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ORIGINAL ARTICLE

Nanofluid heat transfer between two pipes considering Brownian motion using AGM

M. Sheikholeslamia*, M. Nimafarb, D.D. Ganjic

a Department of Mechanical Engineering, Babol University of Technology, Babol, Iran b Department of Mechanical Engineering, Central Tehran Branch, Islamic Azad University, Tehran, Iran c Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran

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. © 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/).

1. 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 con-vective 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 Sheik-holeslami 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 © 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/).

Nomenclature

Ec Eckert number k thermal conductivity

v* dimensionless velocity Greek symbols

cp specific heat capacity r electrical conductivity

B constant applied magnetic field X constant rotation velocity

Pr Prandtl number g aspect ratio

Ha Hartmann number / nanoparticle volume fraction

Nu Nusselt number p fluid density

Re Reynolds number a thermal diffusivity

r* dimensionless radius e dimensionless temperature

Rd radiation parameter i dynamic viscosity

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. Sher-emet 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.

2. Governing formulae

Laminar 2D flow in cylindrical coordinate is studied (Fig. 1). The governing formulae are as follows:

d v 1 dv v

dr2 r dr r2

ÇnfvBg _ dv Pnf dr

knf d / dT

r dr dr

■ ftif

q 3ßR dy 'T -4TcT 3Tc

r — r1 : v(r) — X1r1, T — T1 r — r2 : v(r) — 0, T — T2

(r)nf, (pCp)nf, (pß)f and (pnf) can be introduced as follows: 1 3(rp/rf - 1)/

rf (rp/rf + 2)-(rp/rf - 1)/J (pß)f — 1 - /)(pß)f +(pß)p/,

(PCp)nf — /(pCp )p + (pCp)f(1 - /) ; pnf — $pp + pf(1 - /)

(knf) and (inf) are obtained according to Koo-Kleinstreuer-Li (KKL) model [48]:

3(kp/kf - 1)/

knf —

-(kp/kf - 1)/ + (kp/kf + 2)

Fig. 1 Geometry of the problem.

+ 5/ x 104cpfg(dp, T, /)pJ pb-V ppdp

g'(dp, T, /) — (at + a2Ln(dp) + a5Ln(dp)2 + a3Ln(/)

+ a4ln(dp)Ln(/))Ln(T) + (a6 + a7Ln(dp) + a10Ln(dp)2 + a8Ln(/) + a9 ln(dp)Ln(/)) Rf — dp(1/kpf - 1/kp), Rf — 4 x 10-8 km2/W

Table 1 Thermo physical properties of water and nanoparticles [48].

p(kg/m3) Cp(J/kgk) k(W/m k) dp (nm) r(X m)-1

Pure water 997.1 4179 0.613 - 0.05

CuO 6500 540 18 29 10-10

Table 2 The coefficient values of CuO-Water nanofluids [48].

Coefficient values CuO-Water

ai -26.593310846

a2 -0.403818333

a3 -33.3516805

a4 -1.915825591

a5 6.421858E-02

a6 48.40336955

a7 -9.787756683

a8 190.245610009

a9 10.9285386565

a10 -0.72009983664

Д Present work - Aberkane et al.

0.4 * 0.6

Fig. 2 Comparison of Aberkane et al. [49] and present results of velocity profile, for g = 0.5, Ha = 4.

_ if . kBrownian if

1nf "(1 - /f5 + kf X Pr

All needed coefficients and properties are illustrated in Tables 1 and 2 [48].

The dimensionless forms of above equations are as follows:

d2v* 1 dv*

Ha2 A5 1

ReA± v* dvvl=о (6)

dr*2 ' r* dr* 1(1 - g)2 A2 r*2j v e A2 v dr* (

r* dr* V dr*

A2f dv* v*\2 4 d2e EcPr—2 I —— -- + — Rd——r A4 \dr* r*J 3A4 dr*2

p » A3 * de PrRe—v*—- = 0 A4 dr*

r* = g . v* (r*) = 1, h = 1

r* = 1 : v*(r*) = 0 , e = 0

r = — ,v r2

v r\ rf T - T2

~-;g = Ha = B0d\l ; e = ^-^T ;

"1r1 r2 \ If T1 - T2

PfXr1r2 Pr = 1f(pCp)f Ec= Pf(X1r1

Rd = 4reTc/(ßRkf), A1 = Pnf, A2 = 1nf,

Pfkf " (PC)fDT' _ Inf

Pf ' ~ If'

(pC)nf A =kn£ A = — (pCp )f 4 kf; 5 rf

Nu over the hot cylinder is as follows:

Nu = -A4—

3. Basic Idea of AGM

The general forms of equation with its boundary conditions are as follows:

pk : f(u, u', u'',..., u(m)) =0; u = u(x)

(11) 4. Application of AGM

J u(x) — u0, u'(x)—u1,..., u(m 1)(x)—um^1 at x — 0 yu{x)=uLo, u!(x)—uLl,... u(m-V)(x) — uLm1 at x — L

We assume that the solution of this equation is as follows:

u(x) — — a0 + aix1 + a2x2 + ■■■ + anxn (13)

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.

Firstly, we introduce the residuals:

d!v*+1 ( Ha2 v*_ReAiv*dvl = 0

@r»2 r* @r* 1(1 _ g)2 —2 r*2jv e—2 v dr*

G- - — (r* —\ + EcPr— (—- -Y + — Rd—

r* dr* \ dr*) A4\dr* r* ) 3A4 dr*2

A3 , @e

_ PrRe—3v* — = 0

A4 dr*

We assume that the solutions of these equations are as follows:

v* = £al(r*)', 0 = £bI(r*)'

Pr = 6.8,77 = 0.5, Rd = 0.1, Re = 1, $ = 0.04 Pr = 6.8,77 = 0.5,Ec = 0.01, Re = 1, <j> = 0.04

Fig. 4 Influence of Ha, Re, Ec, Rd on temperature profile.

= 6.8,77 = 0.5,Ec = 0.01, Re =1,0 = 0.04

Pr = 6.8,Ec = 0.01, Rd =0Л,Яе=1,ф = 0.04

Fig. 5 Influence of Re, Ha, Ec, Rd, g on Nusselt number.

According to below equations, all constant parameters can be obtained.

v* = v*(B.C), 0 = 0(B.C),

F(B.C)=0, F (B.C) = 0, (16)

G(B.C) = 0, G' (B.C) = 0

5. 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. 2 proved the accuracy of AGM. Fig. 3 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. 4. 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. 5 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.

6. 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

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 reduces with augment of Hartmann number and radiation parameter. Nusselt number augments with rise of Lorentz forces but it decreases with rise of viscous dissipation.

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