nanofluid heat transfer between two pipes considering brownian motion using agm

Sheik-holeslami and Vajravelu [15] investigated the nanofluid flow and heat transfer in a cavity with variable Lorentz forces.. [5]Mohsen Sheikholeslami Kandelousi, Effect of spatially var

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

bDepartment 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 equa-tion 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 andg 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/)

1 Introduction

Magnetohydrodynamic free convection has several

applica-tions 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

of magnetic field Sheikholeslami and Ellahi [4] utilized LBM

to study Fe3O4-water flow for aim of drug delivery They con-cluded 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 pre-sented by Hayat et al [6] Sheikholeslami [7] studied the ther-mal 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 aug-ments with rise of radiation parameter Noreen et al [9] exam-ined 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

* Corresponding author

(M Sheikholeslami),m.nimafar@gmail.com(M Nimafar)

Peer review under responsibility of Faculty of Engineering, Alexandria

University

H O S T E D BY

Alexandria University Alexandria Engineering Journal

www.elsevier.com/locate/aej www.sciencedirect.com

http://dx.doi.org/10.1016/j.aej.2017.01.032

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

Sheik-holeslami 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

meth-ods 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

sim-ulated by several authors in recent decade [28–37,17,38–47]

The chief aim of this paper is to illustrate the effect of

mag-netic 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,

Eck-ert number, Hartmann number are presented.

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

tnf

@2

v

@r2þ 1 r

@v

@r 

v

r2

 rnfvB2

knf

r

@

@r r

@T

@r

þ lnf

@v

@r 

v r

 @qr

@r ¼ ðqCpÞnfv @T

@r ;

qr¼  4re

3bR

@T4

@y ; T

4ffi 4T3

cT  3T4

r ¼ r1: vðrÞ ¼ X1r1; T ¼ T1

r ¼ r2: vðrÞ ¼ 0; T ¼ T2

ð3Þ ðrÞnf, ðqCpÞnf, ðqbÞnf and ðqnfÞ can be introduced as follows:

rnf

rf

¼ 1 þ 3 ðrp=rf 1Þ/

ðrp=rfþ 2Þ  ðrp=rf 1Þ/ ; ðqbÞnf¼ 1  /ÞðqbÞfþ ðqbÞp/;

ðqCpÞnf¼ /ðqCpÞpþ ðqCpÞfð1  /Þ; qnf¼ /qpþ qfð1  /Þ

ð4Þ

ðknfÞ and ðlnfÞ are obtained according to Koo–Kleinstreuer–Li (KKL) model [48] :

knf¼ 3 ðkp=kf 1Þ/

ðkp=kf 1Þ/ þ ðkp=kfþ 2Þ þ 1

þ 5/  104

cp;fg0ðdp; T; /Þqf

ffiffiffiffiffiffiffiffiffi

jbT

qpdp

s

g0ðdp; T; /Þ ¼ ða1þ a2Ln ðdpÞ þ a5Ln ðdpÞ2

þ a3Ln ð/Þ

þ a4ln ðdpÞLnð/ÞÞLnðTÞ þ ða6þ a7Ln ðdpÞ

þ a10Ln ðdpÞ2

þ a8Ln ð/Þ þ a9ln ðdpÞLnð/ÞÞ

Rf¼ dpð1=kp;eff 1=kpÞ; Rf¼ 4  108km2=W

ð5Þ

Fig 1 Geometry of the problem

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

Nomenclature

v dimensionless velocity

Cp specific heat capacity

B constant applied magnetic field

Pr Prandtl number

Nu Nusselt number

Re Reynolds number

r dimensionless radius

Rd radiation parameter

k thermal conductivity Greek symbols

r electrical conductivity

X constant rotation velocity

g aspect ratio / nanoparticle volume fraction

a thermal diffusivity

h dimensionless temperature

l dynamic viscosity

lnf¼ lf

ð1  /Þ2:5þ kBrownian

kf  lf

Pr All needed coefficients and properties are illustrated in Tables

1 and 2 [48] The dimensionless forms of above equations are as follows:

@2

v

@r2þ 1

r

@v

@r Ha2 ð1  gÞ2

A5

A2þ 1

r2

!

v Re A1

A2v

@v

@r¼ 0 ð6Þ 1

r

@

@r r@h

@r

þ EcPr A2

A4

@v

@r v

r

þ 4 3A4

Rd @2h

@r2

 PrRe A3

A4

v@h

r¼ g : vðrÞ ¼ 1; h ¼ 1

where

r¼ r

r2

; v¼ v

X1r1

; g ¼ r1

r2

; Ha ¼ B0d

ffiffiffiffi

rf

lf

r

; h ¼ T  T2

T1 T2

;

Re ¼ qfXr1r2

lf Pr ¼ lfðqCpÞf

qfkf ; Ec ¼ qfðX1r1Þ2

ðqCpÞfDT ;

Rd ¼ 4reT3=ðbRkfÞ; A1¼ qnf

qf; A2¼ lnf

lf ;

A3¼ ðqCpÞnf ðqCpÞf A4¼ knf

kf; A5¼ rnf

rf

ð9Þ

Nu over the hot cylinder is as follows:

Nu ¼ A4@h

@r





3 Basic Idea of AGM

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

Table 2 The coefficient values of CuO–Water

nanofluids[48]

0

0.2

0.4

0.6

0.8

1

Present work Aberkane et al.

r*

V

Fig 2 Comparison of Aberkane et al.[49]and present results of

velocity profile, forg = 0.5, Ha = 4

Fig 3 Influence of Ha; Re on velocity profile

pk: fðu; u0; u00; ; uðmÞÞ ¼ 0; u ¼ uðxÞ ð11Þ

u ðxÞ ¼ u0; u0ðxÞ ¼ u1; ; uðm1ÞðxÞ ¼ um1 at x ¼ 0

uðxÞ ¼ uL 0; u0ðxÞ ¼ uL 1; uðm1ÞðxÞ ¼ uL m1 at x ¼ L

(

ð12Þ

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

u ðxÞ ¼ X

n

i¼0

aixi¼ a0þ a1x1þ a2x2þ    þ anxn ð13Þ

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.

4 Application of AGM Firstly, we introduce the residuals:

F ¼ @

2

v

@r2þ 1

r

@v

@r Ha2 ð1  gÞ2

A5

A2þ 1

r2

!

v Re A1

A2v

@v

@r¼ 0

G ¼ 1

r

@

@r r@h

@r

þ EcPr A2

A4

@v

@r v

r

þ 4 3A4Rd

@2h

@r2

 PrRe A3

A4

v@h

@r¼ 0

ð14Þ

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

v¼ X

7

i¼0

aiðrÞi; h ¼ X

7

i¼0

Fig 4 Influence of Ha; Re; Ec; Rd on temperature profile

According to below equations, all constant parameters can be

obtained.

v¼ vðB:CÞ; h ¼ hðB:CÞ;

F ðB:CÞ ¼ 0; F0ðB:CÞ ¼ 0;

G ðB:CÞ ¼ 0; G0ðB:CÞ ¼ 0

ð16Þ

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

param-eters 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: Dis-tance between the two pipes reduces with augments of g; 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 mag-netic field KKL Model is selected for nanofluid Governing Fig 5 Influence of Re; Ha; Ec; Rd; g on Nusselt number

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

radia-tion parameter Nusselt number augments with rise of Lorentz

forces but it decreases with rise of viscous dissipation.

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