Magnetic field effects on natural convection flow of a nanofluid in a horizontal cylindrical annulus using Lattice Boltzmann method

International Journal of Thermal Sciences 64 (2013) 240e250

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Magnetic field effects on natural convection flow of a nanofluid in a horizontal cylindrical annulus using Lattice Boltzmann method Hamid Reza Ashorynejad a, Abdulmajeed A. Mohamad b, *, Mohsen Sheikholeslami c a

Faculty of Mechanical Engineering, University of Guilan, Rasht, Islamic Republic of Iran Dept of Mechanical Engineering, CEERE, University of Calgary, Calgary, Canada c Faculty of Mechanical Engineering, Babol University of Technology, Babol, Islamic Republic of Iran b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 12 February 2012 Received in revised form 9 August 2012 Accepted 9 August 2012 Available online 27 September 2012

Effect of static radial magnetic field on natural convection heat transfer in a horizontal cylindrical annulus enclosure filled with nanofluid is investigated numerically using the Lattice Boltzmann method (LBM). The inner and outer cylinder surfaces are maintained at the different uniform temperatures. The surfaces are non-magnetic material. The investigation is carried out for different governing parameters namely, Hartmann number, nanoparticle volume fraction and Rayleigh number. The effective thermal conductivity and viscosity of nanofluid are calculated using the MaxwelleGarnetts (MG) and Brinkman models, respectively. The results reveal that the flow oscillations can be suppressed effectively by imposing an external radial magnetic field. Also, it is found that the average Nusselt number is an increasing function of nanoparticle volume fraction and Rayleigh number, while it is a decreasing function of Hartmann number. Ó 2012 Elsevier Masson SAS. All rights reserved.

Keywords: Radial magnetic field Nanofluid Natural convection in annulus Lattice Boltzmann method

1. Introduction The study of magnetic field effects has attracted attentions of engineers and sciences due to its wide industrial applications, such as in the polymer industry (where one deals with stretching of plastic sheets) and in metallurgical process, where hydro-magnetic techniques are being used. To be more specific, it may be pointed out that many material processes involve the cooling of continuous strips or filaments by drawing them through a quiescent fluid. In all these cases, the properties of the final product depend to a great extent on the rate of cooling by drawing strips in an electrically conducting fluid subject to a magnetic field and the characteristic desired in the final product. Ozoe and Okada [1] reported a threedimensional numerical study investigating the effect of the magnetic field path in a cubical enclosure. Schwab and Stierstadt [2] experimentally examined the convective heat transfer in a horizontal layer of magnetic fluid which is known as the Benard problem. It is possible to control convection of a magnetic fluid by varying the applied magnetic field. Rudraiah et al. [3] investigated numerically the effect of magnetic field on natural convection in a rectangular enclosure. They found that the magnetic field

* Corresponding author. E-mail addresses: [email protected] (H.R. Ashorynejad), [email protected] (A.A. Mohamad), [email protected] (M. Sheikholeslami). 1290-0729/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ijthermalsci.2012.08.006

decreases the rate of heat transfer. Sathiyamoorthy and Chamkha [4] used different thermal boundary conditions to examine the steady state, laminar, two-dimensional natural convection in the presence of inclined magnetic field in a square enclosure filled with a liquid gallium. They found that the heat transfer decreases with an increase in the magnetic field strength and that the vertically and horizontally applied magnetic fields suppress the rate of heat transfer differently. Ece and Buyuk [5] examined the steady state, laminar natural convection flow in the presence of a magnetic field in an inclined rectangular differentially heated enclosure. They found that the magnetic field suppressed the convective flow and the heat transfer rate. Oztop et al. [6] analyzed magnetohydrodynamic buoyancy induced heat transfer and fluid flow in a nonisothermally heated square enclosure using the finite volume method. The bottom wall of the enclosure was heated and cooled periodically (a sinusoidal function) and the top wall was cooled isothermally. Their numerical results showed that the rate of heat transfer decreases by increasing the Hartmann number and amplitude of the sinusoidal function. Buoyancy induced convection in a rectangular cavity with a horizontal temperature gradient in a strong, uniform magnetic field is investigated by Aleksandrova and Molokov [7]. They found that the flow pattern differs significantly by considering the magnetic field orientations. Most of the studies on the natural convection in enclosures with the magnetic effects have considered the electrically conducting fluid with a low thermal conductivity. This, in turn, limits the enhancement of heat

H.R. Ashorynejad et al. / International Journal of Thermal Sciences 64 (2013) 240e250

Nomenclature B0 cs ea Fext feq k gy geq i Ha heq i k Nuave Pr r Ri, Ro Ra T u,v

magnetic flux density speed of sound in lattice scale discrete lattice velocity in direction external force equilibrium distribution acceleration due to gravity equilibrium distribution pffiffiffiffiffi Hartmann number ð ¼ B0 ðRo  Ri Þ= yhÞ equilibrium distribution thermal conductivity (W/m K) average Nusselt number Prandtl number (¼y/a) radial coordinate radius of the inner and outer cylinders Rayleigh number (¼gbDT(RoRi)3/an) fluid temperature velocity components in the x-direction and y-direction

Greek symbols angular coordinate

g

transfer in the enclosure particularly in the presence of the magnetic field. Nanofluids with enhanced thermal characteristics have widely been examined to improve the heat transfer performance of many engineering applications. Khanafer et al. [8] firstly conducted a numerical investigation on the heat transfer enhancement due to adding nanoparticles in a differentially heated enclosure. They found that the suspended nanoparticles substantially increase the rate of heat transfer at any given Grashof number. Ghasemi et al. [9] presented the results of a numerical study on natural convection heat transfer in an inclined enclosure filled with a watereCuO nanofluid. They found that the heat transfer rate is maximized at a specific inclination angle depending on Rayleigh number and solid volume fraction. Abu-Nada et al. [10] investigated natural convection heat transfer enhancement in horizontal concentric annuli filled by nanofluid. They found that for low Rayleigh numbers, nanoparticles with higher thermal conductivity enhance the rate of heat transfer. As discussed earlier, the magnetic field results in the decrease of convective circulating flows within the enclosures filled with electrically conducting fluids, consequently reduces the rate of heat transfer. The addition of nanoparticles to the fluid can improve its thermal performance and enhance the heat transfer mechanism in the enclosure. In some engineering applications, such as the magnetic field sensors, the magnetic storage media and the cooling systems of electronic devices, enhanced heat transfer is desirable, whereas the magnetic field weakens the convection flow field. In order to improve the heat transfer performance of such devices, the use of nanofluids with higher thermal conductivity may be considered as a promising solution. Lattice Boltzmann method is emerged as a powerful tool to simulate fluid flow and heat and mass transfer. There are two Lattice Boltzmann models for simulating MHD flows: the multispeed (MS) model and multi-distribution function (MDF) model. In MS model the equilibrium distribution is transformed in order not only to include the magnetic field force, but also to be equal to the magnetic field vector on a different vector base. This is achieved by allowing an extra degree of freedom and by defining supplementary vectors on each base vector. Combining the base and the supplementary vectors the two vector bases are defined for the momentum and the magnetic field [11,12]. In the MDF model

h Ui wi

l a

4

g m y s q r b

241

magnetic resistivity weighting factor of magnetic field weighting factor aspect ratio (¼Ro/Ri) thermal diffusivity volume fraction of the nanoparticles angle of turn of the semi-annulus enclosure dynamic viscosity kinematic viscosity lattice relaxation time dimensionless temperature fluid density thermal expansion coefficient

Subscripts c cold h hot ave average nf nanofluid f base fluid s solid particles eq equivalent

presented by Dellar [13,14], the Lorentz force can be introduced as a point-wise force, the induction equation is also solved using an LBGK equation by introducing an independent distribution function. MDF models can improve the numerical stability. The accuracy of the MDF models has been verified by several benchmark studies [15e17]. Ehsan Fattahi et al. [18] reported a two-dimensional numerical study to investigate natural convection heat transfer in nanofluids by Lattice Boltzmann method. Later on, Nemati et al. [19] studied natural convection in a lid-driven cavity by adding nanoparticle to the working fluid. Also, they investigated the effect of magnetic field on that system [20]. They found the averaged Nusselt number increases for nanofluids when increasing the solid volume fraction, while, in the presence of a high magnetic field, this effect decreases. The present study, LBM is used to examine the natural convection in a horizontal cylindrical annulus enclosure filled with a watereAg nanofluid which is subjected to a radial-applied magnetic field. The effective thermal conductivity and viscosity of nanofluid are calculated using the MaxwelleGarnetts (MG) and Brinkman models, respectively. In addition, the MDF model was used for simulating the effect of uniform magnetic field. 2. Problem definition and mathematical model 2.1. Problem statement The schematic diagram of the physical model is shown in Fig. 1. A two-dimensional natural convection in a horizontal cylindrical annulus is simulated with an inner radius Ri and an outer radius Ro. The angle, g, is measured counterclockwise from the upward vertical plane through the center of outer cylinders. The inner and outer cylinder surfaces are maintained at different uniform temperatures Th and Tc, respectively (Th > Tc). Also, it is assumed the walls are insulating with a radial magnetic field. l ¼ Ro/Ri denotes radial aspect ratio. 2.2. The Lattice Boltzmann method The LB model used here is the same as that employed by Mohamad et al. [21,22]. The thermal LB model utilizes three

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where Dt denotes lattice time step, ci is the discrete lattice velocity in the direction i, Fk is the external force in direction of lattice velocity, sv and sC denotes the lattice relaxation time for the flow and temperature fields, respectively. The kinetic viscosity, y, and the thermal diffusivity, a, are defined in terms of their respective relaxation times, i.e. y ¼ c2s ðsv  1=2Þ and a ¼ c2s ðsC  1=2Þ, respectively. Note that the limitation 0.5 < s should be satisfied for both relaxation times to ensure the positivity of viscosity and thermal diffusivity. Furthermore, the local equilibrium distribution function determines the type of problem that needs to solve. It also models the equilibrium distribution functions for flow and temperature fields respectively. In this study, the density distribution function, feq i , was modified to consider the magnetic effect:

" fieq ¼ wi r 1 þ

Fig. 1. Geometry of the problem.

ci $u 1 ðci $uÞ2 1 u2 þ  2 c4s 2 c2s c2s

# þ

  wi B2 c2 2  ðc$BÞ 2 2c2s (3)

distribution functions, f, g and B, for the flow, temperature and magnetic fields, respectively. It is used with modeling of movement of fluid particles to capture macroscopic fluid quantities such as velocity, pressure, temperature and magnetic field. In this approach, the fluid domain discretized to uniform Cartesian cells. Each cell holds a fixed number of distribution functions, which represent the number of fluid particles moving in the discrete directions. The D2Q9 model was used and values of w0 ¼ 4/9 for w14 ¼ 1/9 for jc14 j ¼ 1 and jc0 j ¼ 0 (for the static particle), pffiffiffi w5e9 ¼ 1/36 for jc59 j ¼ 2 are assigned in this model (Fig. 2(a)). The density and distribution functions i.e. the f, g and B are calculated by solving the Lattice Boltzmann equation (LBE), which is a special discretization of the kinetic Boltzmann equation. After introducing BGK approximation, the general form of lattice Boltzmann equation with external force is: For the flow field [23]:

fi ðx þ ci Dt; t þ DtÞ ¼ fi ðx; tÞ þ

i Dt h eq f ðx; tÞ  fi ðx; tÞ þ Dtci Fk sv i

(1) For the temperature field:

gi ðx þ ci Dt; t þ DtÞ ¼ gi ðx; tÞ þ

i Dt h eq gi ðx; tÞ  gi ðx; tÞ sC

(2)

eq

gi

  c $u ¼ wi T 1 þ i 2 cs

(4)

where B is the magnetic field, wi is weighting factor, cs is the speed of sound and defined by cs ¼ c/O3. In this study D2Q9 model was used for both the flow and temperature. Thus weighting factors for both are:

8 4 > > for i ¼ 0 > > 9 > > > < 1 wi ¼ for i ¼ 1  4 > >6 > > > > > 1 : for i ¼ 5  8 36

(5)

Similarly, the density equilibrium function (feq i ), for calculating the magnetic field, magnetic equilibrium function are considered as follows [13]:

    1 eq hix ¼ Ui Bx þ 2 eix uy Bx  ux By cs

Fig. 2. (a) Discrete velocity set of two-dimensional nine-velocity (D2Q9) model; (b) Curved boundary and lattice nodes.

(6)

H.R. Ashorynejad et al. / International Journal of Thermal Sciences 64 (2013) 240e250



 1 eiy ux By  uy Bx 2 cs

heq ¼ Ui By þ iy

 

(7)

For simulating magnetic field, D2Q5 model is used, which also utilized by Dellar [13,17,20]. Thus Ui , which is the weighting factor of magnetic field, is defined as [13]:

8 1 > > <3 Ui ¼ > > :1 6

i ¼ 0

for

(8) i ¼ 14

for

For solving the velocity and magnetic fields, the following equation is considered [13]:

hi ðx þ ci Dt; t þ DtÞ ¼ hi ðx; tÞ þ

i Dt h eq hi ðx; tÞ  hi ðx; tÞ sm

(9)

The magnetic resistivity, like kinetic viscosity, y, and the thermal diffusivity, a, defined in terms of its respective relaxation time h ¼ c2s ðsm  1=2Þ. In order to incorporate buoyancy force into the model, the force term in Eq. (1) needs to calculate as below in the vertical direction (y):

F ¼ 3wi gy bq

(10)

For natural convection, the Boussinesq approximation is assumed and radiation heat transfer is assumed to be negligible. To ensure that the code works in a near incompressible regime, the characqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi teristic velocity of the flow for the natural ðVnatural h bgy DTHÞ regime must be small compared with the fluid speed of sound. In the present study, the characteristic velocity selected as 0.1 of the sound speed. Finally, macroscopic variables calculate with the following formula:

r¼

X i

fi ; ru ¼

X

ci fi ; T ¼

i

X i

gi ; Bx ¼

X i

hix ; By ¼

X

hiy

(11)

i

243

shown in Fig. 3. The link between the fluid node xf and the wall of the node xw intersects the physical boundary at x b. The fraction       intersected link in the fluid region is D ¼ xf  xw =xf  xb . To calculate the post-collision distribution function ~f a ðxb ; tÞ based upon the surrounding nodes information, a ChapmaneEnskog expansion for the post-collision distribution function on the right-hand side of Eq. (1) is conducted as:

 ~f ðx ; tÞ ¼ ð1  cÞ~f x ; t þ cf * ðx ; tÞ þ 2wa r 3 e $u w a f a b a b c2 a

(12)

where

  3  fa* ðxb ; tÞ ¼ faeq xf ; t þ wa r xf ; t 2 ea $ ubf  uf ; c  ð2D  1Þ 1 ubf ¼ uff ¼ u xff ; t ; c ¼ ; if 0  D  s2 2 1 3 ð2D  1Þ 1 ð2D  3Þuf þ ubf ¼ uw ; c ¼ ; if  D  1 s  1=2 2D 2D 2 (13) In the above, ea h ea; uf is the fluid velocity near the wall; uw is the velocity of solid wall and ubf is an imaginary velocity for interpolations. 2.3.2. Curved boundary treatment for temperature Following the work of Mei et al. [26] the non-equilibrium parts of temperature distribution function can be defined as

ga ðxb ; tÞ ¼ gaeq ðxb ; tÞ þ g aneq ðxb ; tÞ

(14)

Substituting Eq. (14) into Eq. (2) leads to:



1 neq ~a ðxb ; t þ DtÞ ¼ g aeq ðxb ; tÞ þ 1  g g ðxb ; tÞ s a

(15)

s

2.3. Boundary conditions 2.3.1. Curved boundary treatment for velocity For treating velocity and temperature fields with curved boundaries, the method proposed by Guo et al. [24,25] has been used. An arbitrary curved wall separating a solid region from fluid is

Obviously, both g aeq ðxb ; tÞ and g aneq ðxb ; tÞ are needed to calculate ~a ðxb ; t þ DtÞ. In Eq. (15) the equilibrium part is defined the value of g as:

 3 gaeq ðxb ; tÞ ¼ wa Tb* 1 þ 2 ea $u*b c

(16)

Fig. 3. Comparison of the present work and (a) equivalent thermal conductivity on inner and outer cylinder with experimental data of Kuehn and Goldstein [30] for Pr ¼ 0.71 and Ra ¼ 5  105; (b) isotherms experimental study by Laboni and Guj [31] for Ra ¼ 0.9  105.

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Table 1 Thermo-physical properties of water and nanoparticles [27].

Pure water Silver (Ag)

r (kg/m3)

Cp (J/kg K)

k (W/m K)

b  105 (K1)

997.1 10,500

4179 235

0.613 429

21 1.89

where T*b is defined as a function of Tb1 ¼ [Tw þ (D  1)Tf]/D and Tb2 ¼ [2Tw þ (D  1)Tff]/(1 þ D)

if D  0:75 if D  0:75

Tb* ¼ Tb1 ; Tb* ¼ Tb1 þ ð1  DÞTb2 ;

(17)

and u*b is defined as a function of ub1 ¼ [uw þ (D  1)uf]/D and ub2 ¼ [2uw þ (D  1)uff]/(1 þ D)

u*b

if D  0:75 if D  0:75

u*b ¼ ub1 ; ¼ ub1 þ ð1  DÞub2 ;

(18)

The non-equilibrium part in Eq. (15) is defined as:

  neq ga ðxb ; tÞ ¼ Dga xf ; t þ ð1  DÞga xff ; t neq

neq

(19)

Fig. 4. Comparison of the temperature on axial midline between the present results and numerical results by Khanafer et al. [8] f ¼ 0.1 and Pr ¼ 6.8 (CueWater).

Here w denotes the magnetic induction, bf is the imaginary magnetic for interpolations and eb h eb. 2.3.3. Curved boundary conditions for the magnetic field The ChapmaneEnskog expansion for the post-collision distribution function on Eq. (9) is expressed as [16]:

 o ~ ~ ðx ; t þ DtÞ ¼ ð1  c Þh h m b x f ; t þ Dt þ cm hb ðx b ; t þ DtÞ b b h  i 6Ub ea $Low þ Ub zm Bw  Bf  s Bbf  Bf (20) where

h i hob ðxb ; t þ DtÞ ¼ Ub Bbf þ 3eb Lobf Lo ¼ uB  Bu

(21)

Lobf ¼ Loff

sb þ ð1  cm Þðsb  1Þ  cm if 0 < D < 1 > 2 > Ds > > > > > 2D  1 > ; ¼ sb  2

zm ¼

Bbf ¼

1

Lobf ¼

zm cm

1

1



D 1

Bf þ 

1

D

Lof þ

Bw

1

Low

D D sb þ ð1  cm Þðsb  1Þ  cm ¼ Dþs 2D  1 ¼ sb

6

(22)

Rudraiah [3] Gr=2 10 5 Present Study Rudraiah [3] Gr=2 10 4 Present Study

5

9 >> >> = >> >> ;

incompressible; no-chemical reaction; negligible viscous dissipation; negligible radiative heat transfer;

4

Nuavr

cm

For pure fluid in absence of nanoparticles in the enclosures, the governed equations are Eqs. (1)e(18). However for modeling the nanofluid, the governing equations should be modified by taking into the consideration the variation of the fluid thermal conductivity, density, heat capacitance and thermal expansion. The fluid is a water-based nanofluid containing Ag (silver) nanoparticles. The nanofluid is a mixture of a two components. In simulation the following assumptions were considered: (i) (ii) (iii) (iv)

9 > > > > > > > > =

Bbf ¼ Bff

2.4. The Lattice Boltzmann model for nanofluid

if

1 D1 2

(23)

Table 2 Comparison of the average Nusselt number Nuave for different grid resolution at Ra ¼ 105, Ha ¼ 60 and 4 ¼ 0.06. Mesh size 80  80 100  100 120  120 140  140 160  160 180  180 Nuave 2.1670 2.2612 2.3512 2.3449 2.3389 2.3342

3

2

1

0

0

20

40

60

80

100

Ha Fig. 5. Comparison of average Nusselt number versus at different Grashof number under various strengths of the magnetic field between the present results and numerical results by Rudraiah et al. [3].

Fig. 6. Comparison of the streamlines (left) and isotherms (right) contours between Ag-water nanofluid (f ¼ 0.06) (-$$-) and pure fluid (f ¼ 0)(e) for different values of Ra at l ¼ 3, Ha ¼ 0 and Pr ¼ 6.2.

Fig. 7. Comparison of the streamlines (red) and isotherms (black) contours for different values of Ha at l = 3, Ra ¼ 10 and f ¼ 0.06. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 8. Isotherms (black) and streamlines (red) contours for different values of Hartmann number when l ¼ 3, Ra ¼ 105 for Ag-water nanofluid. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

(v) nano-solid-particles and the base fluid are in thermal equilibrium and no slip occurs between them.

The viscosity of the nanofluid containing a dilute suspension of small rigid spherical particles is (Brinkman model [29]):

The thermo-physical properties of the nanofluid are given in Table 1 [27]. The effective density rnf, the effective heat capacity (rCp)nf and thermal expansion (rb)nf of the nanofluid are defined as [28]:

mnf ¼

rnf ¼ rf ð1  4Þ þ rs 4

The effective thermal conductivity of the nanofluid can be approximated by the MaxwelleGarnetts (MG) model as [10]:



rCp

 nf

    ¼ rCp f ð1  4Þ þ rCp s 4

ðrbÞnf ¼ ðrbÞf ð1  4Þ þ ðrbÞs 4

(24) (25)

(26)

where 4 is the solid volume fraction of the nanoparticles and subscripts f,nf and s stand for base fluid, nanofluid and solid, respectively.

mf ð1  4Þ2:5

 ks þ 2kf  24 kf  ks kn f  ¼ kf ks þ 2kf þ 4 kf  ks

(27)

(28)

In order to compare total heat transfer rate, Nusselt number is used. Local Nusselt numbers and the average Nusselt number are defined on inner and outer cylinder as:

H.R. Ashorynejad et al. / International Journal of Thermal Sciences 64 (2013) 240e250

247

Fig. 9. Effects of the nanoparticle volume fraction, Rayleigh number and Hartmann number on jjjmax.

Nui ¼ ¼

 kn f vT  Ri ln ðRo =Ri Þ  ; vr r¼Ri kf  kn f vT  Ro ln ðRo =Ri Þ  ; kf vr r¼Ro

    maxgrid Gnþ1  Gn   107

Nuo

Nu ¼

kn f 1 kf 2p

Z2p

ðNuÞgdg

0

(29) And the average Nusselt number is:

Nuave ¼

Nui þ Nuo 2

(31)

where n is the iteration number and G stands for the independent variables. Also, to validate the numerical simulation, equivalent thermal conductivity Keq is obtained for a natural convection in concentric horizontal annulus and the results are compared with those of Kuehn and Goldstein [30] in Fig. 3. The average equivalent heat conductivity is defined for inner and outer cylinder by:

(30)

3. Grid testing and code validation To test and assess grid independence of the solution scheme, numerical experiments are performed as shown in Table 2. Different mesh combinations were used for the case of Ra ¼ 105, Ha ¼ 60 and 4 ¼ 0.06 for Agewater nanofluid. The developed code was tested for grid independence by calculating the average Nusselt number on the inner wall of the semi-annulus enclosure. It is found that a grid size of 120  120 ensures the grid independent solution for this case. The convergence criterion for the termination of all computations is:

Keqi ¼ 

ln ðlÞ pðl  1Þ

Zp 0

vT dq ; vr

l$ln ðlÞ pðl  1Þ

Zp

Keqo ¼ 

0

vT dq vr

(32)

Furthermore, comparisons of isotherms between the present work and experimental study of Laboniai and Guj [31] at the different Rayleigh numbers are shown in Fig. 3(b). Also, validation test was carried for natural convection in an enclosure filled with Cuewater for different Grashof numbers and compared with results of Khanafer et al. [8]. Figs. 4 and 5 show the effect of a transverse magnetic field on natural convection flow inside a rectangular enclosure which are compared with the result of

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Fig. 10. Effects of the nanoparticle volume fraction, Rayleigh number and Hartmann number for Ag-water (f ¼ 0.06) nanofluids on average Nusselt number.

Rudraiah et al. [3]. All of the previous comparisons indicate the present LBM code is accurate and produces dependable results. 4. Results and discussion The effect of static radial magnetic field on the buoyancy-driven, convective heat transfer in a horizontal cylindrical annulus enclosure filled with nanofluid is investigated numerically using the Lattice Boltzmann method (LBM). The cylindrical annular enclosure is filled with working fluid, Agewater nanofluid. The thermophysical properties of Ag nanoparticles and based fluid are summarized Table 1 [27]. Calculations were performed for a few values of volume fraction of nanoparticles (4 ¼ 0%, 2%, 4% and 6%), Rayleigh number (Ra ¼ 103, 104 and105) and Hartmann number (Ha ¼ 0, 20, 40, 60) for Prandtl number, Pr ¼ 6.8, and radius ratio, l ¼ 3. The effect of nanoparticles on the streamlines and isotherms is shown in Fig. 6. The velocity components of nanofluid increase because of an increase in the energy transport in the fluid with the increasing of volume fraction. Thus, the values of stream functions indicate that the strength of flow increases with increasing the volume fraction of nanofluid (Fig. 6). The sensitivity of thermal boundary layer thickness to volume fraction of nanoparticles is related to the increased thermal conductivity of the nanofluid. In fact, higher values of thermal conductivity are accompanied by

higher values of thermal diffusivity. The high value of thermal diffusivity causes a drop in the temperature gradients and accordingly increases the thermal boundary thickness. The increase in thermal boundary layer thickness reduces the Nusselt number; however, according to Eq. (29), the Nusselt number is a multiplication of temperature gradient and the thermal conductivity ratio (conductivity of the nanofluid to the conductivity of the base fluid). Since the reduction in temperature gradient due to the presence of nanoparticles is much smaller than the thermal conductivity ratio, therefore an enhancement in Nusselt is taken place by increasing the volume fraction of nanoparticles (Fig. 6). As Rayleigh number increases the buoyancy force increases and overcomes the viscous force and the heat transfer is dominated by convection at high Rayleigh number. Moreover, the isotherms are distorted at higher Rayleigh numbers due to the stronger convection effects, with increasing magnitude of the velocity circulating in the enclosure. The effects of intensity of magnetic field on the streamlines (red) and isotherms (black) contours for different values of Ha at l ¼ 3, Ra ¼ 104 and 4 ¼ 0.06 are shown in Fig. 7. As seen, increase of the Hartmann number causes the flow strength decreases considerably. As the Hartmann number increases, the streamlines are distorted. At Ha ¼ 60, it can be seen that primary eddy divide into three secondary eddies which are rotate in the same direction. Pattern of the isotherms is affected strongly by changing intensity of magnetic field. There is high temperature gradient at the bottom

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249

Fig. 11. Variation of average Nusselt number ratio with solid volume fraction.

of cylinder in the absent of magnetic field. With increasing Hartmann number, the thermal boundary layer thickness increases at the bottom of inner cylinder. The convection is suppressed at the higher Hartmann number. It causes that the plume on the top of the inner circular wall disappears and isothermal lines become concentric and parallel between the cylinders. It is shown that convective heat transfer becomes weaker and causes the heat transfer mostly dominated by conduction between the cylinders. Effects of increasing Hartmann number at different volume fractions of nanoparticles at l ¼ 3, Ra ¼ 105 are shown in Fig. 8. When Hartmann number increases the primary vortex divides in to two smaller vortexes which rotate in opposite direction. In addition, isotherms at the upper half of the outer cylinder are slightly squeezed and the temperature value at the vertical centerline is lower than that at the same height close to the vertical centerline. Thus primary plume is divided into three plumes as shown in Fig. 8. A third plume appears above the top of the inner cylinder with reverse direction owing to the two secondary vortices newly generated over the upper part of the inner cylinder. Also, Fig. 8 demonstrates that as volume fraction of nanoparticles increases the division of primary vortex occurs in higher Hartmann number. Fig. 9 shows the effects of the nanoparticle volume fraction, Rayleigh number and Hartmann number on absolute values of stream function (jjjmax ). The absolute values of stream function

increases as nanoparticle volume fraction or Rayleigh number increases, while it decreases as Hartmann number increases. Also, it can be found that effect of nanoparticle volume fraction for high magnetic field at high Rayleigh number is more pronounced than at low Rayleigh number. Fig. 10 shows the effects of the nanoparticle volume fraction, Rayleigh number and Hartmann number for Agewater (4 ¼ 0.06) nanofluids on average Nusselt number. It was found that the Nusselt number is an increasing function of nanoparticle volume fraction and Rayleigh number, while it is a decreasing function of Hartmann number. Fig. 11 illustrates that variation of average Nusselt number ratio ðNu* ¼ ðNuave j4 Þ=ðNuave j4¼0 ÞÞ with solid volume fraction for range of controlling parameters. The average Nusselt number ratio monotonically increases as volume fraction increases.

5. Conclusions The effect of static radial magnetic field on natural convection heat transfer in a horizontal cylindrical annulus enclosure filled with nanofluid is investigated numerically using Lattice Boltzmann method (LBM). The effects of Hartmann number, nanoparticle volume fraction and Rayleigh number on the flow and heat transfer

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characteristics were examined. From the investigation, some conclusions can be summarized as follows: a. Lattice Boltzmann method based on double-population is a suitable approach for simulating thermo-magnetic in the geometry with curved boundaries. The method can simulate the velocity, temperature and magnetic fields with second order accuracy. b. As volume fraction of nanoparticles increases the partition of primary vortex occurs in higher Hartmann number. c. The absolute values of stream function increases as nanoparticle volume fraction or Rayleigh number increases, while it decreases as Hartmann number increases. d. Average Nusselt number is an increasing function of nanoparticle volume fraction and Rayleigh number, while it is a decreasing function of Hartmann number.

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