Alexandria Engineering Journal (2017) xxx, xxx–xxx H O S T E D BY Alexandria University Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com ORIGINAL ARTICLE Nanofluid heat transfer between two pipes considering Brownian motion using AGM M Sheikholeslami a,*, M Nimafar b, D.D Ganji c a Department of Mechanical Engineering, Babol University of Technology, Babol, Iran Department of Mechanical Engineering, Central Tehran Branch, Islamic Azad University, Tehran, Iran c Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran b Received 28 October 2016; revised 19 January 2017; accepted 23 January 2017 KEYWORDS Nanofluid; Rotating cylinders; Magnetic field; KKL model; Radiation; AGM Abstract Nanofluid flow between two circular cylinders is studied in existence of magnetic field KKL model is applied for nanofluid Thermal radiation effect has been considered in energy equation AGM is selected for solving ODEs Semi analytical procedures are examined for various active parameters namely; aspect ratio, Hartmann number, Eckert number and Reynolds number Results indicate that temperature gradient enhances with rise of Ha, Ec and g but it reduces with augment of Re Velocity reduces with rise of Lorentz forces but it augments with rise of Reynolds number Ó 2017 Faculty of Engineering, Alexandria 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/) Introduction Magnetohydrodynamic free convection has several applications In recent decade, nanotechnology has been offered as novel method for heat transfer improvement MHD nanofluid free convection in a tilted enclosure has been presented by Sheremet et al [1] They showed that augment of titled angle causes convective heat transfer to enhance 3D MHD free convective heat transfer was examined by Sheikholeslami and Ellahi [2] using LBM Their results revealed that Lorentz forces causes temperature gradient to reduce Ismael et al [3] investigated Lorentz forces effect on nanofluid flow in an enclosure with moving walls Their outputs indicated that the impact of Lorentz forces reduces with change in direction * Corresponding author E-mail addresses: m.sheikholeslami1367@gmail.com (M Sheikholeslami), m.nimafar@gmail.com (M Nimafar) Peer review under responsibility of Faculty of Engineering, Alexandria University of magnetic field Sheikholeslami and Ellahi [4] utilized LBM to study Fe3O4-water flow for aim of drug delivery They concluded that the velocity gradient reduces with rise of magnetic number Influence of non-uniform Lorentz forces on nanofluid flow style has been studied by Sheikholeslami Kandelousi [5] He concluded that improvement in heat transfer reduces with rise of Kelvin forces New model for nanofluid flow was presented by Hayat et al [6] Sheikholeslami [7] studied the thermal radiation effect on nanofluid flow in a cavity with tilted elliptic inner cylinder Influence of thermal radiation on magnetohydrodynamic nanofluid motion has been reported by Sheikholeslami et al [8] They concluded that nanofluid concentration gradient augments with rise of radiation parameter Noreen et al [9] examined the motion of nanofluid in a bent channel They showed that curvature can enhance the longitudinal velocity MHD Fe3O4-water flow in a wavy cavity was examined by Sheikholeslami and Chamkha [10] Influence of magnetic field on force convection was reported by Sheikholeslami et al [11] Their outputs illustrated that higher lid velocity include more http://dx.doi.org/10.1016/j.aej.2017.01.032 1110-0168 Ó 2017 Faculty of Engineering, Alexandria 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: M Sheikholeslami et al., Nanofluid heat transfer between two pipes considering Brownian motion using AGM, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.032 M Sheikholeslami et al Nomenclature k thermal conductivity Greek symbols r electrical conductivity X constant rotation velocity g aspect ratio / nanoparticle volume fraction q fluid density a thermal diffusivity h dimensionless temperature l dynamic viscosity Eckert number dimensionless velocity specific heat capacity constant applied magnetic field Prandtl number Hartmann number Nusselt number Reynolds number dimensionless radius radiation parameter Ec và Cp B Pr Ha Nu Re rà Rd sensible Kelvin forces effect Bondareva et al [12] utilized Buongiorno’s mathematical model for magnetic field effect on transient free convection Sheikholeslami and Rokni [13] studied the effect of Lorentz forces on free convection in a semi annulus Bondareva et al [14] was utilized Heatline analysis for simulating MHD free convection in an open cavity Sheikholeslami and Vajravelu [15] investigated the nanofluid flow and heat transfer in a cavity with variable Lorentz forces Sheremet et al [16] investigated magnetic field on free convection of wavy open cavity They presented the effect of corner heater on nanofluid flow Nonlinear equations can be solved via semi analytical methods There are several semi analytical methods such as DTM [17–19], HPM [20,21], HAM [22], ADM [23,24], OHAM [25,26] and etc One of the new powerful semi analytical approaches is AGM Mirgolbabaee et al [27] used AGM for Duffing-type nonlinear oscillator They showed the accuracy of this paper Nanofluid flow and heat transfer have been simulated by several authors in recent decade [28–37,17,38–47] The chief aim of this paper is to illustrate the effect of magnetic field on nanofluid hydrothermal treatment between two pipes AGM is utilized to solve this problem The roles of the aspect ratio, radiation parameter, Reynolds number, Eckert number, Hartmann number are presented Governing formulae Laminar 2D flow in cylindrical coordinate is studied (Fig 1) The governing formulae are as follows: @ v @v v rnf vB20 @v tnf ỵ ẳv 1ị qnf @r2 r @r r2 @r 2 knf @ @T @v v @q @T r ỵ lnf ; r ẳ qCp ịnf v @r @r r @r r @r @r qr ¼ À 4re @T4 ; T ffi 4T3c T À 3T4c 3bR @y 2ị r ẳ r1 : vrị ẳ X1 r1 ; T ẳ T1 r ẳ r2 : vrị ẳ 0; T ẳ T2 3ị rịnf , qCp ịnf , qbịnf and ðqnf Þ can be introduced as follows: rnf 3ðrp =rf 1ị/ ; ẳ1ỵ rp =rf ỵ 2ị rp =rf 1ị/ rf qbịnf ẳ /ịqbịf ỵ qbịp /; qCp ịnf ẳ /qCp ịp ỵ qCp ịf /ị; qnf ẳ /qp ỵ qf À /Þ ð4Þ ðknf Þ and ðlnf Þ are obtained according to Koo–Kleinstreuer–Li (KKL) model [48]: knf ¼ 3ðkp =kf 1ị/ ỵ1 kp =kf 1ị/ ỵ kp =kf þ 2Þ sffiffiffiffiffiffiffiffiffi jb T þ 5/  10 cp;f g ðdp ; T; /Þqf qp dp g0 dp ; T; /ị ẳ a1 ỵ a2 Lndp ị þ a5 Lnðdp Þ2 þ a3 Lnð/Þ þ a4 lnðdp ịLn/ịịLnTị ỵ a6 ỵ a7 Lndp ị 5ị ỵ a10 Lndp ị2 ỵ a8 Ln/ị ỵ a9 lndp ịLn/ịị Fig Table Rf ẳ dp 1=kp;eff 1=kp ị; Geometry of the problem Rf ¼  10À8 km2 =W Thermo physical properties of water and nanoparticles [48] Pure water CuO qðkg=m3 Þ Cp ðJ=kg kÞ kðW=m kÞ dp ðnmÞ rðX mÞÀ1 997.1 6500 4179 540 0.613 18 – 29 0.05 10À10 Please cite this article in press as: M Sheikholeslami et al., Nanofluid heat transfer between two pipes considering Brownian motion using AGM, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.032 Nanofluid heat transfer between two pipes lf lnf ¼ Coefficient values CuO–Water a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 À26.593310846 À0.403818333 À33.3516805 À1.915825591 6.421858EÀ02 48.40336955 À9.787756683 190.245610009 10.9285386565 À0.72009983664 All needed coefficients and properties are illustrated in Tables and [48] The dimensionless forms of above equations are as follows: ! à @ v @v Ha2 A5 A1 @v ỵ ỵ v Re v ẳ 6ị Ã2 r @r @r A2 @r ð1 À gÞ A2 r /ị 2:5 ỵ kBrownian lf kf Pr Table The coefficient values of CuO–Water nanofluids [48] 2 @ A2 @và và @2h @h ỵ EcPr r ỵ Rd rà @rà @rà 3A4 @rÃ2 A4 @rà rà A3 @h À PrRe và à ¼ A4 @r rà ¼ g : v r ị ẳ 1; h ẳ r ẳ : v r ị ẳ 0; h ¼ ð8Þ where Present work Aberkane et al 0.8 7ị r ẳ r r v r1 rf T À T2 ;v ¼ ; g ¼ ; Ha ¼ B0 d ; ;h ¼ r2 X1 r1 r2 lf T1 À T2 lf ðqCp Þf qf Xr1 r2 qf X1 r1 ị2 ; Pr ẳ ; Ec ẳ lf qf kf qCp ịf DT q l Rd ẳ 4re T3c =bR kf ị; A1 ẳ nf ; A2 ¼ nf ; qf lf Re ¼ 0.6 V 0.4 A3 ẳ qCp ịnf knf rnf A4 ẳ ; A5 ẳ qCp ịf kf rf Nu over the hot cylinder is as follows: @h Nu ¼ ÀA4 à @r r ẳg 0.2 0.2 0.4 r* 0.6 0.8 9ị ð10Þ Basic Idea of AGM Fig Comparison of Aberkane et al [49] and present results of velocity profile, for g = 0.5, Ha = Fig The general forms of equation with its boundary conditions are as follows: Influence of Ha; Re on velocity profile Please cite this article in press as: M Sheikholeslami et al., Nanofluid heat transfer between two pipes considering Brownian motion using AGM, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.032 M Sheikholeslami et al pk : fðu; u0 ; u00 ; ; umị ị ẳ 0; ( u ẳ uxị 11ị uxị ẳ u0 ; u0 xị ẳ u1 ; ; um1ị xị ẳ um1 uxị ẳ uL0 ; u0 xị ẳ uL1 ; um1ị xị ẳ uLm1 at at xẳ0 xẳL 12ị We assume that the solution of this equation is as follows: uxị ẳ n X xi ẳ a0 ỵ a1 x1 ỵ a2 x2 ỵ ỵ an xn 13ị iẳ0 The larger n makes larger accuracy of the solution By inserting Eq (13) into (11), the residual can be obtained According to boundary conditions and values of residual at boundaries, the constant parameters in Eq (12) can be obtained Fig 4 Application of AGM Firstly, we introduce the residuals: ! @ và @và Ha2 A5 A1 @v F ẳ ỵ ỵ v Re v ẳ r @r @r A2 @r ð1 À gÞ A2 r à @ @h A2 @v v @2h ỵ Rd G ẳ r ỵ EcPr r @r @r 3A4 @rÃ2 A4 @rà rà A3 @h À PrRe và à ¼ A4 @r ð14Þ We assume that the solutions of these equations are as follows: và ¼ 7 X X r ịi ; h ẳ bi r Þi i¼0 ð15Þ i¼0 Influence of Ha; Re; Ec; Rd on temperature profile Please cite this article in press as: M Sheikholeslami et al., Nanofluid heat transfer between two pipes considering Brownian motion using AGM, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.032 Nanofluid heat transfer between two pipes Fig 5 Influence of Re; Ha; Ec; Rd; g on Nusselt number According to below equations, all constant parameters can be obtained và ¼ và ðB:CÞ; h ¼ hðB:CÞ; FðB:CÞ ¼ 0; F0 B:Cị ẳ 0; GB:Cị ẳ 0; G0 B:Cị ẳ ð16Þ Results and discussion CuO-water nanofluid flow in an annulus is studied analytically using AGM Horizontal magnetic field and thermal radiation impacts are taken into account Influences of effective parameters are depicted as graphs AGM outputs are verified with those of reported by Aberkane et al [49] Fig proved the accuracy of AGM Fig depicts the influence of Ha; Re on velocity profile As inertial forces dominate viscous forces, velocity augments As electromagnetic forces dominate viscous force, velocity decreases So velocity enhances with rise of Re but it reduces with rise of Ha: Impacts of Ha; Re; Rd; Ec on temperature profile are illustrated in Fig Lorentz forces reduce the velocity and generate secondary flow, and in turn temperature reduces with rise of Ha: Temperature gradient near the inner pipe reduces with rise of Ha; Rd but it enhances with rise of Re: Higher Ec provides higher viscous dissipation, so temperature augments with increase of Ec: Fig shows the impacts of Re; Ec; Rd; g; Ha on Nu Nusselt number augments with rise of Ha; Rd but it decreases with rise of Re; Ec: Distance between the two pipes reduces with augments ofg; so rate of heat transfer enhances with increase in aspect ratio Conclusions Effect of radiation heat transfer on nanofluid heat transfer between two pipes is examined in existence of horizontal magnetic field KKL Model is selected for nanofluid Governing Please cite this article in press as: M Sheikholeslami et al., Nanofluid heat transfer between two pipes considering Brownian motion using AGM, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.032 formulae are solved by means of AGM Roles of aspect ratio, Reynolds number, Eckert number and Hartmann number are illustrated as graphs Results revealed that temperature enhances with rise of Eckert number and Reynolds number but it 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Nanofluid heat transfer between two pipes. .. al., Nanofluid heat transfer between two pipes considering Brownian motion using AGM, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.032 Nanofluid heat transfer between two pipes. .. al., Nanofluid heat transfer between two pipes considering Brownian motion using AGM, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.032 Nanofluid heat transfer between two pipes