Sidik et al. Nanoscale Research Letters 2013, 8:178http://www.nanoscalereslett.com/content/8/1/178

NANO EXPRESS Open Access

Simulation of forced convection in a channel withnanofluid by the lattice Boltzmann methodNor Azwadi Che Sidik*, Maysam Khakbaz, Leila Jahanshaloo, Syahrullail Samion and Amer Nordin Darus

Abstract

This paper presents a numerical study of the thermal performance of fins mounted on the bottom wall of ahorizontal channel and cooled with either pure water or an Al2O3-water nanofluid. The bottom wall of the channelis heated at a constant temperature and cooled by mixed convection of laminar flow at a relatively lowtemperature. The results of the numerical simulation indicate that the heat transfer rate of fins is significantlyaffected by the Reynolds number (Re) and the thermal conductivity of the fins. The influence of the solid volumefraction on the increase of heat transfer is more noticeable at higher values of the Re.

Keywords: Nanofluid, Volume fraction, Nusselt number, Reynolds number, Lattice Boltzmann method

BackgroundRecent years have eyewitnessed a blossom flourishing inthe evolvement of electronics, communications, and auto-computing industries, and this bearing is irrefutably con-tinuing in this century. The cooling of electrical, mechan-ical, and electronic components has become troublesomein today's fast-growing technologies. Inasmuch as the sig-nificance of heat exchangers in tremendous engineeringapplications, the subject of potential heat transfer enhance-ment in these devices has received sizeable attention inpractice and research. On account of the fact that theconsistency of the electronic components commodiouslyincreases, conspicuous lack of heat transfer enhancementboth in macro- and microscales of channels is realized.Encountering a fluid flow by utilizing transverse surfaces ina channel is a prevalent method that is used to intensifythe rate of heat transfer from heated surfaces.Alamyane and Mohamad [1] studied the forced convec-

tion heat transfer in a channel with extended surfaces. Theeffects of the Reynolds number (Re) and the fin height andspacing on the fluid flow and the heat transfer were exam-ined. Yang et al. [2] simulated the forced convection in aparallel plate channel. Constant temperature was consid-ered in both upper and lower walls, and a transverse objectwas located at the lower channel wall. The effects of theReynolds number, the thermal conductivity ratio of the

* Correspondence: azwadi@fkm.utm.myFaculty of Mechanical Engineering, Universiti Teknologi Malaysia, UTMSkudai, Johor 81310, Malaysia

© 2013 Sidik et al.; licensee Springer. This is anAttribution License (http://creativecommons.orin any medium, provided the original work is p

fluid, and the fin profile area on the fluid flow and the heattransfer rate were analyzed. The study results showed thatthe heat transfer enhancement with an increment of theReynolds number and the thermal conductivity ratio of thefluid at various fin profiles. Yang et al. [3] numerically inves-tigated the effect of mix convection heat transfer in aninclined parallel plate channel with a transverse object atthe bottom wall. In this research, the effects of thermal con-ductivity, Reynolds number, the fin profile, and the channelinclination on the heat transfer rate at various Richardsonnumbers were examined. They discovered that the aceaspect ratio of the fin was related to the fin with utmostheat transfer at various Reynolds and Richardson numbers.Young and Vafai [4] observed the impact of controlling

parameters on the cooling of heated channels withmounted objects. Concentrating on the effect of alteringthe dimensions of the object, the thermal conductivity, theheating method, and the Re was embraced. They deducedthat the fluid flow and heat transfer are affected by thegeometry and material of the object, and a correlation forthe average Nusselt number was proposed as a function ofthe controlling parameters.Meinders and Hanjalic [5] experimentally investigated

the effect of the cubes' arrangement on the turbulentfluid flow. They comprehended that the flow stream wasaffected by the distance between the objects owing tothe fact of augmenting the flow velocity. Moreover,amelioration in velocity distribution and heat transferthan the staggered distribution case was found for flow

Open Access article distributed under the terms of the Creative Commonsg/licenses/by/2.0), which permits unrestricted use, distribution, and reproductionroperly cited.

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over inline cubes. Yan et al. [6] experimentally investigatedthe influence of short surface-mounted objects at the top ofa flat plate on the heat transfer enhancement. Scrutinizingwas done on the effect of varies cross sections, spacing andnumbers of objects, and the Reynolds number. Theyperceived that the heat transfer was incremented when theheight of the object is comparatively equal to half of thechannel height.In an experimental investigation by Yuan et al. [7], the

heat transfer and friction characteristics of a channelwhich were attached by winglets were examined. Heattransfer from the channel was achieved to be noticeablyaugmented by using winglets in comparison with conven-tional channels with rectangular transverse objects. For ahigh Reynolds number, the heat transfer was enhanced bya factor of 2.7 to 6 times of the smooth channel.Utilizing nanofluids for the purpose of enhancing the

heat transfer in thermal systems is another alternativetechnique [8]. The thermal performance of different typesof nanofluids has been the subject of many recent studieson forced, natural, and mixed convection problems. Severalexplorations have studied natural convection of nanofluidsin cavities [9,10]. They argued that the addition ofnanoparticles in the fluid indisputably increase the naturalconvection heat transfer.Chein and Huang [11] analyzed the cooling of two silicon

microchannel heat sinks with a water-Cu nanofluid. Theheat transfer and fraction coefficients were based on thetheoretical models and the experimental correlations. Theyrealized that the heat transfer performance of microchannelswas greatly improved when nanofluids were added into basefluid as coolants without any extra pressure drop.Recently, Santra et al. [12] numerically investigated the

effect of water-Cu nanofluid through parallel plate channelin laminar forced convection. A cold nanofluid was sentthrough the channel, and the walls of the channel wereisothermally heated. The effects of the Reynolds numberand the solid volume fraction on the heat transfer werestudied by considering the fluid to be Newtonian and non-Newtonian. They observed that the rate of heat transferincreased with an increase of the Reynolds number and thesolid volume fraction. The increase in the heat transfer wasapproximately the same for both scenarios.

Figure 1 A schematic plot of flow in a channel.

The lattice Boltzmann method (LBM) is another numer-ical method that is often used to simulate flow problems.LBM have been used for more than 2 decades as an alter-native numerical technique. In LBM, it is intended tomodel fluids as a collection of particles, which successivelyundergo collision and propagation over a discrete latticemesh. Several lattice Boltzmann models have beenproposed for the incompressible Navier–Stokes equations.A collision model was proposed by Bhatnagar et al. [13] tosimplify the analysis of the lattice Boltzmann equation,which leads to the so-called lattice BGK model. Remarkableefforts have been conducted by many researchers that madethis numerical method more attractive for fluid dynamicsmodeling, e.g., [14,15]. For more details about LBMand its application, kindly refer to the aforementionedpublications.Most of the researches cited above considered the heat

transfer enhancement by adding either the fin or usingnanofluids. The main objective of this study is to examineboth of these effects on the heat transfer performance. Ingeneral, previous works were performed to investigate dif-ferent cases of nanofluid flow and heat transfer in channelswith mounted objects by focusing on changing geometries,arrangement, and dimensions of the objects. However,more efforts are needed in order to optimize the controllingparameters for best heat transfer enhancement.

MethodsProblem definitionThe geometry of the problem is shown in Figure 1. A coldmixture of base fluid (water) and the nanoparticles (alu-mina) is forced to flow into a channel that is heated from itsbottom and kept at a constant high temperature, while thetop wall is insulated. The channel aspect ratio is fixed at L/H = 15. The Prandtl number is taken as 7.02, and the Reyn-olds numbers are 10, 50, and 100, whereas the extendedsurfaces' height to space ratio l/S is 0.2, and the ratio be-tween the objects' height to the channel's height l/H is 0.2.The flow is assumed as Newtonian, laminar, two-

dimensional, and incompressible. In addition, it is assumedthat the cold mixture of base fluid (water) and the solidspherical nanoparticles (alumina) is in thermal equilibrium,and it flows at the same velocity as a homogenous mixture.

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Numerical simulationThe D2Q9 LBM model is used to simulate fluid flow intwo-dimensional channel with uniform grid size of δx × δy.The lattice Boltzmann equation (known as LBGK equation)with single relaxation time can be expressed as [13]

fi xþ eiΔt; t þ Δtð Þ � fi x; tð Þ ¼ � 1τf

fi � f eqi� � ð1Þ

which can be reformulated as

fi xþ eiΔt; t þ Δtð Þ ¼ ωf feqi þ 1� ωf

� �fi ð2Þ

where ωf ¼ 1τf

and τf as the single relaxation time of the

fluid, fi represents the particle distribution function, ei is theparticle streaming velocity, and f eqi is the local equilibriumdistribution function. For D2Q9 model f eqi is given by [8]

f eqi ¼ ρωi 1þ 3ei⋅uþ 92

ei⋅uð Þ2 � 32u2

� �ð3Þ

where ρ is the density of the fluid and ωi is the weightfunction, which has the values of ω0 ¼ 4

9, ωi ¼ 19 for i = 1

to 4, and ωi ¼ 136 for i = 5 to 8. The macroscopic fluid flow

velocity in lattice units is represented by u. In the LBM,the fluid macroscopic quantities such as density, ρ, andflow momentum, ρu are calculated using the distribution

function fi, and given by ρ ¼X8i¼0

fi and ρu ¼X8i¼0

ei fi , res-

pectively. The streaming speed for particles in coordinate(x and y) directions (i.e., 1 to 4, see Figure 2) can beexpressed as ei = cos(π/2 (i − 1)), sin(π/2 (i − 1)), whereasparticles in diagonal directions (i.e., 5 to 8 in Figure 2) havevelocities of ei ¼

ffiffiffi2

pcos π=4 2i� 9ð Þð Þ; sin π=4 2i� 9ð Þð Þð Þ ;

however, the particle in the lattice center is at rest and hasno streaming speed, i.e., e0 = 0.The thermal part is simulated using another distribu-

tion function for the temperature. For instance, g is usedto simulate the distribution function of the dependentvariable (temperature) in the lattice Boltzmann equation,and an approach similar to that used to simulate thefluid flow is utilized to simulate the temperature

Figure 2 A schematic plot showing the thermal boundary conditions

distribution. In addition, the algorithm suggested bySucci [15] is adopted throughout this work. The kineticequation for the temperature distribution function withsingle relaxation time is given by:

gi xþ eiΔt; t þ Δtð Þ � gi x; tð Þ ¼ � 1τt

gi � geqi� � ð4Þ

which can be written in the form

gi xþ eiΔt; t þ Δtð Þ ¼ ωtgeqi þ 1� ωt½ �gi ð5Þ

Where gi represents the temperature distribution func-tion of the particles, geqi is the local equilibrium distribu-tion function of the temperature, and ωt ¼ 1

τt, where τt is

the single relaxation time of the temperature distribu-tion. Thus, the equilibrium distribution function of thethermal part is given by [15]:

geqi ¼ ϕ x; tð Þωi 1þ ei⋅uC2s

� �ð6Þ

where, ϕ is the macroscopic temperature and C2s is the

speed of sound. The diffusion coefficient can be obtained

as a function of the relaxation time and given by Δx2Δt

1ω � 1

2

� �. The macroscopic temperature is then computed from:

ϕ x; tð Þ ¼X8i¼0

gi ¼X8i¼0

geqi ð7Þ

A uniform lattice of 100 × 1,500 is used to perform allof the simulations. However, the number of lattices wasdoubled to test the grid dependency results.Since the inlet velocity of the flow is specified, the

inward distribution functions should be computed at theboundary. In the D2Q9 model, the values of the distri-bution functions pointing out of the domain at the inletboundary (i.e., f3, f6, f7 in Figure 2) are known from thestreaming step, and the only unknowns are (f1, f5, f8) aswell as the fluid density ρ. Following the work of Zouand He [16], the inlet density and the distributionfunctions can be obtained from:

ρin ¼1

1� uinf0 þ f2 þ f4 þ 2 f3 þ f6 þ f7ð Þ½ � ð8Þ

of the problem.

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The unknown distribution functions are calculatedusing

f1 ¼ f3 þ 23ρux

f5 ¼ f7 � 12

f2 � f4ð Þ þ 16ρux þ 1

2ρuy

f8 ¼ f6 þ 12

f2 � f4ð Þ þ 16ρux þ 1

2ρuy

ð9Þ

An extrapolation scheme is used to simulate the outletflow condition, which can be represented as fi(Nx, t) = fi(Nx − 1, t), i = 3, 6, 7. The bounce-back scheme is usedto specify the boundary conditions on solid surfaces (no-slip boundary), in which the distribution functionspointing to the fluid are equal to those pointing out ofthe domain. The thermal boundary conditions for thiscase are given in Figure 2. For constant wall temperature(the lower wall temperature is constant), the unknownfunctions are obtained using the following equation [15]:

gi ¼ ϕwall ωi þ ωiþ2ð Þ � giþ2 for i ¼ 2; 5 and 6 ð10Þ

The left-hand boundary (channel inlet) is kept at a con-stant temperature (Dirichlet boundary condition) and setto a dimensionless value of zero. Therefore, the resultedequations of the unknown distribution functions on the leftboundary are given by gi(0, t) = − gi + 2(0, t), for i = 1, 5,and 8. For the adiabatic boundary condition, the gradientof the dependent variable normal to the boundary shouldbe zero, i.e., ∂ ϕ/∂ y = 0. The distribution functions arefound to be in the following form [15]:

Figure 3 Velocity and profiles at different cross sections.

X8i¼0

gi x;Ny þ 1� � ¼ X8

i¼0

gi x;Ny� � ð11Þ

A second-order extrapolation similar to the one givenin [17] is used to obtain the values of the unknown dis-tribution functions for the right-hand side boundary(channel outlet) as follows:

gi Nx; tð Þ ¼ 2gi Nx � 1; tð Þ � gi Nx � 2; tð Þfor i¼ 3; 6; and 7: ð12Þ

The local Nusselt number (Nux) is computed using thefollowing equation:

Nux ¼LC ∂ϕ=∂yð Þy¼0

ϕwall � ϕmð13Þ

where Lc is the characteristic length and ϕwall is the wallconstant temperature. The mean temperature ϕm isgiven by:

ϕm ¼

ZuϕdAZudA

ð14Þ

The effective density of the nanofluid is

ρnf ¼ 1� φð Þρf þ φρs ð15Þ

where ϕ is the solid volume fraction. The effectivedynamic viscosity of the nanofluid given by Brinkman[18] is

Figure 4 Temperature profiles at different cross sections.

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μnf ¼μf

1� φð Þ2:5 ð16Þ

The thermal diffusivity of the nanofluid is

αnf ¼ keffρcp� �

nf

ð17Þ

The heat capacitance of the nanofluid is

ρcp� �

nf ¼ 1� φð Þ ρcp� �

f þ ρcp� �

s ð18Þ

keff is the effective thermal conductivity of the nanofluidand is determined using the model proposed by Patelet al. [19]. For the two-component entity of sphericalparticle suspension, the model gives:

Figure 5 Velocity and temperature profiles at different Re.

keff ¼ kf 1þ ksAs

kf Afþ cksPe

As

kf Af

� �ð19Þ

where ks and kf are the thermal conductivities ofdispersed Al2O3 nanoparticles and pure water.

Pe ¼ usdsαf

ð20Þ

where us is the Brownian motion velocity of thenanoparticles given by:

us ¼ 2kbTπμf d2

sð21Þ

where kb = 1.3087×10−23JK−1 is the Boltzmann constant.

Figure 6 Streamlines and isotherms for the Al2O3-water nanofluid and pure water at different Reynolds number. (A) Streamline plots at(a) Re = 10, (b) Re = 50, and (c) Re = 100. (B) Isotherm plots at Re = 10 and (a) ϕ = 0.0 and (b) ϕ = 0.05. (C) Isotherm plots at Re = 50 and (a)ϕ = 0.0 and (b) ϕ = 0.05. (D) Isotherm plots at Re = 100 and (a) ϕ = 0.0 and (b) ϕ = 0.05.

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Results and discussionCode validation and computational resultsFor the purpose to ensure that the obtained results areproper and that the code is free of errors, a flow of coldair in a two-dimensional heated channel was taken as abenchmark test. Both upper and lower walls wereheated. The comparisons were carried up between thedimensionless velocity and temperature fields at different

locations in the channel as shown in Figures 3 and 4.The obtained results were found to be identical to theresults of [20].Figure 5 shows the effect of Reynolds on the

temperature profiles at the same cross sections forRe = 10, 50, and 100. The figures depicted that thetemperature profiles are less sensitive to the change inReynolds compared to the velocity profiles.

Table 1 Average Nusselt number for various Reynoldsnumber and solid volume fraction

Reynoldsnumber

Average Nusseltnumber

ϕ = 0.0 ϕ = 0.03 ϕ = 0.05

Re = 10 Nuave 2.712 2.826 2.965

Re = 50 Nuave 5.294 5.683 5.919

Re = 100 Nuave 10.252 10.797 11.109

Figure 7 Average Nusselt number for various Re.

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The effects of the Reynolds number and the solidvolume fraction on the heat transfer, isotherms, andstreamlines are studied. Figure 6 presents the stream-lines and the isotherms for the Al2O3-water nanofluid(ϕ = 0.05) and pure water at different Reynolds number(Re = 10, 50, and 100).The streamlines show that as the Reynolds number

increases, the vortices that are formed behind the finsbecome larger and stronger. This can be more clearlyillustrated in Figure 5 where the horizontal velocity in themiddle section between fins is presented. At Re = 10, thevelocity is consistently positive. However, as the Reynoldsnumber increases, the flow velocity becomes negative.This is an indication of flow reversal. The strong vortex athigh numbers enhances the heat transfer from leftface objects to right face objects and the wall betweenthe two fins. This difference, however, becomes noticeableat higher Re.At low Reynolds numbers, the conduction is the

dominating mechanism of heat transfer. Therefore,the isotherms stretch above the fins and take a largearea in the channel. As Re increases, the convectionbecomes the dominating mechanism, and the strongcold inlet flow pushes the isotherms near the bottomwall. The comparison between the isotherms of thenanofluid and pure water shows that in each point ofthe channel, the nanofluid temperature is higher thanthe pure water. It is due to the nanofluid's higherthermal conductivity.The current investigation is wrapped with the analysis

of the effect of the Reynolds number and percentage ofnanoparticle volume fraction on the heat transferenhancement in the channel. Figure 7 and Table 1display values of average Nusselt number at various

Reynolds numbers and solid volume fraction from 0%to 5%. These figures demonstrate that the Nusseltnumber increases with the Reynolds number forvalues of volume fraction tested in the present study.For example, at Re = 100, in the addition of volumefraction of 5%, the average Nusselt number increasesabout 17%. High Reynolds number results in highenergy transport through the fluid and cause irregularmotion of nanoparticle. The higher solid volumefraction further stimulates the flow and contributes tohigher Nusselt number as shown in the figure. Thepresence of nanoparticles also increases the rate ofheat transfer by conduction mode through the flow.

ConclusionsLBM was applied to simulate forced convection heattransfer in two-dimensional channel including extendedsurfaces to investigate the effect of changing differentparameters such as Reynolds number (10, 50, and 100)and nanofluid (Al2O3) volume fractions (0.0, 0.03, and0.05). The results showed that as the Reynolds numberincreases, the rate of heat transfer also increases. Theformation of vortices both in front and behind theobjects enhances the heat transfer process. As the solidvolume fraction increases, the heat transfer is enhanced

Sidik et al. Nanoscale Research Letters 2013, 8:178 Page 8 of 8http://www.nanoscalereslett.com/content/8/1/178

for all values of the Reynolds numbers. This enhance-ment is more significant at high Reynolds numbers. Theheat transfer rate of the fins increases with the thermalconductivity ratio of the fin to pure water. This enhance-ment has a finite limit. At this limit, the temperature atall surfaces of the fins approach the wall temperature. Inthis condition, the fins behave like constant temperatureof heat sources.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsMK, LJ, and SS conceived the study and checked the grammar of themanuscript. NACS and AND drafted the manuscript. All authors read andapproved the final manuscript.

AcknowledgmentsThis research is financially supported by the Ministry of Higher Education ofMalaysia through Fundamental Research Grant Scheme, FRGS Vot no. 4L074.

Received: 11 February 2013 Accepted: 29 March 2013Published: 17 April 2013

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doi:10.1186/1556-276X-8-178Cite this article as: Sidik et al.: Simulation of forced convection in achannel with nanofluid by the lattice Boltzmann method. NanoscaleResearch Letters 2013 8:178.

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