Commun Nonlinear Sci Numer Simulat 15 (2010) 1501–1510

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Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier .com/locate /cnsns

Heat transfer enhancement of copper-water nanofluidsin a lid-driven enclosure

M. Muthtamilselvan a, P. Kandaswamy a,b,*, J. Lee b

a UGC-DRS Center for Fluid Dynamics, Department of Mathematics, Bharathiar University, Coimbatore 641046, Indiab Department of Mechanical Engineering, Yonsei University, Seoul, Republic of Korea

a r t i c l e i n f o

Article history:Received 14 December 2007Received in revised form 29 May 2009Accepted 10 June 2009Available online 14 June 2009

PACS:64.75.Ef66.10.cd67.30.ef

Keywords:Mixed convectionNanofluidsFinite volume method

1007-5704/$ - see front matter Crown Copyright �doi:10.1016/j.cnsns.2009.06.015

* Corresponding author. Address: UGC-DRS CenCoimbatore 641046, India. Tel.: +91 422 2422222x4

E-mail address: pgkswamy@yahoo.co.in (P. Kand

a b s t r a c t

A numerical study is conducted to investigate the transport mechanism of mixed convec-tion in a lid-driven enclosure filled with nanofluids. The two vertical walls of the enclosureare insulated while the horizontal walls are kept at constant temperatures with the topsurface moving at a constant speed. The numerical approach is based on the finite volumetechnique with a staggered grid arrangement. The SIMPLE algorithm is used for handlingthe pressure velocity coupling. Numerical solutions are obtained for a wide range ofparameters and copper-water nanofluid is used with Pr ¼ 6:2. The streamlines, isothermplots and the variation of the average Nusselt number at the hot wall are presented anddiscussed. It is found that both the aspect ratio and solid volume fraction affect the fluidflow and heat transfer in the enclosure. Also, the variation of the average Nusselt numberis linear with solid volume fraction.

Crown Copyright � 2009 Published by Elsevier B.V. All rights reserved.

1. Introduction

The fluids that are traditionally used for heat transfer applications such as water, mineral oils and ethylene glycol have arather low thermal conductivity and do not meet the growing demand as an efficient heat transfer agent. Taking into accountthe rising demands of modern technology, including chemical production, power stations and microelectronics, there is aneed to develop new types of fluids that will be more effective in terms of heat exchange performance. Nanofluids arenew kind of heat transfer fluids containing a small quantity of nanosized particles (usually less than 100 nm) that are uni-formly and stably suspended in a liquid. These nanofluids appear to have a very high thermal conductivity and may be ableto meet the rising demand as an efficient heat transfer agent. Scientists and engineers have started showing interest in thestudy heat transfer characteristics of these nanofluids. But a clear picture about the heat transfer through these nanofluids isyet to emerge.

Khanafer et al. [1] developed a numerical model to determine natural convection heat transfer in nanofluids. The nano-fluid in the enclosure was assumed to be in single phase. The effect of suspended nanoparticles on buoyancy-driven heattransfer process was analyzed. It was observed that the heat transfer rate increases as the particle volume fraction is in-creased at any given Grashof number. More recently, Tiwari and Das [2] investigated numerically heat transfer augmentationin a lid-driven square cavity filled with nanofluids. They considered three cases depending on the direction of the moving

2009 Published by Elsevier B.V. All rights reserved.

ter for Fluid Dynamics, Department of Mathematics, Bharathiar University, Maruthamalai Road,10; fax: +91 422 2425706.aswamy).

Nomenclature

Ar aspect ratio, L/Hg gravitational acceleration, m=s2

Gr Grashof number, gbDhH3

m2fH enclosure height, m

kf thermal conductivity of the fluid, W/m Kks thermal conductivity of the solid, W/m KL enclosure length, mNuavg average Nusselt numberp pressure, N=m2

Pr Prandtl number, mf

af

Re Reynolds number, UpHmf

Ri Richardson number, Gr=Re2

T dimensionless temperature, h�hchh�hc

U;V dimensionless velocities in X- and Y-direction, respectivelyUp lid velocity, m=sUc dimensionless velocity in X-direction at mid-plane of the cavityu;v velocities in x- and y-direction, respectively, m=sVc dimensionless velocity in Y-direction at mid-plane of the cavityX;Y dimensionless Cartesian coordinatesx; y Cartesian coordinates, m

Greek symbolsa effective thermal diffusivity, m2=sbf coefficient of thermal expansion of fluid, K�1

bs coefficient of thermal expansion of solid, K�1

Dh temperature differenceh temperature, �Cl effective dynamic viscosity, kg/m sm effective kinematic viscosity, m2=sq fluid density, kg=m3

v solid volume fraction

Subscriptsavg averagec cold walleff effectivef fluidh hot wallnf nanofluids solid

1502 M. Muthtamilselvan et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1501–1510

walls. Also they found that both the Richardson number and the direction of the moving walls affect the fluid flow and heattransfer in the cavity.

Kim et al. [3] analytically investigated the instability in natural convection of nanofluids. They proposed a new factor thatdescribes the effect of nanoparticle addition on the convective instability and heat transfer characteristics of a base fluid.Their results show that as the density and heat capacity of nanoparticles increase and the thermal conductivity and shapefactor of nanoparticles decrease, the convective motion of a nanofluid sets in easily.

All the experimental results have demonstrated the enhancement of the thermal conductivity by addition of nanoparti-cles. Eastman et al. [4] used pure copper nanoparticles of less than 10 nm sized and achieved 40% increase in thermal con-ductivity for only 0.3% volume fraction of the solid dispersed in ethylene glycol. They showed the particle size effect andpotential of nanofluids with smaller particles. Xuan and Li [5] experimentally investigated flow and heat transfer character-istics for copper-water based nanofluids through a straight tube with a constant heat flux at the wall. Their results show thatthe nanofluids give substantial enhancement of heat transfer rate compared to pure water. For an up to date review of heattransfer in nanofluids one may refer to Das et al. [6] and Wang and Mujumdar [7].

Nanofluids can be considered as the next generation heat transfer fluids, as they offer exciting new possibilities to en-hance heat transfer performance compared to pure liquids. Large enhancement of conductivity was achieved with a small

M. Muthtamilselvan et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1501–1510 1503

v ¼ 0:08 concentration of particles that completely maintained the Newtonian behavior of the fluid. The experimental dataavailable in literature, such a Newtonian behavior was experimentally confirmed for Al2O3/water suspensions as well as forCuO/water suspensions [8]. Prasad and Das [9] have studied the lid-driven cavity flow with different aspect ratio and used inordinary fluids. They observed a Hopf bifurcation at Gr ¼ �105 for aspect ratio 2. Recently, Wang et al. [10] investigatednumerically the effective thermal conductivity enhancement of carbon fiber composites.

But so far no study has been attempted to investigate the heat transfer characteristics of the nanofluids contained in lid-driven cavities with different aspect ratios. Hence, in this paper the effect of aspect ratio in lid-driven cavity filled with nano-fluid is investigated numerically. While applying nanofluids for commercial cooling, Tzeng et al. [11] studied the effect ofnanofluids when used as engine coolants. CuO and Al2O3 and antifoam were individually mixed with automatic transmissionoil. The experimental platform was a real-time four-wheel-drive transmission system. This problem may be encountered in anumber of electronic equipment cooling and MEMS applications [12].

2. Mathematical analysis

We consider a steady two-dimensional flow of nanofluid contained in a rectangular enclosure of height H and lengthL as shown in Fig. 1. It is assumed that the top wall is moving from left to right at a constant speed U0 and is main-tained at a constant temperature Th. The bottom wall is maintained at a constant temperature TcðTh > TcÞ. The verticalside walls are considered to be adiabatic. The present geometry is favored in forced convection as well as mixed con-vection; using nanofluids could produce considerable enhancement of heat transfer coefficient that increased withincreasing the nanoparticle volume fraction. The nanofluid in the enclosure is Newtonian, incompressible and laminar.The nanoparticles are assumed to have uniform shape and size. Also, it is assumed that both the fluid phase and nano-particles are in the thermal equilibrium state and they flow with the same velocity. The physical properties of the nano-fluid are considered to be constant except the density variation in the body force term of the momentum equationwhich is satisfied by the Boussinesq’s approximation. Under the above assumptions the system of equations governingthe two-dimensional motion of a nanofluid is:

@u@xþ @v@y¼ 0 ð1Þ

u@u@xþ v @u

@y¼ � 1

qnf ;0

@p@xþ

leff

qnf ;052u ð2Þ

u@v@xþ v @v

@y¼ � 1

qnf ;0

@p@yþ

leff

qnf ;052v þ 1

qnf ;0½vqs;0bs þ ð1� vÞqf ;0bf �gðh� hcÞ ð3Þ

u@h@xþ v @h

@y¼ anf52h ð4Þ

Up

g

x

y

T

T h

c

Adiabatic

H

Adiabatic

L

Fig. 1. Flow configuration and coordinate system.

1504 M. Muthtamilselvan et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1501–1510

where

anf ¼keff

ðqCpÞnf ;0ð5Þ

and the effective viscosity as given by Brinkman [13] is

leff ¼lf

ð1� vÞ2:5ð6Þ

The effective density of the nanofluid at reference temperature is

qnf ;0 ¼ ð1� vÞqf ;0 þ vqs;0 ð7Þ

and the heat capacitance of nanofluid is

ðqCpÞnf ¼ ð1� vÞðqCpÞf þ vðqCpÞs ð8Þ

as given by Xuan and Li [5]. The effective thermal conductivity of the solid–liquid mixture is given by Khanafer et al. [1] asfollows

keff

kf¼ ðks þ 2kf Þ � 2vðkf � ksÞðks þ 2kf Þ þ vðkf � ksÞ

ð9Þ

Introducing the following dimensionless variables and parameters

X ¼ xH; Y ¼ y

H; U ¼ u

Up; V ¼ v

Up; T ¼ h� hc

hh � hc

Gr ¼ gbDhH3

m2f

; P ¼ p

qU2p

; Re ¼ UpHmf

; Pr ¼ mf

af

the governing equations may be written in the dimensionless form as

@U@Xþ @V@Y¼ 0 ð10Þ

U@U@Xþ V

@U@Y¼ �

qf ;0

qnf ;0

@P@Xþ 1

Releff

mf :qnf ;052U ð11Þ

U@V@Xþ V

@V@Y¼ �

qf ;0

qnf ;0

@P@Yþ 1

Releff

mf :qnf ;052V þ

vqsbs þ ð1� vÞqf bf

qnf ;0bfRiT ð12Þ

U@T@Xþ V

@T@Y¼ anf

af

1Pr:Re

52T ð13Þ

The dimensionless boundary conditions, used to solve Eqs. (10)–(13) are

U ¼ 1; V ¼ 0; T ¼ 1 ðY ¼ 1ÞU ¼ V ¼ 0; T ¼ 0 ðY ¼ 0Þ

U ¼ V ¼ 0;@T@X¼ 0 ðX ¼ 0;1Þ

The average Nusselt number is a dimensionless heat transfer coefficient which is a measure of the ratio of the heat trans-fer rate to the rate at which heat would be conducted within the fluid under a temperature gradient. It is the quantity of heattransfer between a solid body and the fluid and is measured with the help of a dimensionless heat transfer coefficient. It isgiven by

Nuavg ¼ �keff

kf

1Ar

Z Ar

0

@T@Y

dX

3. Method of solution and code validation

Numerical solutions to the governing equations are secured by employing the finite volume computational procedureusing staggered grid arrangement with the SIMPLE algorithm as given in Patankar [14]. The convective terms in the interiorpoints are discretized by using the deferred QUICK scheme and central difference scheme was used adjacent to the bound-aries. The resulting algebraic equations are solved by using tridiagonal matrix (TDMA) algorithm. The pseudo-transient ap-

M. Muthtamilselvan et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1501–1510 1505

proach is followed for the numerical solution as it is useful for situation in which the governing equations give rise to sta-bility problems, e.g., buoyant flows [15]. Euclidean norm of the residual is taken as convergence criteria for each dependentvariable in the entire row field [16]. The iteration is carried out until the normalized residuals of the mass, momentum andtemperature equation become less than 10�7.

In the present study the grid independent test is conducted for v ¼ 8% and Ri ¼ 1. This test is performed using 11� 21,21� 41, 41� 81 and 81� 161 grids. It can be seen from Fig. 2 that the mid-plane velocity profile remains almost the samefor grids finer than 41� 81 and heavily depends on the grid size for less finer grids. Therefore, considering both the accuracyand the computational time involved computations were performed with 41� 81 uniform grid.

The validation of the present computational code has been verified for Rayleigh numbers between 103 and 106. Table 1compares the results of the present study for special cases with those of de Vahl Davis [17], Manzari [18] and Wan et al. [19].The computed results are in very good agreement with the benchmark solutions. This effects provided credence to the accu-racies of the present numerical solutions.

X

V

0 0.5 1-0.5

0

0.5

11X21

21X4141X8181X161

Fig. 2. Average Nusselt number for different mesh sizes at v ¼ 8% and Ri ¼ 1.

Table 1Comparison of Nusselt number (Nu) and the corresponding ordinate (Max., maximum; Min., minimum; Av., average).

Ra Nu Ref. [17] Ref. [18] Ref. [19] Present study

103 Max. 1.50 1.47 1.501 1.508(0.092) (0.109) (0.08) (0.094)

Min. 0.692 0.623 0.691 0.69(1.0) (1.0) (1.0) (1.0)

Av. 1.12 1.074 1.117 1.118

104 Max. 3.53 3.47 3.579 3.545(0.143) (0.125) (0.13) (0.145)

Min. 0.586 0.497 0.577 0.582(1.0) (1.0) (1.0) (1.0)

Av. 2.243 2.084 2.254 2.248

105 Max. 7.71 7.71 7.945 7.833(0.08) (0.08) (0.08) (0.08)

Min. 0.729 0.614 0.698 0.721(1.0) (1.0) (1.0) (1.0)

Av. 4.52 4.3 4.598 4.546

106 Max. 17.92 17.46 17.86 18.642(0.038) (0.039) (0.03) (0.031)

Min. 0.989 0.716 0.9132 0.959(1.0) (1.0) (1.0) (1.0)

Av. 8.8 8.743 8.976 8.975

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4. Results and discussion

The present computations are carried out for copper-water as working fluid with Prandtl number of 6.2. The aspect ratioAr is considered for five different values 0.25, 0.5, 1.0, 2.0 and 4.0. It is worth to note that, in this study the aspect ratioAr ¼ L=H is equal to the ratio of cold wall length to adiabatic wall length. The effect of solid volume fraction is investigatedin the range of 0–8%. If the concentration exceeds the maximum level of 8%, sedimentation takes place. Therefore, the inves-tigation is of no consequence when the concentration level exceeds 8%.The thermophysical properties of fluid and solidphase was given by Khanafer et al. [1]. The results are presented in Figs. 3–7 in the form of streamlines and isotherms.

Fig. 3 shows the streamlines and isotherms in square enclosure ðAr ¼ 1Þ for various values of solid volume fraction. Thisfigure indicates that the natural convection effect is comparable with forced convection effect. The fluid flow in a two-dimensional lid-driven cavity is characterized by a primary circulating cell (major cell) on the top of the enclosure generatedby the lid and a weaker anticlockwise rotating cell near the bottom. As can be seen from Fig. 3, the streamlines collapse to-gether toward the left top corner where the sliding top wall impinges on the vertical left wall which is a characteristic of alid-driven cavity flow problem. The main cell is generated by the lid dragging the adjacent fluid. It is observed from thestreamlines that the bottom cell size is decreasing when v is increased to 8%. This is due to an increase in the volume fractionas a result of high-energy transport through the flow associated with the irregular motion of the ultrafine particles. The iso-therm plots indicate that there is some variation in the temperatures with increasing value of volume fraction. It is interest-ing to note that, with the increased concentration inside the fluid, nanofluid helps in minimizing the natural convectioneffect and increase the forced convection effect.

The streamlines and isotherms for Ar ¼ 2 and 4 representing shallow enclosures are displayed in Figs. 4 and 5, respec-tively. Similar to the case of square enclosure, a clockwise rotating major cell is observed at the top of the enclosure forall volume fractions. A minor cell is observed at the lower left region for both values of the aspect ratio of the enclosures.The size of the minor cell decreases with an increase in v. In this case the isotherms are clustered near the right bottom sur-face of the enclosure, which indicates steep temperature gradient in the vertical direction in this region. In the remainingarea of the cavity, the temperature gradients are very small due to the mechanically-driven circulations. Consequently,the temperature differences in this interior region are very small. Figs. 4 and 5 indicate that mixed convection is the dom-inating mode in the enclosure. When v is increased from 0% to 8%, the nanofluid does not have appreciable effect on the flowfield. Comparing Figs. 3–5, the thicken boundary layer is observed in the right bottom corner when Ar ¼ 4. This thickenboundary layer is reduced when the aspect ratio of the cavity decreases. The average Nusselt number Table 2, shows thebetter heat transfer when Ar ¼ 2.

Figs. 6 and 7 show the influence of aspect ratio and solid volume fraction on flow and heat transfer in enclosures havingthe aspect ratios Ar ¼ 0:5 and 0.25, respectively, which are considered to be tall enclosures. In Fig. 6, streamlines show that

0.9375

0.3125

0.9375

0.1875

Fig. 3. Streamlines and isotherms for Ar ¼ 1.

0.94

0.38

0.63

0.94

0.38

Fig. 4. Streamlines and isotherms for Ar ¼ 2.

0.88

0.25

0.88

0.13

Fig. 5. Streamlines and isotherms for Ar ¼ 4.

0.94

0.31

0.94

0.19

Fig. 6. Streamlines and isotherms for Ar ¼ 0:5.

M. Muthtamilselvan et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1501–1510 1507

the main circulation fills the top of the enclosure and three weaker counter clockwise rotating cells occur one bellow theother in the remaining part of the enclosure. In this case weakened flow activities due to the stratification suggest the for-mation of a separate cell in the lower part of the cavity. When v is increased from 0% to 8%, nanofluids help in minimizing thenatural convection effect and precipitate in merging three weaker cells into two. The isotherms indicate that the conductionis dominating in the bottom half of the enclosure. The fluid is well circulated in the top half of the enclosure. When aspectratio is decreased to 0.25 (Fig. 7), fluid is stagnant at the bottom of the enclosure. The upper cells grows in size and strengthand moves towards the bottom of the enclosure as v increases. The corresponding isotherm distributions are nearly parallel

0.94

0.38

0.94

0.31

Fig. 7. Streamlines and isotherms for Ar ¼ 0:25.

Table 2Comparison of the average Nusselt number (Nuavg ) for different aspect ratio and various solid volume fractions.

Ar A ðv ¼ 0%Þ B ðv ¼ 2%Þ % Increasea D ðv ¼ 4%Þ % Increaseb E ðv ¼ 6%Þ % Increasec F ðv ¼ 8%Þ % Increased

0.25 0.38 0.40 5.26 0.41 7.89 0.43 13.16 0.44 15.790.5 0.82 0.86 4.88 0.91 11.0 0.96 17.07 1.02 24.391 2.26 2.40 6.19 2.56 13.27 2.73 20.80 2.91 28.762 2.73 2.93 7.33 3.13 14.65 3.36 23.08 3.59 31.504 3.20 3.36 5.00 3.53 10.31 3.70 15.62 3.89 21.56

a B�AA � 100.

b D�AA � 100.

c E�AA � 100.

d F�AA � 100.

1508 M. Muthtamilselvan et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1501–1510

to the horizontal wall of the cavity, indicating that most of the heat transfer is carried out by conduction. The convectiveactivities are appreciable in a small region at the top of the enclosure where a mechanically induced movement occurs.

The horizontal velocity profiles at mid-section of the enclosure for various volume fraction and aspect ratio are displayedin Fig. 8. The numerical result of the present study indicates that the flow pattern of a nanofluid changes remarkably with thevolume fraction of nanoparticles. As the volume fraction increases, the velocity components of the nanofluid increase as aresult of an increase in the energy transport through the fluid. High velocity peaks of the horizontal velocity componentsare observed for high volume fractions.

The average Nusselt number along the hot wall from the numerical results for various values of aspect ratio and volumefraction is shown in Fig. 9. This figure shows a linear variation of the average Nusselt number with the solid volume fraction.Also it can be clearly seen that the heat transfer increases with increasing v. The variation of the average Nusselt numberwith aspect ratio and v are shown in Table 2. It is observed from this table that, when v is 8% and Ar ¼ 2 the increase inthe rate of heat transfer is the maximum (31.5%). The average Nusselt number, a measure of heat transfer is optimizedagainst the solid volume fraction of nanofluids. When the volume fraction is increases from 0% to 8% the heat transfer is alsoincreases see Fig. 9. When the solid volume fraction is above 8% sedimentation sets in and the fluid loses the Newtoniancharacter.

5. Conclusions

Mixed convection in a lid-driven enclosure filled with nanofluids is studied numerically. Results for various parametricconditions are presented and discussed. From the above study, the following conclusions are made:

� The inclusion of nanoparticles into the base fluid has produced an augmentation of the heat transfer coefficient, whichincreases appreciably with an increase of nanoparticles volume concentration.

� Nanofluids are capable to change the flow pattern.

X

V

0 0.5 1-0.4

0

0.4

Ar = 1

8%

0%2%4%6%

X

V

0 1 2-0.5

0

0.5

8%

0%2%4%6%

Ar = 2

X

V

0 2 4

-0.5

0

0.5

8%

0%2%4%6%

Ar = 4

X

V

0 0.5 1-0.01

0

0.01

Ar = 0.5

8%

0%2%4%6%

X

V

0 0.5 1-0.0004

0

0.0004

8%

0%2%4%6%

Ar = 0.25

Fig. 8. Velocity profiles at mid-plane of the cavity.

M. Muthtamilselvan et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1501–1510 1509

� When Ar ¼ 0:25, the fluid is stagnant in the bottom of the enclosure and recirculating cells appear in the remaining part ofthe enclosure.

� When v is 2% the increase in the rate of heat transfer is approximately 5% for all aspect ratios and when v is 8% theincrease is approximately 20% for all aspect ratios except Ar ¼ 0:25.

χ

Nu

0

1

2

3

4

5

avg

Ar=0.25

Ar=4

Ar=2

Ar=1

Ar=0.5

8%0% 2% 4% 6%

Fig. 9. Variation of average Nusselt number at the heated wall with v.

1510 M. Muthtamilselvan et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1501–1510

The study of nanofluids is still at its early stage, it seems very difficult to have a precise idea on the way the use of nanopar-ticles acts in natural convection heat transfer and complementary works are needed to understand the heat transfer char-acteristics of nanofluids and identify new and unique applications for these fields.

References

[1] Khanafer K, Vafai K, Lightstone M. Int J Heat Mass Transfer 2003;46:3639–53.[2] Tiwari RK, Das MK. Int J Heat Mass Transfer 2007;50:2002–18.[3] Kim J, Kang YT, Choi CK. Phys Fluids 2004;16:2395–401.[4] Eastman JA, Choi SUS, Li S, Yu W, Thompson LJ. Appl Phys Lett 2001;78:718–20.[5] Xuan Y, Li Q. ASME J Heat Transfer 2003;125:151–5.[6] Das SK, Choi SUS, Patel HE. Heat Transfer Eng 2006;27:3–19.[7] Wang X-Q, Mujumdar AS. Int J Thermal Sci 2007;46:1–19.[8] Putra N, Roetzel W, Das SK. Heat Mass Transfer 2003;39:775–84.[9] Prasad YS, Das MK. Int J Heat Mass Transfer 2007;50:3583–98.

[10] Wang M, Kang Q, Ning P. Appl Therm Eng 2009;29:418–21.[11] Tzeng SC, Lin CW, Huang KD. Acta Mech 2005;179:11–23.[12] Nguyen CT, Roy G, Gauthier C, Galanis N. Appl Thermal Eng 2007;27:1501–6.[13] Brinkman HC. J Chem Phys 1952;20:571–81.[14] Patankar SV. Numerical heat transfer and fluid flow. Washington, DC: Hemisphere; 1980.[15] Versteeg HK, Malalasekera W. An introduction to computational fluid dynamics: the finite volume method. Malaysia: Longman Group Ltd.; 1995.[16] Van Doormaal JP, Raithby GD. Numer Heat Transfer Part A 1984;7:147–63.[17] de Vahl Davis D. Int J Numer Meth Fluids 1983;3:249–64.[18] Manzari MT. Int J Numer Meth Heat Fluid Flow 1999;9:860–87.[19] Wan DC, Patnaik BSV, Wei GW. Numer Heat Transfer Part B 2001;40:199–228.