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ORIGINAL ARTICLE

Ferrofluid convective heat transfer under the influence of external magnetic source

M. Sheikholeslamia*, D.D. Ganjib

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

Received 4 September 2016; revised 28 October 2016; accepted 6 November 2016

KEYWORDS

Nanofluid; Natural convection; Magnetic source; CVFEM; Sinusoidal wall

Abstract Ferrofluid convective heat transfer in a cavity with sinusoidal cold wall is examined under the influence of external magnetic source. The working fluid is Fe3O4-water nanofluid. Single phase model is used to estimate the behavior of nanofluid. Vorticity stream function formulation is utilized to eliminate pressure gradient source terms. New numerical method is chosen namely Control volume base finite element method. Influences of Rayleigh, Hartmann numbers, amplitude of the sinusoidal wall and volume fraction of Fe3O4 on hydrothermal characteristics are presented. Results indicate that temperature gradient enhances as space between cold and hot walls reduces at low buoyancy force. Lorentz forces cause the nanofluid velocity to reduce and augment the thermal boundary layer thickness. Nusselt number augments with rise of buoyancy forces but it decreases with augment of Lorentz forces.

© 2016 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

New kinds of fluid were required to reach more efficient performance in new days. Nanofluid was proposed as innovative way to enhance heat transfer. Teamah and Shehata [1] reported the Lorentz forces effect on free convection in trapezoidal enclosure. Hsiao [2] investigated electrical MHD nanofluid flow over a plate. He utilized FDM for simulation. Sheikholeslami and Ganji [3] presented various applications of nanofluid in their review paper. Sheremet et al. [4] simulated the unsteady

* Corresponding author.

E-mail addresses: mohsen.sheikholeslami@yahoo.com, m.sheikhole-slami1367@gmail.com (M. Sheikholeslami), ddg_davood@yahoo.com (D.D. Ganji).

Peer review under responsibility of Faculty of Engineering, Alexandria University.

MHD flow in an enclosure. They used FDM to simulate that paper. Kandasamy et al. [5] examined the reaction of nanofluid versus chemical reaction. Ahmad and Mustafa [6] investigated the rotating nanofluid flow induced by an exponentially stretching. Their results revealed that temperature gradient reduces with augment of angular velocity. Awais et al. [7] simulated the slip effect on nanofluid motion in the presence of magnetic field. Hussein et al. [8] analyzed the natural convection of nano-fluid in T-shaped cavity. They concluded that temperature gradient decreases with augment of heat source length. Radiation heat transfer over a sensor surface has been studied by Hamzah et al. [9]. They indicated the 30% augmentation in Nusselt number with use of nanofluid.

Selimefendigil and Oztop [10] examined nanofluid conjugate conduction-convection mechanism in a titled cavity. They proved that temperature gradient rises with augment of Grashof number. Sheikholeslami and Ellahi [11] selected

http://dx.doi.org/10.1016/j.aej.2016.11.007

1110-0168 © 2016 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

B magnetic induction a thermal diffusivity

Ec Eckert number X&W dimensionless vorticity & stream function

H the magnetic field strength 0 dimensionless temperature

~g gravitational acceleration vector P fluid density

Nu Nusselt number l dynamic viscosity

Ha Hartmann number r electrical conductivity

T fluid temperature

Ra Rayleigh number Subscripts

V, U vertical and horizontal dimensionless velocity nf nanofluid

Y, X vertical and horizontal space coordinates f base fluid

loc local

Greek symbols c cold

b thermal expansion coefficient

lo magnetic permeability of vacuum

LBM to simulate Lorentz force influence on nanofluid temperature distribution. They depicted that temperature gradient reduces with augment of Hartmann number. Kefayati [12] considered second law analysis for nanofluid laminar natural convection in a permeable enclosure. He proved that irreversibilities augment as Rayleigh number enhances. MHD nanofluid free convective hydrothermal analysis in a tilted wavy enclosure was presented by Sheremet et al. [13]. Their results illustrated that change of titled angle causes con-vective heat transfer to enhance. Influence of non-uniform Lorentz forces on nanofluid flow style has been studied by Sheikholeslami Kandelousi [14]. He concluded that improvement in heat transfer reduces with rise of Kelvin forces. Sheik-

holeslami et al. [15] examined about the impact of radiation of nanofluid free convective heat transfer in existence of magnetic field. They showed that rate of heat transfer decreases with augment of Lorentz forces. Malvandi et al. [16] analyzed the fluid flow on a sheet. They presented the thermodynamic optimization of this problem. Several authors investigated about nanofluid heat transfer augmentation [17-33,1].

The purpose of this article was to investigate impact of magnetic source on hydrothermal behavior of nanofluid in a cavity with sinusoidal cold wall. CVFEM is chosen to simulate this paper. Effects of Rayleigh and Hartmann numbers, volume fraction of Fe3O4 on hydrothermal treatment are considered.

(c) Hy (x,y )

Figure 2 Contours of the (a) magnetic field strength H; (b) magnetic field intensity component in x direction Hx; (c) magnetic field intensity component in y direction Hy.

Table 1 Thermo-physical properties of water and nanoparticles.

P (kg/m3) Cp (J/kg K) k (W/m K) dp (nm) r (X m)"1

Pure water 997.1 4179 0.613 - 0.05

Fe3O4 5200 670 6 47 25,000

Table 2 Comparison of the average Nusselt number Nuave along lid wall for different grid resolutions at Ra = 105 , a = 0.3, / = 0.04, Ha = 20, Ec = 10~5 and Pr = 6.8.

51 x 151 61 x 181 71 x 211 81 x 241 91 x 271 101 x 301

2.759311 2.766715 2.769861 2.775131 2.776103 2.779106

Figure 3 Comparison of center line temperature between the present results and numerical results by Khanafer et al. [36] Gr = 104, / = 0.1 and Pr = 6.8(Cu-Water).

2. Problem statement

An enclosure with hot inner walls is considered. Fig. 1(a) depicts the boundary conditions. To obtain the shape of the left sinusoidal wall profile, the following equation should be used:

1 + sin (nH - P/2))}

Magnetic source has been considered as shown in Fig. 2. Hx, Hy, H can be calculated as follows [34]:

Hx -(y - b)[(a - x)2 + (b - y)] 2P;

_ _ _ 2 _ 2 -1

Hy-(a - x)[(a - x) +(b - y)] 2n,

H - \[hI + H2.

3. Simulation method

3.1. Governing formulation

2D laminar nanofluid flow and forced convective heat transfer are taken into account. The governing PDEs can be considered as follows:

d2u d2u\ dP nl

!f + @X-2) - dX - rnfB2U + fByv

d v d2v\ dP

du dv o

du du \ ^

dfxu + äyv) -(qnf)

pnf{lrxu + dTyv) -+fdx-+W2) - dy + By"nfBxu

- BxrnfBxv + (T - Tt)ßnfgpnf (pCp)f(v dT + u I) - OnfiBxV - Byu)2 + knf(fi + 0)

+f 2(fx)2 + 2( IX)2 + ( t + dy)2},

B - 10 H.

pnf ; (pCp)nf, a.nf; ßnf, Inf ; knf and o„f are calculated as

Pnf - Pf(1 - /)+Ps

(PCP)nf - (PCP)f(1 - /) + (PCp)s/;

anf-TPCp)nf ;

ßnf - ßf(1 - /)+ßs/.

(8) (9)

(10) (11)

Table 3 Average Nusselt number versus at different Grashof number under various strengths of the magnetic field at Pr = 0.733.

Ha Gr - 2 x 104 Gr - 2 x 105

Present Rudraiah et al. [37] Present Rudraiah et al. [37]

0 2.5665 2.5188 5.093205 4.9198

10 2.26626 2.2234 4.9047 4.8053

50 1.09954 1.0856 2.67911 2.8442

100 1.02218 1.011 1.46048 1.4317

ARTICLE IN PRESS

Ferrofluid convective heat transfer 5

Figure 4 Isotherms (up) and streamlines (down) contours for different values of Hartmann number and amplitude of the sinusoidal wall when Ra = 103, / = 0.04.

ARTICLE IN PRESS

6 M. Sheikholeslami, D.D. Ganji

a=0.1 a=0.3

Figure 5 Isotherms (up) and streamlines (down) contours for different values of Hartmann number and amplitude of the sinusoidal wall when Ra = 104, / = 0.04.

I 0.95

I 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15

I 0.95 I 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 I 0.15

24 23 22 21 20 19

I 0.95 I 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 — 0.25

I—I 0.2

I—I 0.15

H °.l

I—I 0.05

15 14 13

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

13 12 11 10 9 8 7

5 4 3 2 1

Figure 6 Isotherms (up) and streamlines (down) contours for different values of Hartmann number and amplitude of the sinusoidal wall when Ra = 105, / = 0.04.

If —

knf —

(1 - /)"

ks + 2kf — 2/(kf — ks ks + 2kf + /(kf — ks)

— 1 +

-21 — — 1

Dimensionless parameters are defined as follows:

_ (H,Hx,Hy)

Ho t__p

(X; Y) —

(a, b)—^, (H, Hx, Hy U_uL V_vL 0_ T—Tc P_ -

U f ' V f ' H (q"L/kf) ' P Pf(af/L)2

So final equations are as follows:

@U @V_ 0 @X + OY — 0'

TT dU Tr „

— U H--V= Pr

@XU ^ BY

'lnf/1.

Pnf=Pf-

@2u d2 u a y2 + ax-2

r nf/ r f

Pnf/Pf

h2u—HxHyv) — '

v@v i n@v — Pr(aV i a2A [fyl v ar + U ax — Pr{by2 + ax2) [Pnf/Pfj

—Ha- Prf ] (HlV — HxHyU) — @Y + Ra Pf,

v3H 4- t T3H— i bH

v by ^ U ax — I by2 ^ ax2

'(PCP)n ,

(pCp)f

+Ha2 Ec

rnf rf

(PCP)nf

.(pCpf.

{VHx — UHyg

(PCP)nf

.(PCP 'f.

Ec{ 2(S)2 + 2(|

and dimensionless parameters are

Raf — gbL

\q"L/kf)/{a.fOf), Prf = Of/a.f, Ha = L^^y^Jlf; Ec = f)/[(PCp)fDTL2]

The thermo-physical properties of Fe3O4 and water are presented in Table 1 [34]. Pressure gradient source terms discard by vorticity stream function.

X —-' W — - ' x — — — + — ' (u' v)— hf-'

a.f a.f ay ox \Oy ax

According to Fig. 1, boundary conditions are

on left wall 0 — 0.0 on all walls W — 0.0 on right wall 0 — 1.0

on other walls @0 — 0.0

Nuiocai' Nuave along left wall are as follows:

— 14

— 12

1.8 1.6

Ô 0.4

- Ha = 0

■ s \ '—^ ^ ----Ha = 5

's. \ — -------Ha = 10

----------- Ha = 20

- ^vOs. \ Xs. \\

- Ha = 0

----Ha = 5

-------Ha = 10

----------- Ha = 20

"......... i ......

- - Ha = 0

- ----Ha = 5

_ -------Ha = 10

1 ----------- Ha 20

Figure 7 Effects of Hartmann number, amplitude of the sinusoidal wall and Rayleigh number on local Nusselt number Nuioc along right wall.

Ra = 10 ,0 = 0.04

Ra = 105 ,a = 0.3

log(Ra)

Ra = 105,Ha = 20

a = 0.3,Ha = 20

Nu.....

log(Ra)

log(Ra)

0 = 0.04,Ha = 20

0 = 0.04,a = 0.3

Ra = 105,0= 0.04

Ra = 105 ,Ha = 20

Figure 8 Effects of Hartmann number, amplitude of the sinusoidal wall and Rayleigh number on local Nusselt number Nuave along right wall.

M. Sheikholeslami, D.D. Ganji

log(Ra)

Ha » «»-

0 = 0.04,Ha = 20

log(Ra)

0= 0.04,a = 0.3

7Î5| |3 051391 13 085631 |3t

Ra = 105 ,a = 0.3

log(Ra)

1 = 0.3,Ha = 20

Fig. 8 (continued)

Nuloc=[k-A

kf ) dn

Nuave = ~ Nuloc dS 2 j0

3.2. Numerical procedure

Linear interpolation is utilized for approximation of variables in the triangular element which is considered as building block (Fig. 1(b)). Algebretic equations are solved via Gauss-Seidel method. More details exist in reference book [35].

4. Mesh independency and validation

For selecting the mesh size to access mesh independency outputs, different grids are tested. As shown in Table 2, a grid size of 71 x 211 should be chosen. Fig. 3 shows the accuracy of FORTRAN code for nanofluid heat transfer [36]. Table 3 illustrates that our code is validated for magnetohydrodynamic heat transfer [37].

5. Results and discussion

A cavity filled with nanofluid in existence of magnetic source is studied. Cold wall has sinusoidal shape. Various amounts of Hartmann number (Ha = 0-20), Rayleigh number

(Ra — 103, 104 and 105), amplitude of the sinusoidal wall (a = 0.1-0.3) and volume fraction of Fe3O4 (/ — 0 and 0.04) have been considered. Ec and Pr are 10-5 and 6.8, respectively.

Figs. 4-6 depict the impacts of amplitude of the sinusoidal wall, Rayleigh and Hartmann numbers on streamlines and isotherms. At a = 0.1, one main eddy appears. As amplitude of the sinusoidal wall augments, the main eddy turn into two eddies. By applying magnetic field, Lorentz forces generate. This force causes the nanofluid flow to retard. Increasing Lorentz forces causes eddies to stretch vertically. In weak buoyancy forces, the domination mode is conduction. So reducing distance between cold and hot walls, rate of heat transfer augments but opposite behavior can be seen for stronger buoyancy forces. As buoyancy forces enhances, eddies become stronger and isotherms become nonparallel together. So temperature gradient near the right wall enhances. As Lorentz force augments, the velocity reduces and this force makes isotherms to become parallel together.

Figs. 7 and 8 depict the influence of the Fe3O4 volume fraction, amplitude of the sinusoidal wall, Lorentz and buoyancy forces on Nuloc, Nuave. The correlation for Nuave corresponding to active parameters is as follows:

Nuave — -2.85 + 9.12a + 1,33(log(Ra)) - 4.34/ + 1.79 x 10-4Ha - 2.6a(log(Ra)) - 0.13a/ - 1.6 x 10-16aHa + 1.9(log(Ra))/- 1.15 x 10-17(log(Ra))Ha

- 1.18 x 10-15/Ha + 3.5a2 - 7.5 x 10-4(log(Ra))2

- 0.45/2 - 8.9 x 10-6Ha2

As buoyancy forces augment, temperature gradient near the right wall enhances and in turn Nuave enhances with augment of Rayleigh number. Enhancing Lorentz forces leads to reduce Nuave due to domination of conduction mode. Adding Fe3O4 nanoparticle into water makes Nusselt number to enhance. Nuave enhances with rise of a in low Ra but opposite changes occur for high Ra.

6. Conclusions

Ferrofluid heat transfer in a cavity with sinusoidal cold wall is presented under the impact of external magnetic field. CVFEM has been chosen as simulation method. Isotherms and streamlines are depicted for different values of volume fraction of Fe3O4, amplitude of the sinusoidal wall, Rayleigh and Hartmann numbers. Results indicate that impact of adding Fe3O4 is more sensible for lower buoyancy forces. Temperature gradient enhances with rise of amplitude of the sinusoidal wall when buoyancy force is weak. Temperature augments with rise of Lorentz forces but it reduces with augment of volume fraction of Fe3O4 and buoyancy forces.

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