7.323 A=2 6.954 A=3 6.523 A=4 6.138

28.4 %

Table 2

Streamlines and temperature contours at Ra = 104, Pr = 0.71 and A = 1. (Top) Mohammad et al. [22]. (Bottom) Present

44.1 %

Figure 3 study.

7

11 10 9 8

20

40

60

80

100

120

Node Number

Figure 4 Mesh grid independence treatment for a special case study for a cavity with A = 1.

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M. Jafari et al.

0.15

T

T

0.28

u , v

-0.15

-2

0

u , v (10 )

0.32

0.24

T u v 0.2

0

50000

100000

-0.3 150000

Iteration

Figure 5

Table 3

Schematic of applied numerical convergence treatment to get convergent condition in an arbitrary point at a special case study.

CPU-time of calculations at different case studies.

Aspect ratio

A=1 A=2 A=3 A=4

Rayleigh number Ra = 103

Ra = 104

Ra = 105

4957 10,583 16,061 21,810

5177 11,006 16,285 21,374

5608 11,573 16,451 20,092

explained in details. The target problem is investigated for three types of convectional fluids which are named as Water, Water–Cu and Water-SWCNT nanofluids. The volume concentration of added nanoparticles to Water, as base fluid of Water/Cu and Water/SWCNT nanofluids, is one of the important parameters which must be studied to clear the role of these nanoparticles on both thermal and fluid fields of the base fluid. Besides studying of the volume fraction of nanoparticles (0 6 u 6 1), different values of aspect ratio of the cavity (1 6 A 6 4) and Rayleigh number (103 6 Ra 6 105) are assumed as important geometrical and physical parameters at the main problem. The enhancement of the heat transfer of the nanofluids may be caused by the suspended nanoparticles increasing the thermal conductivity of fluids, and the chaotic movement of ultrafine particles increases fluctuation and turbulence of the fluids, accelerating the energy exchange process. Therefore, current study tries to make a comparison between SWCNT and Cunanoparticles to answer this question that ‘‘which kind of these nanoparticles can present a better heat transfer enhancement’’. At following subsection, the effects of the nanoparticles on thermal and fluid fields at the problem domain are discussed. Then the values of average Nusselt number are presented to clarify the role of the nanoparticles on the convection rate. 5.1. Nanoparticles effects The presence of nanotube particles with high thermal conductivity in suspensions has different effects on temperature and

flow fields. The effect of added nanotubes and Cu nanoparticles on flow and temperature fields of the Water (the base fluid) is presented in Fig. 6 for the cavity with A = 1 at different values of Rayleigh number. In the present study to be sure that nanofluids behave completely same as a single phase fluid, the volume fraction value of added nanotubes is selected in a range less than 1% to make a dilute suspension, therefore it can be acceptable that the overall shape of the contours is same for both pure fluid and nanofluid with added nanotube particles. Also, the maximum value of stream-function in the flow field is presented in this figure for Water, Water–Cu and Water-SWCNT nanofluids at different case studies. The presented values for maximum stream-function show that adding the 1% volume fraction of SWCNT-nanoparticles increases the maximum value of stream-function about 57%, 59% and 75% respectively for Ra = 103, 104 and 105 when the Cu nanoparticles change the value of maximum stream function about 10%, 11% and 14% respectively for Ra = 103, 104 and 105. The increase of stream-function values shows that the nanoparticles have more effect on fluid flow at a high Rayleigh number. This result can be due to the importance of the fluid density at a high Rayleigh number when natural convection is completely dominant. Although, the Cu nanoparticles have higher density, the results show that SWCNT-nanoparticles have more positive effects on flow field of the base fluid. To explore this phenomenon, it is important to notice that the extra ordinary thermal conductivity can play an important role on flow and thermal fields of convectional flows when the dominant regime is natural convection. In natural convection regime, there is a double-way couple between flow and thermal field. This doubleway connection causes that the role of thermal conductivity be more important in comparison with the forced connection. At the forced convection the connection between flow and thermal field is a one-way connection which means that just flow fields make changes in thermal field and the thermal field has no effect on the flow field. For more exploration, the variation pattern of velocity components of SWCNT-nanofluid at horizontal mid-section of the cavity is plotted in Fig. 7 for different values of Rayleigh

Please cite this article in press as: Jafari M et al., Lattice Boltzmann simulation of natural convection heat transfer of SWCNT-nanofluid in an open enclosure, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.012

Lattice Boltzmann simulation of SWCNT-nanofluid in an open enclosure

(a)

( Ψmax)Water=-0.150

( Ψmax)Cu=-0.165

Water

(b)

( Ψmax)Water=-0.613

(c)

( Ψmax)Water=-1.426

( Ψmax)SWCNT=-0.235

SWCNT

Cu

SWCNT

Cu ( Ψmax)Cu=-1.626

Water

T: 0.0

Cu

SWCNT

Water

Cu

SWCNT

Water

Cu

SWCNT

( Ψmax)SWCNT=-2.975

SWCNT

Cu

0.1

Water

( Ψmax)SWCNT=-0.975

( Ψmax)Cu=-0.684

Water

9

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 6 Streamlines and temperature contours of the base fluid, SWCNT and Cu-nanofluid (u = 1%) at A = 1. (a) Ra = 103 (b) Ra = 104 (c) Ra = 105.

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M. Jafari et al.

(a) 0.1

(a)

Ra=103 Ra=104 Ra=105

14

0

12

Ra=103 -0.1

4

Ra=10 5

Ra=10

-0.2

Nuave=7.910+1.876 ϕ

10

Nuave

u(10 -2)

16

8

Nuave=3.381+0.788 ϕ

6

-0.3

ϕ=0.2% ϕ=0.6% ϕ=1.0%

-0.4

4

Nuave=1.214+0.274 ϕ

2 -0.5

0

0.2

0.4

0.6

0.8

0

1

X/H

(b) 0.06

(b)

ϕ=0.2% ϕ=0.6% ϕ=1.0%

0.05

0

0.2

0.4

0.6

0.8

1

0.8

1

16 3

Ra=10 Ra=104 Ra=105

14 12

0.04

10

Ra=103

0.03

Nuave

v

ϕ

Ra=104

0.02

Nuave=7.880+7.607 ϕ

8

Nuave=3.365+3.278 ϕ

6

Nuave=1.208+1.119 ϕ

Ra=105 4

0.01

2 0

0

0.2

0.4

X/H

0.6

0.8

1

Figure 7 Effect of different volume fractions of nanotubes on uVelocity (a) and v-Velocity (b) in horizontal mid-section of the cavity at different Ra when A = 1.

number and volume fraction of the added nanotubes when the aspect ratio is equal to unity. It is visible that adding nanoparticles to the base fluid has more obvious effects at a higher Rayleigh number. Investigation of the role of SWCNT-nanoparticles on convection rate in an open enclosure is main aim of the present study. Therefore, the values of average Nusselt number (Nuave) aspect to the volume fraction of SWCNT and Cu nanoparticles are presented in Fig. 8 for different values of Rayleigh number. Also, some correlations which show the relation between average Nusselt number and volume fraction of nanoparticles is presented in this figure. It is found that the presence of a low value of Carbon nanotubes obviously enhances the heat transfer rate. According to the presented results, adding the 0.6% volume fraction of Cu-particles enhances the average Nusselt number about 13.3%, 13.8% and 14% at Ra = 103, Ra = 104 and Ra = 105, respectively, and these values are about 54%, 56.6% and 56.7% for the SWCNT-particles with

0

0

0.2

0.4

ϕ

0.6

Figure 8 Average Nusselt number at different Rayleigh numbers and volume fractions of Cu-nanoparticles (a) and SWCNTs (b) for the cavity with A = 1.

same volume fraction when the cavity aspect ratio is equal to unity. Improvement of thermal conductivity of the base fluid and increasing of flow momentum are some of the advantages of nanoparticles presence in the base fluid. These increases in flow momentum and thermal conductivity lead to heat transfer enhancement of the base fluid. As presented before, the volume fraction value of added single-walled Carbon nanotubes is less than 1% that means significant effects on convection heat transfer and its flow are observed by adding a low value of added nanotubes which is due to extraordinary thermal conductivity and other properties of the Carbon nanotubes. The thermo-physical properties of the added nanoparticles specify the magnitudes of heat transfer enhancement of the base fluid. The values of average Nusselt number increase about 22% and 92%,respectively, by adding the 1% volume fraction of Cu and SWCNT-nanoparticles when A = 1 and Ra = 103, these values for Ra = 104 are about 23% and 96% and also for

Please cite this article in press as: Jafari M et al., Lattice Boltzmann simulation of natural convection heat transfer of SWCNT-nanofluid in an open enclosure, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.012

Lattice Boltzmann simulation of SWCNT-nanofluid in an open enclosure

Figure 9 A = 4.

11

Streamlines and temperature contours of the base fluid and SWCNT-nanofluid (u = 1%) at Ra = 105. (a) A = 2 (b) A = 3 (c)

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M. Jafari et al.

(a)

10 5

Ra=10 " Curve Fit Ra=104 " Curve Fit 3 Ra=10 " Curve Fit

9 8 7 6

Nuave

Ra = 105 are 24% and 98%. It is observed that SWCNTnanoparticles make better enhancement in heat transfer rate in comparison with Cu-nanoparticles. The main reason of this matter is higher thermal conductivity of SWCNT nanoparticles. Although, higher density of added nanoparticles can make a better enhancement of flow strength, the couple of flow and thermal fields cause that the thermal conductivity has a powerful role on flow strength in the natural convection. Therefore, the SWCNT-nanoparticles have more effects on values of stream function even with lower density in comparison with Cu-nanoparticles.

Nuave= 1.177 A2- 8.294 A+ 14.956

5

Nuave= 0.634 A2- 4.073 A+ 6.715

4

Nuave= 0.132 A2- 0.897 A+ 1.959

3

5.2. Aspect ratio effects

2

At studying of convection problems in 2D cavities, aspect ratio of a cavity must be mentioned as characteristic geometrical

1 0

1

2

3

4

A

(a)

0.1

(b) 10 5

Ra=10 " Curve Fit Ra=104 " Curve Fit 3 Ra=10 " Curve Fit

9

0

8 7 6

Nuave

-2

u (10 )

-0.1

-0.2

5

0

1

2

3

Nuave= 0.138 A2- 0.941 A+ 2.049

3

A=1 A=2 A=3 A=4

-0.4

2 1 0

4

X/H

(b)

Nuave= 0.664 A2- 4.265 A+ 7.025

4

-0.3

-0.5

Nuave= 1.233 A2- 8.687 A+ 15.653

1

3

4

A

0.06

(c)

A=1 A=2 A=3 A=4

0.05

10 5

Ra=10 " Curve Fit Ra=104 " Curve Fit 3 Ra=10 " Curve Fit

9 8 7

0.04

Nuave

6

v

2

0.03

Nuave= 1.392 A2- 9.825 A+ 17.717

5

Nuave= 0.758 A2- 4.871 A+ 7.995

4

0.02

Nuave= 0.160 A2- 1.089 A+ 2.330

3 0.01

2 1

0 0

1

2

3

4

X/H

Figure 10 Pattern of u and v-velocity components of SWCNTnanofluid (u = 1%) in horizontal mid-section of the cavity with different aspect ratios when Ra = 105.

0

1

2

3

4

A

Figure 11 The effects of aspect ratio on average Nusselt number at different Rayleigh numbers. (a) Water (b) Cu–Water (u = 0.2%) (c) SWCNT-Water (u = 0.2%).

Please cite this article in press as: Jafari M et al., Lattice Boltzmann simulation of natural convection heat transfer of SWCNT-nanofluid in an open enclosure, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.012

Lattice Boltzmann simulation of SWCNT-nanofluid in an open enclosure parameter of the cavity. In this section, the aspect ratio is studied in different graphs and contours. In this way, the streamlines and temperature contours of the base fluid and SWCNT-nanofluid (u = 1%) in the cavity with different

(a)

1

Α=1 (ϕ=0) Α=1 (ϕ=1%) Α=2 (ϕ=0) Α=2 (ϕ=1%) Α=3 (ϕ=0) Α=3 (ϕ=1%) Α=4 (ϕ=0) Α=4 (ϕ=1%)

0.8

y/H

0.6

0.4

0.2

0

values of aspect ratio are presented in Fig. 9 when Ra = 105. The overall view of this figure shows that the increase of cavity’s aspect ratio leads to reduce of flow strength. The distance between hot and cold side in an enclosure is one of most important parameter of natural convection regime, the increase of this distance changes the natural convection regime to a conduction-like manner in the problem domain. To observe of the pattern of velocity components in midsection of the cavity with different aspect ratios, some graphs which show the velocity components along the cavity’s length are illustrated in Fig. 10. The effect of aspect ratio on flow structure and heat transfer rate in the problem domain is sensible at these figures. For more clarification of the leading role of the aspect ratio on heat transfer as the target outcome, the values of Nuave at different values of Rayleigh number are plotted in Fig. 11 for cavities with various aspect ratios. Also some practical correlations

(a) 0

5

10

15

20

16

A=1 A=2 A=3 A=4

25

14

Nu 1

Α=1 (ϕ=0) Α=1 (ϕ=1%) Α=2 (ϕ=0) Α=2 (ϕ=1%) Α=3 (ϕ=0) Α=3 (ϕ=1%) Α=4 (ϕ=0) Α=4 (ϕ=1%)

0.8

Nuave= 7.910 + 1.876 ϕ

10

Nuave= 2.836 + 0.638 ϕ

8

Nuave= 0.900 + 0.169 ϕ

6

y/H

0.6

12

Nuave

(b)

13

Nuave= 0.529 + 0.90 ϕ

4

0.4

2 0.2

0 0

5

10

15

20

(c)

(b)

1

Α=1 (ϕ=0) Α=1 (ϕ=1%) Α=2 (ϕ=0) Α=2 (ϕ=1%) Α=3 (ϕ=0) Α=3 (ϕ=1%) Α=4 (ϕ=0) Α=4 (ϕ=1%)

0.8

y/H

0.6

0

0.2

0.4

25

Nu

0.4

0.6

0.8

1

A=1 A=2 A=3 A=4

12

Nuave= 7.880 + 7.607 ϕ

10

Nuave= 2.827 + 2.714 ϕ

8

Nuave= 0.898 + 0.713 ϕ

6

Nuave= 0.530 + 0.319 ϕ

4

0.2

ϕ%

16 14

Nuave

0

2 0

0

5

10

15

20

25

Nu

Figure 12 The Nusselt number distribution along the hot wall of the cavity with different aspect ratios for the base fluid and SWCNT-nanofluid (u = 1%). (a) Ra = 103 (b) Ra = 104 (c) Ra = 105.

0

0

0.2

0.4

ϕ%

0.6

0.8

1

Figure 13 Average Nusselt number at different aspect ratios and volume fractions of nanoparticles at Ra = 105. (a) Cu-nanoparticles (b) SWCNTs.

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14 are added to this figure. Paying an attention to the presented results shows that by increasing of the aspect ratio, the rate of natural convection heat transfer reduces for all studied values of Rayleigh number and fluids (see Fig. 11). To delineate the details of heat transfer reduction by increasing of the cavity’s aspect ratio, the Nusselt number distribution along the left hot wall is presented in Fig. 12 for different physical conditions. Changing of aspect ratio affects on heat transfer rate especially for a high Rayleigh number. Decreasing of heat transfer by increasing of the aspect ratio is due to increase of the distance between hot wall and open side. Increase of hot and cold walls distance leads to decrease of both convection and conduction heat transfer in the problem domain. At high aspect ratios, the heat transfer pattern is likely to be conductive that is because of the big gap existence between hot and cold walls therefore it is found that Rayleigh number loses its effects on heat transfer rate at high aspect ratio of the cavity. Finally, for more observation about the effects of nanoparticles effects on convection rate at different physical conditions, the values of Nuave aspect to the volume fraction of Cu and SWCNT nanoparticles, participated with needed correlations, are plotted in Fig. 13 at different aspect ratios. Taking an overview to the plotted data and especially the presented correlations emphasize that the nanoparticles have better performance at the cavity with lower aspect ratio. 6. Conclusion The effect of Single Walled Carbon Nanotubes and Cu nanoparticles on laminar natural convection heat transfer in an open-end enclosure was studied numerically. The double population approach in lattice Boltzmann Method was used at this study and the Bounce Back Method with second order accuracy in both flow and temperature field is applied to model solid boundaries. The problem was investigated at different aspect ratios of the cavity (1 6 A 6 4) and the volume fractions of SWCNT and Cu-nanoparticles (0 6 u 6 1) when Rayleigh number varies from 103 to 105. Some of most important results that have been achieved in this study are as follows:  Single Walled Carbon Nanotubes and Cu-nanoparticles can enhance both flow strength (stream-function) and convection rate (Nusselt number) in an open end enclosure.  Results show that adding a low value of SWCNT to the base fluid led to significant enhancement of convection heat transfer.  The heat transfer rate is at closed relation with thermal conductivity of the suspensions therefore the use of nanoparticles with better thermal conductivity leads to better heat transfer enhancement at base fluid.  Make a comparison between SWCNT and Cu-nanoparticles shows that the SWCNT-nanoparticle has better performance to enhance convection rate.  The aspect ratio of the cavity plays an important role on natural convection heat transfer. An increase of this parameter leads to heat transfer reduction in a 2D deep open cavity.  Rayleigh number loses its importance on natural convection in the high aspect ratios.

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Mohammad Jafari He is a faculty member of Mechanical Engineering Department at Lorestan University. He received his B.Sc. degree from school of Mechanical Engineering in solid-design subfield at Razi University and his M.Sc. degree in Energy Conversion subfield from school of Mechanical Engineering at Babol University of Technology. Currently, he is a PhD candidate in Energy Conversion subfield from school of Mechanical Engineering at Babol University of Technology. His main research interests are Computational Fluid Dynamics, Lattice Boltzmann Method and Mechanics of Fluid in Nano/Microscale. Currently he works on development of new idea to enhance performance of heat exchanger devices.

Mousa Farhadi He is an associated Professor of Mechanical Engineering Department of Babol University of Technology. Large Eddy Simulation of complex fluid dynamics problems is one of his main research interests. A comprehensive researches on heat transfer enhancement in heat exchangers by new passive methods are an under studying topic in his laboratory at Babol University of Technology.

Sanaz Akbarzade She received his B.Sc. degree from school of Mechanical Engineering in solid-design subfield at Mazandaran University and his M.Sc. degree in Energy Conversion subfield from school of Mechanical Engineering at Babol University of Technology. She is currently a PhD student at Semnan University. Her main research interests are thermal characteristics of different types of nanofluids.

Mahboubeh Ebrahimi She received his B.Sc. degree from school of Mechanical Engineering in solid-design subfield at Mazandaran University and his M.Sc. degree in Energy Conversion subfield from school of Mechanical Engineering at Babol University of Technology. Investigation of the effects of adding nanoparticles to heat exchangers is the main research interests of her.

Please cite this article in press as: Jafari M et al., Lattice Boltzmann simulation of natural convection heat transfer of SWCNT-nanofluid in an open enclosure, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.012