Variable magnetic field (VMF) effect on the heat transfer of a half-annulus cavity filled by Fe3O4-water nanofluid under constant heat flux

Accepted Manuscript Variable Magnetic Field (VMF) Effect on the Heat Transfer of a Half-Annulus Cavity Filled by Fe3O4-Water Nanofluid Under Constant Heat Flux M. Hatami, J. Zhou, J. Geng, D. Jing PII: DOI: Reference:

S0304-8853(17)31787-0 https://doi.org/10.1016/j.jmmm.2017.10.110 MAGMA 63326

To appear in:

Journal of Magnetism and Magnetic Materials

Received Date: Revised Date: Accepted Date:

8 June 2017 16 October 2017 27 October 2017

Please cite this article as: M. Hatami, J. Zhou, J. Geng, D. Jing, Variable Magnetic Field (VMF) Effect on the Heat Transfer of a Half-Annulus Cavity Filled by Fe3O4-Water Nanofluid Under Constant Heat Flux, Journal of Magnetism and Magnetic Materials (2017), doi: https://doi.org/10.1016/j.jmmm.2017.10.110

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Variable Magnetic Field (VMF) Effect on the Heat Transfer of a Half-Annulus Cavity Filled by Fe3O4-Water Nanofluid Under Constant Heat Flux M. Hatamia, J. Zhoub, J. Gengb, D. Jingb,1 a

Department of Mechanical Engineering, Esfarayen University of Technology, Esfarayen, North Khorasan, Iran b

International Research Center for Renewable Energy, State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China

Abstract In this paper, the effect of a variable magnetic field (VMF) on the natural convection heat transfer of Fe3O4-water nanofluid in a half-annulus cavity is studied by finite element method using FlexPDE commercial code. After deriving the governing equations and solving the problem by defined boundary conditions, the effects of three main parameters (Hartmann Number (Ha), nanoparticles volume fraction (φ) and Rayleigh number (Ra)) on the local and average Nusselt numbers of inner wall are investigated. As a main outcome, results confirm that in low Eckert numbers, increasing the Hartmann number make a decrease on the Nusselt number due to Lorentz force resulting from the presence of stronger magnetic field. Keywords: Variable magnetic field (VMF); Half-annulus cavity; Nanofluid; Hartmann; Nusselt. 1. Introduction Magnetic nanofluids (MNFs), as a new class of nanofluids also be called smart fluids or functional fluids, are stable colloidal suspensions of sub-domain magnetic particles (iron, nickel, cobalt, and their oxides such as magnetite) with a particle size of less than 20 nm dispersed in a

1

Corresponding author: Tel/Fax:+98-915-718-6374 E-mails: [email protected] (D. Jing) [email protected], [email protected] (M. Hatami)

1

base fluid such as water, oil, ethylene glycol, etc. In order to keep the stability of suspension, a monolayer of insoluble surfactant is usually used [1-3]. Nomenclature B magnetic induction Cp specific heat at constant pressure Ec Eckert number En heat transfer enhancement Grf Grashof number g gravitational acceleration vector

Hx , Hy H Ha k Nu Pr Ra T u,v U,V x,y X,Y

Greek symbols α thermal diffusivity φ volume fraction γ magnetic field strength at the source σ electrical conductivity µ dynamic viscosity µ0 magnetic permeability of vacuum ( = 4 π×10 −7 Tm/A) of the magnetic field υ kinematic viscosity

components intensity the magnetic field strength Hartmann number thermal conductivity Nusselt number Prandtl number ( = υf / αf ) Rayleigh number fluid temperature velocity components in the x -direction and y -direction dimensionless velocity components in the X -direction and Y -direction space coordinates dimensionless space coordinates

ψ ɵ ρ β ω and Ω Subscripts c h

stream function dimensionless temperature fluid density thermal expansion coefficient vorticity and dimensionless vorticity

ave

average

loc nf

local nanofluid

cold hot

MNFs are attracted much attention of researchers in recent years, because they can be controlled by external magnetic field (MF) and have widely range of applications such as dynamic sealing, heat dissipation, damping and doping technological materials [4-6]. Most published studies on ferrofluids have focus on enhancing heat transfer by experimental and numerical methods [7-10]. Shahsavar et al. [11] investigated the performance of convective heat transfer of ferrofluid with carbon nanotubes under laminar flow and alternating magnetic fields. Their results showed that the convective heat transfer performance of the case with magnetic field has been improved

2

about 20.5% compared to the case without magnetic field. Moreover, the effects of magnetic field were more noticeable in the hybrid nanofluids with higher volume concentrations and lower Reynolds number. Goharkhah et al. [12] studied convective heat transfer characteristic of magnetite nanofluid under constant and alternating magnetic field, experimentally. They chose three different volume fractions of φ=1，1.5，2% as coolant and the uniformly heated tube with 9.8 mm diameter and 629 mm length was chosen as the test section, and local convective coefficients was measured at developed region. Their results indicated that the convective coefficient has improve approximately 13.5% by using magnetite ferrofluid with φ=2% at Re=1200, compared to the DI-water. Laminar forced convective heat transfer of ferrofluids under oscillating magnetic field is also investigated experimentally by Yarahmadi et al. [13]. They showed that convective heat transfer through the circular tube increased after using an oscillatory magnetic field compare to without magnetic field, however, it is worth notice that heat transfer decreased after a constant magnetic field was applied. As another main outcome, the maximum enhancement was 19.8% in the local convective heat transfer by using the oscillating magnetic field compared with the case without magnetic field. Some researchers also studied the effect of magnetic field on the enhancement of nanofluids heat transfer in minichannel heat sink [14]. Their study show that with the magnetic field, the Nusselt number tends to increase, also the Nusselt number increases with increasing the magnetic field strength. Other similar investigations are presented in the literature [15-18]. There are also some numerical investigations in heat transfer of magnetic fluids presented in the following. Larimi et al. [19] researched on forced convection heat transfer in a channel under non-uniform transverse magnetic field. They studied the heat transfer, pressure loss and wall shear stress, respectively, by using two-phase mixture model and control volume technique. 3

Their results showed that external magnetic field has a significant effect on promoting the heat transfer, especially in the ribbed region. In addition, the magnetic field could increase pressure drop and surface friction coefficient along the channel, especially at low Reynolds number. Fadaei et al. [20] investigated three-dimensional forced-convection heat transfer of magnetic nanofluids in a pipe under laminar flow and equilibrium magnetization. It was found that the fluid mixing could be intensified that leads to an increase in the Nu number value along pipe length and the fully developed Nu number value can be increased by 96%. The laminar forced convection heat transfer of Fe3O4-water nanofluid in mini channels is also studied numerically in [21] which an alternating magnetic field with frequencies ranging from 0 to 10 Hz is applied to the heat transfer region. The valuable result is obtaining an optimum frequency of magnetic field for heat transfer enhancement, also magnetic field is more important at lower Re numbers. Sheikholeslami et al. [22] investigated convective heat transfer of magnetic nanofluid in existence of non-uniform magnetic field and the effects of thermophoresis and Brownian motion are taken into account by using two phase model. The results showed that Nusselt number has a reverse relation with Hartmann and Lewis numbers. In addition, the friction coefficient had direct relation with Hartmann and Squeeze numbers [23]. A similar research also studied numerically on the heat transfer and MHD boundary layer flow of nanofluids [24], their results indicated that thermal boundary layer thickness increases with Brownian motion and thermoresis parameters. Recently, several researchers investigated about magnetic fluid and heat transfer such as [25-28]. Most recently, Rashad et al. [29] studied the effect of magnetic field and internal heat generation on the free convection flow in a rectangular cavity filled with porous medium while saturated with copper nanofluid. It was found that the average Nusselt number decreases as the Hartmann number or the solid volume fraction increases, while the opposite behavior

4

occurred with the increase in magnetic field inclination angle. Results also revealed that increase in the Hartmann number can lead to decrease in maximum value of stream function, while it enhances the maximum temperature. Zhou et al. [30] investigated the heat transfer in a microchannel heat sink and tried to optimize its geometry by numerical methods. Also, Hatami and Safari [31] used the same optimization method (RSM) for optimizing the natural convection in a wavy-wall enclosure as well as Hatami et al. [32] study for a circular-wavy cavity optimization. Gibanov et al. [33] analyzed the convective heat transfer of ferrofluid under the variable magnetic field. Also, Bondareva et al. [34] based on their study investigated the effect of porous material, magnetic field and nanofluid in the same time on heat transfer of an inclined cavity as well as Sheremet et al. [35] study. By a short refer to above papers; it can see that most of the natural convection studies are investigated on the cavities due to large applications in solar collectors. Circle and annulus cavities are sections of fluid conveying tube in concentrating parabolic solar collectors while square and wavy cavities are more usable in flat plate solar collectors. In present study, the main purpose is investigation of variable magnetic field effect on the Fe3O4-water nanofluid heat transfer in a half annulus cavity due to its large applications mentioned above. The magnetic intensity is changed radially in the geometry and its effect on the nanoparticles heat transfer and motions is analyzed. 2. Problem Description In this study, as shown in Fig. 1, a half annulus 2D cavity is considered under the constant heat flux and variable magnetic field effect. A magnetic source is located by placing a magnetic wire

5

vertically on to the xy- plane at the point (a , b ) . The components of the magnetic field intensity

(H x ,H y ) and the magnetic field strength H are considered in the following forms HX =

γ 1 2 2π x − a + y − b

Hy = −

(

) (

)

2

γ 1 2 2π x − a + y − b

(

2

) (

2

H = HX + Hy =

γ 2π

)

( y − b) 2

(1)

( x − a)

(2)

1 2

( x − a ) + ( y − b)

(3)

2

where γ is the magnetic field strength at the source (the wire) and ( a , b ) is the position where the source is located (in this study it is located in center of half-annulus as shown in Fig. 1). The boundary condition of the outer wall is insulation, inner wall is constant heat flux and two side walls are in cold temperature (Tc). The nanofluid used in this analysis is Fe3O4-water with thermal properties of Table 1 and assumed to be Newtonian, incompressible and laminar flow. Using the Boussinesq approximation the governing equations for a steady incompressible twodimensional laminar nanofluid flow are [26], ∂v ∂u + =0 ∂y ∂x

 ∂u ∂u  ∂P ρnf  v + u  = − + µ nf ∂x  ∂x  ∂y

(4)

 ∂ 2 u ∂ 2u  2  2 + 2  − σ nf By u + σ nf Bx By v ∂ x ∂ y  

 ∂ 2v ∂ 2v   ∂v ∂v  ∂P ρnf  v + u  = − + ρnf βnf γ ( T − Tc ) + µ nf  2 + 2  + σ nf Bx By u − σnf Bx2v ∂x  ∂y  ∂y  ∂x ∂y 

6

(5)

(6)

( ρC )

p nf

2  ∂ 2T ∂ 2T   2 + 2  + σ nf ( uBy − vBx ) ∂y   ∂x 2 2   ∂u  2  ∂v   ∂v ∂u   + µnf 2   + 2   +  +    ∂x   ∂y ∂x     ∂x 

 ∂T ∂T  +u v  = k nf ∂x   ∂y

(7)

By = µ0 H y

where ρnf, (ρCp)nf αnf,βnf ,µnf,knf,σnf are defined as

ρnf = ρs φ + ρf (1 − φ )

(8)

( ρCP )nf = ( ρCP ) s φ + ( ρCP ) f (1 − φ )

(9)

α nf =

knf

( ρCP )nf

(10)

βnf = β f (1 − φ ) + βs φ

(11)

µ nf =

k nf kf

=

µ nf

(1 − φ )

(12)

2.5

−2φ ( k f − k s ) + 2k f + ks

(13)

φ ( k f − k s ) + 2k f + k s

σ  3  s −1 φ σ  σnf  f  = 1+ σf  σs   σs  + 2 −  − 1 φ   σ   σf   f 

(14)

The following dimensionless variables are introduced: Θ=

T − TC uR vR a b y x ,U= ,V= , a= , β= , Y = , Χ= , ′′ R R R R αf αf (q R / kf )

7

(15)

P=

pR 2 ρf ( α f )

, ( H , H x, H y ) = 2

(H, H

X

, Hy

H0

),H

0

( )

= H a, 0 =

γ 2π b

,

Using(15) , Eqs, (4)-(7) can be written in dimensionless form as ∂V ∂U + =0 ∂Y ∂X

V

(16)

 µ µ   ∂ 2U ∂ 2U  ∂U ∂U +U = + Pr  nf f   2 + 2  ∂Y ∂X ∂Y   ρnf ρf   ∂X

(17)

σ σ  ∂P , − Ha 2 Pr  nf f  ( H y2U − H x H yV ) − ∂X  ρnf ρf 

V

 µ µ   ∂ 2V ∂ 2V  ∂V ∂V +U = Pr  nf f   2 + 2  ∂Y ∂X ∂Y   ρnf ρf   ∂X

−

(18)

σ σ  β ∂P − Ha 2 Pr  nf f  ( H y2V − H x H yU ) + Ra Pr  nf ∂Y  ρnf ρf   β f

   k nf    ∂ 2Θ ∂ 2 Θ  ∂Θ ∂Θ  k f  U +V = +  ∂X ∂Y  ( ρC p )   ∂X 2 ∂Y 2  nf    ( ρC p ) f     σ nf   σ  2 f  {UH y + VH x } + Ha 2 Ec   ( ρC p )nf     ( ρC p ) f     µ nf  2 2  µ    ∂U  2  ∂V   ∂U ∂V   f  2  + Ec  + 2 + +  ∂Y   ∂X ∂Y          ( ρC p ) nf    ∂X     ( ρC p ) f  With dimensionless parameters 8

  Θ,  (19)

Ra f =

gβ f R 3 ( q′′ R k f

) , Pr = υ

(α υ ) (α µ ) Ec = ( ρC ) ( q R k ) R  f

, Ha = Rµ0 H 0

σf µf

,

(20)

f

′′

p

αf

f

f

f

f

f

2

 

The thermo-physical poperties of Fe3O4 and water are presented in Table 1. The stream function,

vorticity, local and average Nusselt numbers are difined as

Ω=

ωR 2 ψ ∂ψ ∂ψ ∂u ∂v , ψ= , ν=− , u= , ω= − + αf αf ∂x ∂y ∂y ∂x

On the cold side walls

ψ = 0.0

On all the insulated walls

∂Θ ∂n = 0.0

On the heat flux

∂Θ ∂n = −1.0

k Nuloc =  nf  kf 

Nuave =

1 π Ri

(22)

Θ = 0.0

On all walls

1  Ri Θ

∫

π

0

(21)

(23)

(24)

Nulox rdθ

3. Numerical Method

A finite element method is characterized by a variational formulation, a discretization strategy, one or more solution algorithms and post-processing procedures. A typical work out of this method involves (1) dividing the domain of the problem into a collection of subdomains, and (2) systematically recombining all sets of element equations into a global system of equations for the final calculation. In the finite element method, the solution region is considered as built up of 9

many small, interconnected sub-regions called finite elements. The solution of a general continuum problem by the finite element method always follows an orderly step-by-step process [32]. 4. Results and Discussions

As described above, the main goal of this study is finding the effect of Variable Magnetic Field (VMF) on the heat transfer of a half-annulus (see Fig. 1) filled by Fe3O4-Water nanofluid (with detail thermal properties of Table 1) using finite element method (FEM). In first step, to validate the results of FEM, a square geometry based on Sheikholeslami et al. [26] is modeled and the results are compared. As seen the value of temperatures and stream lines are in a good agreement. Furthermore, Table 2 shows a comparison of this method with FDM in different Hartmann numbers and shows a good agreement for the obtained average Nusselt number by Rashad et al. [29]. Also, Fig. 2 is presented to compare the results of stream lines and temperature contours for φ=0.04, Ha=10 and Ra=104. In addition, mesh independency study presented in Table 3 pointed out that 40×62 or 50×78 grid number is suitable for this scale with acceptable results. After this validation and confirmation, the half-annulus mesh is generated as Fig. 1-b and the solutions are performed to show the effect of different parameters on the heat transfer treatment. Fig. 3 shows the variable magnetic strength in different directions. Totally, the intensity of the magnetic is varied radially and near the source has strength force while the outer walls are under lower magnetic effect. In Fig. 4, the effect of the Hartmann number on the contours is presented which approximately is related to the magnetic intensity when φ=0.02, Ec=0.00001, Ra=103. The main difference in this figure is on the streamline values, as seen by increasing the Ha, the values on the stream lines reduced significantly while the center of the vortexes moved upward due to lower magnetic effect near the upper wall. This reduction on the 10

stream line may be due to Lorentz force resulting from the presence of the magnetic field which slowdown the fluid motion and by reduction in flows motion, the heat transfer will be reduced as Fig. 5 confirmed that. In the zoomed area on Fig. 5, it can be seen the maximum local Nusselt number is occurred when Ha=0 and by increasing Ha, Nu decreased as shown in Fig. 5. Fig. 6 and 7 are depicted to show the effect of Fe3O4 nanoparticles volume fraction on the heat transfer treatment. It is obvious that increasing the Fe3O4 nanoparticles volume fraction enhance the heat transfer due to their high thermal conductivity, so cooling process will be better and temperature contours will be decreased by increasing it. In highlighted area of Fig. 7, it is completely reasonable that in higher φ, Nusselt number has larger values. As know, Rayleigh number in convection heat transfers shows the strength of convection heat transfer due to gravity effect. Fig. 8 evaluates the Ra effect when Ha=80, Ec=0.00001, φ=0.02 and confirms that increasing in Ra makes very important effects on both temperature and stream lines values. Both temperatures and steam line values arises by increasing the Ra. From Fig. 9 outcome, it can be concluded that it makes a valuable increase on the Nusselt number. It must be mentioned in all these cases Eckert number also have a significant effect in these treatment and most of the outcomes are valid for lower (10e-5) Eckert numbers. To make a better perception on the interaction effect of these three parameters (Ha, φ and Ra) on the heat transfer, average Nusselt number is defined and the results of 24 different cases presented in Table 4 are compared in Fig. 10. From this figure, it is observed that cases number 8, 9, 10, 12, 19, 20 and 21 have larger average Nusselt number among others. By an analysis, it can be found that all these cases have the maximum nanoparticle volume fraction, so this parameter has more effect on the heat transfer in defined ranges for parameters. Fig. 11 also confirms this conclusion that the effect of φ is more than Ra due to larger slope in 3D graph. In 11

addition, this figure says increasing the Ra can improve the average Nusselt number. From Fig. 12 the comparison of φ and Ha is possible and the larger slope of φ confirms that it has larger effect also in comparison by Ha number while Ha has a vice versa effect on Nusselt number. 5. Conclusion

In this study, natural convection heat transfer of Fe3O4-water nanofluid in a half-annulus was studied under a variable magnetic field (VMF) effect by finite element method using FlexPDE commercial code. The effect of three main parameters on the local and average Nusselt number of outer wall is investigated. Hartmann Number (Ha), nanoparticles volume fraction (φ) and Rayleigh number (Ra) were the parameters under study. Outcomes show that although results depends on Eckert number, but generally increasing the φ and Ra enhance the heat transfer while increasing the Ha makes reduction in Nusselt values. Acknowledgments

The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (No. 51422604, 21276206) and the National 863 Program of China (No. 2013AA050402). This work was also supported by the China Fundamental Research Funds for the Central Universities. References

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[25] M Sheikholeslami，T Hayat，A Alsaedi. Numerical study for external magnetic source influence on water based nanofluid convective heat transfer. Int. J Heat Mass Transfer, 106(2017) 745-755. [26] M. Sheikholeslami, K. Vajravelu, Nanofluid flow and heat transfer in a cavity with variable magnetic field, Applied Mathematics and Computation 298 (2017) 272–282 [27] M. Mustafa, Junaid Ahmad Khan. Numerical study of partial slip effects on MHD flow of nanofluids near a convectively heated stretchable rotating disk. J. Mol. Liq. 2017. [28] M. Mustafa, Junaid Ahmad Khan. Effect of an alternating nonuniform magnetic field on ferrofluid flow and heat transfer in a channel. J Magn Magn Mater, 362(2014) 80-89. [29] A.M. Rashad, M.M. Rashidi, G. Lorenzini, S.E. Ahmed, A.M. Aly, Magnetic field and internal heat generation effects on the free convection in a rectangular cavity filled with a porous medium saturated with Cu–water nanofluid. Int. J. Heat Mass Transfer 104 (2017) 878–889. [30] Zhou, J., Hatami, M., Song, D., & Jing, D. (2016). Design of microchannel heat sink with wavy channel and its time-efficient optimization with combined RSM and FVM methods. International Journal of Heat and Mass Transfer, 103, 715-724. [31] Hatami, M., and H. Safari. "Effect of inside heated cylinder on the natural convection heat transfer of nanofluids in a wavy-wall enclosure." International Journal of Heat and Mass Transfer 103 (2016): 1053-1057. [32] Hatami, M., D. Song, and D. Jing. "Optimization of a circular-wavy cavity filled by nanofluid under the natural convection heat transfer condition." International Journal of Heat and Mass Transfer 98 (2016): 758-767. [33] Gibanov, Nikita S., Mikhail A. Sheremet, Hakan F. Oztop, and Osama K. Nusier.

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"Convective heat transfer of ferrofluid in a lid-driven cavity with a heat-conducting solid backward step under the effect of a variable magnetic field." Numerical Heat Transfer, Part A: Applications 72, no. 1 (2017): 54-67 [34] Bondareva, Nadezhda S., Mikhail A. Sheremet, Hakan F. Oztop, and Nidal Abu-Hamdeh. "Heatline visualization of MHD natural convection in an inclined wavy open porous cavity filled with a nanofluid with a local heater." International Journal of Heat and Mass Transfer 99 (2016): 872-881. [35] Sheremet, Mikhail Aleksandrovich, H. F. Oztop, and I. Pop. "MHD natural convection in an inclined wavy cavity with corner heater filled with a nanofluid." Journal of Magnetism and Magnetic Materials 416 (2016): 37-47

17

Tables Table 1. Thermo-physical properties of water and nanoparticles materials

Pure water Fe3O4

ρ 997.1 5200

Cp 4179 670

k 0.613 6

18

β 21×10-5 1.3×10-5

σ 0.05 25000

Table 2. Validation of the code with the results of FDM by Rashad et al. [29] for γ=π/6 and φ=0.05

Ha 0 10 50

FDM from Ref. [29] 0.5044985 0.4999705 0.4999513

19

FEM current study 0.497578 0.495633 0.495634

Table 3 Mesh independency study Grid Number Nusselt Number

20×32

30×48

40×62

50×78

6.304816

6.433254

6.458288

6.458565

20

Table 4. Different cases under study for finding the interaction between parameters

Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Rayleigh (Ra)

φ

100000 1000 1000 50500 100000 1000 50500 100000 50500 1000 50500 100000 100000 100000 50500 100000 50500 1000 1000 1000 1000 100000 1000 100000

0.02 0.02 0.02 0.02 0.05 0.05 0.05 0.08 0.08 0.08 0.05 0.08 0.02 0.02 0.05 0.02 0.05 0.02 0.08 0.08 0.08 0.08 0.02 0.08

Hartmann (Ha) Eckert Number (Ec)

20 80 20 50 50 50 50 80 50 20 50 20 80 20 80 80 20 80 80 20 80 20 20 80

21

0.001 0.00001 0.00001 0.000505 0.000505 0.000505 0.001 0.00001 0.000505 0.001 0.00001 0.00001 0.00001 0.00001 0.000505 0.001 0.000505 0.001 0.00001 0.00001 0.001 0.001 0.001 0.001

Figures

Fig. 1 Geometry, boundary conditions and generated mesh considered in this study

22

(a)

(b) Fig. 2 Validation the code for stream lines (right) and temperatures (left), a) results by Sheikholeslami et al. [26], b) present study for φ=0.04, Ha=10 and Ra=104

23

0 -0.

5

1 .8

2 .2

5

3

3

1 .5

3. 4

0 .5 1

1.

8 3 . 4 .2

-1 .

- 1. - 1 5

1. 4

1 .4

1 .8

1

-2 .

-4

0. 6

- 4 .5

0.6

2.5

1

2

1.8

(a)

(b) 1. 8 2. 2

1 .8

2. 6

4. 2 5

4 .2 2 .2

2. 6

4.6

1 .8

8

5

3.

3

2.

2

3 3. 4

3. 8

2. 6 3

1 .8

0. 2

2.2

2 .6

(c) Fig. 3 Variable Magnetic Field applied on the problem a) Hx, b) Hy and c) H

24

-2

5

5

Ha=0 0. 25

7 00 -0 .

05

-0

.0

02

5

0.

0.1

0. 0 . 1 070.0 9

55

00

0.10.12 3 0. 15

01

-0.0025

35

00

-0.0 055

9

02

00

04

0.

01

0.0

07

0. 0

.0 -0

.0 -0

6

0. 00 0

5

5

0.

0.1. 134 0

0 .0

-0 .

-0.0 0

0.

00

08

0 .0 2 0.01

0.0 02

0.

08

2

0. 0 .16 0. 1 9

-0 .0

05

0.11

00

0. 1

0.11

0.0 02

0.03

Ha=20 2

2 00 0.

8 0 .0

-0 -0.0016 .0 01

00

00.15 .1 0. 114

6 -0 .0 00 4

0 .1

0.11

0. 1 0 .0 9

01

5

.0

8

0 .0

-0

0.0014

0 .1 4

.0

0. 12 0 .1

4

06 0. . 04 00.0 3

06

01

-0

13

0 .2

001 4

0.0002

0.

8 0. 0 0.

. 002 0

08

9

0. 02 0.0 1

0.

0.0

07

00

0 .0 07

-0 .0 00 4 -0.0

0.

0.1

0.0008 0.0002

0.0 1

0

Ha=40

-0.0006

5

0

0.

3

5

00 01

0 04

-0 .0 0

0.0 1

0 .02

-0 .0 00 3

6 00 0. 0

15 0.0 00

0 .0 0.809 0 0.1 1 .1

0.

15

0.

0. 0

5

4

00

15

0 .0

0. 0

00

06 00 0.00..0 1 42

0.0 3

6

.0

0.

6

00 0

-0

0. 14

0 .0 5 . 00

-0 .

-0 .0 0 01

0. 00075

6

18

5

0 .1

0.

0.14 0.12 0.11 0. 1 .0 8 0

0 .0

13

9

7

-0.0

0

0 .0

0. 0

-0.00015 003

0.00015 03 0.00 45 0 0 0. 00.00 06

12

0.0003

0.

-0.00045

0.

8 00 0 0 .

03

0 0

Ha=80 0.

5 -0

00.0 . 09 0. 10.11 8

6E

13 0. 15

0

8

0.

0 01 2 6E -0 -0 5 .0 00 24

01

8

.00

01

-0

0.03

00

5

4

0.

2 01 00 0. 05 6E-

0. 14 18 0.

0.14 0.12 8 0.11 0 . 1 0 .0

0 6 .0 0. 0 2 1 0 .00.0

5 0. 0

-6E-0

9 0 .0 0 5 .00. 60 0. 07 . 042 0.001

6

- 0. 0

0.00 012

12

0 .0

0 .0 0. 1

E-0 5 2 60.0001 8 00 0.0 1 24 00

0

-0.

00

01 2

0

Fig. 4 Effect of Hartmann on temperature (left) and stream lines (right) when φ=0.02, Ec=0.00001, Ra=103

25

Fig. 5 Effect of Hartmann number on the local Nusselt number of outer wall

26

0.1 2

0. 12

0. 0

0. 11

07

6 0. 0 0. 02

0.0 2

00

5 0.0 4 0 . 0

0.

0 .03

00

0.0 0

18

1 0

.0

6 0. 00

00

-0 . 0

02

02 2

- 0. 0

0 14

-0 .

00

06

-0.0002

0. 0

0.14 0.1 6

0.

0 . 10.1 2 3 0. 0 08 . 1

0. 01

0 .0

01 .0 -0 06 -0 .0 0

4

2

0. 0

0..11 0

0.03

0.14 5 7 0. 1 0. 1

0 .1 1

0. 0 7 0 .0 6 0.0 5

2

00 0. 6 00 1

9

09

3

-0.0002

0.

1 0.

0. 0 0 22 0 .0 01 4

φ=0.00

φ=0.02 6

4 01 0 .0

-0.00 14

00 0.

0 .1 0.0 9 08 0.

-0.000

02

2

2 0 .0

00

0.11

0. 1

00.15 .1 0.1 14

2

01

00

.0

0 .1 4

-0

8

06

-0 .0

8

02

0. 12 0 .1

06 0 .4 03 . 0 0 0.

01

8

.0 0

13

8

0.

0 .0

5 0. 0

-0.001

-0

0.

0 .0

0.2

1

6

9

0. 02 0.01

0

. 07

00

00

0. 0 07

0. 0

0.1

0.

0.

1 00 0. 0.0022

0.01

0. 12 0. 0 0.1 9

0 .0 4

5

06 0. 0 3 . 0 2 .001 00.

3

0.00

0.13

0. 0

0 .000 4

0. 0

0.13 5 0. 1

9 00. .008

0.02 0. 01

0 .17

0. 0

0.14

0. 0 0. 0 6 5

04

008 0. 0 01 6

0. 08

0. 12

0. 07

- 0 . -0 . 0 00 00 08 04

0.0008

0.11

0. 1

0.11

φ=0.04

0.0

-0 .

0

012

00

16

-0 .00 12

φ=0.08

1

0. 0 0 . 0 00 00 2 6

0. 12

4 0 .0

0.

00

-0 . 0

2 00

0 .0 3 0 .0 2

0.0006

0 6 0. 05

-0.001

-0

.0

0.0

01

4

00

8 0 .0

7 0.

.1 7

.0 -0

0.1 8 0. 0

0.140 .10 .2 13

0 .0

07 0. 0 00..00364

0.1 50

2 6 00 0 0 .0 . 0 1 - 0 -00. 00 01 8 -0 .0

0 .1

0.

0 .0 2

2

0. 0 9

9 0. 0

0.11

006

Fig. 6 Effect of nanoparticles volume fraction on temperature (left) and stream lines (right) when Ha=20, Ec=0.00001, Ra=103

27

Fig. 7 Effect of nanoparticles volume fraction on the local Nusselt number of outer wall

28

Ra=103

01 8 0 .0 0

01

00

5 0.

01

8

05

6 E- 0

00

05

-6 E-

08 0.

0.12 0. 0

9

0 .16

0

0 2 01 4 02

-0 .

00

2

-0 .

Ra=104

6 E-

0 .0 2

0065 00. .0.01

.0 0

0

.0

1 00

5

6

2

0 E-6

0.01

0. 2

0.15

0 .0

1 0 .1 8 0.1 0.0

0.03

0. 0 5

01

12

14

0 . 04

00

000

0.

7

0.

- 0.

7 0. 0

-6 E -0 5 -0

0.13

9 0. 0

8

0 .0

0. 1

2

0 .1 1

0

6E -0 5

0

0 12

8 01

0. 0

06

0.0

2

0. 0

00.0 .098 0 .10.11

01

01 0.2 00

13 0. 15 0.

0

8

0

0 .0 3

6 .04 0.0 0 2 01 0 .00.

01 - 0.04 2 00

5

24

-0 .

0. 0

00

00 6

0 .0 0 5 .00. 60 00.07 .1042 0.0

0.

- 0. 0

6

0.14 18 0.

0.14 0.1 2 8 0.11 0. 1.09 0. 0 0

0 .1

-0.00-0. 0 181 2 006 00 -0 .

06

18

0 0. 0

0.00

0.

- 0. 0

01 2

0.0006

-0.

000

6

0

Ra=105

0

0.1 2

6 0.00

0. 0

24 0.0 18

0. 01

13 0.

15 0.

00.1. 3 00. 012 .0 89 0.10.11

12

0

0 0

12

0 .0 6

8 0 .1

2

0.

2

0. 15 0.1 0. 1 40 1 .1

0 .1

0. 1 5

7 0.1

8

0 0.0 3 0.02

24

-0.1

4 0 .2

8

0. 04

8 0. 0

.1

00..00 56

0.

0 .1 9 0

06

-0 .

0

0 .1 3

16

-0 .

-0

7

-0.12

6 0. 0 8 0 .1

0.1 8

1

0 .0

0. 1

0 .0 4 0.0 3 0.01

2 0.

6

6

0. 0 0 9 . 0. 0 1 8

12

00 -0.

-0 .0

0.14 3

0.1

0.

- 0. 0

0.006

0.15

Ra=106

0

0 .0

0 .0 3

8

6 0. 0 040.0 2 1 00..0

5

01

0. 0

06

0.0 0 4 .0 6 0.00.0 12

0. 14

0.14 0. 12 08 0. 1 0.

5

0. 1 6 0. 18

-0 .0

0.11

0 .0

- 0.

9

7

-0.01-0. 0 2 06 -0 - 0. .01 02 8 4

6

2

0 .0

0 .0

0 0 .0 8 1 0 0.

- 0. 1

2

- 0. 0

0.06

6

0

Fig. 8 Effect of Rayleigh number on temperature (left) and stream lines (right) when Ha=80, Ec=0.00001, φ=0.02

29

Fig. 9 Effect of Rayleigh number on the local Nusselt number of outer wall

30

20 18

Average Nusselt Number

16 14 12 10 8 6 4 2 0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1617 18 19 20 21 22 23 24

Case Number

Fig. 10 Effect of different cases on the average Nusselt number of inner wall

31

Fig. 11 Interaction between the parameters on average Nusselt number, a) Ha and φ, b) Ra and γ

32

Fig. 12 Interaction between the parameters on average Nusselt number, a) Ha and φ, b) Ra and γ

33

Tables Caption Table 1.

Thermo-physical properties of water and nanoparticles materials

Table 2.

Validation of the code with the results of FDM by Rashad et al. [29] for γ=π/6 and φ=0.05

Table 3.

Mesh independency study

Table 4.

Different cases under study for finding the interaction between parameters

34

Figures Caption Fig. 1

Geometry, boundary conditions and generated mesh considered in this study

Fig. 2

Validation the code for stream lines (right) and temperatures (left), a) results by Sheikholeslami et al. [26], b) present study for φ=0.04, Ha=10 and Ra=104 Variable Magnetic Field applied on the problem a) Hx, b) Hy and c) H

Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 7

Effect of Hartmann on temperature (left) and stream lines (right) when φ=0.02, Ec=0.00001, Ra=103 Effect of Hartmann number on the local Nusselt number of outer wall Effect of nanoparticles volume fraction on temperature (left) and stream lines (right) when Ha=20, Ec=0.00001, Ra=103 Effect of nanoparticles volume fraction on the local Nusselt number of outer wall

Fig. 9

Effect of Rayleigh number on temperature (left) and stream lines (right) when Ha=80, Ec=0.00001, φ=0.02 Effect of Rayleigh number on the local Nusselt number of outer wall

Fig. 10

Effect of different cases on the average Nusselt number of inner wall

Fig. 11

Interaction between the parameters on average Nusselt number, a) Ha and φ, b) Ra and γ Interaction between the parameters on average Nusselt number, a) Ha and φ, b) Ra and γ

Fig. 8

Fig. 12

35

Highlights

•

Effect of a variable magnetic field (VMF) on the natural convection is studied.

•

Fe3O4-water nanofluid in a half-annulus cavity is considered as the working fluid.

•

Finite element method using FlexPDE commercial code is applied on the problem.

•

Increasing the Hartmann number makes a decrease on the Nusselt number.

36