MHD free convection in an eccentric semi-annulus filled with nanofluid

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JTICE-871; No. of Pages 13 Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx

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MHD free convection in an eccentric semi-annulus filled with nanofluid M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji * Department of Mechanical Engineering, Babol University of Technology, Babol, Iran

A R T I C L E I N F O

A B S T R A C T

Article history: Received 18 December 2013 Received in revised form 11 March 2014 Accepted 16 March 2014 Available online xxx

In this study magnetohydrodynamic effect on free convection of nanofluid in an eccentric semi-annulus filled is considered. The effective thermal conductivity and viscosity of nanofluid are calculated by the Maxwell–Garnetts (MG) and Brinkman models, respectively. Lattice Boltzmann method is applied to simulate this problem. This investigation compared with other works and found to be in excellent agreement. Effects of the Hartmann number, nanoparticle volume fraction, Rayleigh numbers and position of the inner circular cylinder on flow and heat transfer characteristics are examined. Also a correlation of Nusselt number corresponding to active parameters is presented. The results show that Nusselt number has direct relationship with nanoparticle volume fraction and Rayleigh number but it has inverse relationship with Hartmann number and position of inner cylinder at high Rayleigh number. Also it can be concluded that heat transfer enhancement increases with increase of Hartmann number and decreases with augment of Raleigh number. ß 2014 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Keywords: MHD Nanofluid Eccentric annulus Natural convection Lattice Boltzmann method

1. Introduction With the rising request for capable cooling systems, particularly in the electronics industry, more effective coolants are required to keep the temperature of electronic components below safe limits. The low thermal conductivity of conventional heat transfer fluids such as water and oils is a primary limitation in enhancing the performance and the compactness of such systems. Solid typically has a higher thermal conductivity than liquids. An innovative and new technique to enhance heat transfer is using solid particles in the base fluid (i.e. nanofluids) in the range of sizes 10–50 nm. Khanafer et al. [1] performed a numerical investigation on the heat transfer enhancement due to adding nano-particles in a differentially heated enclosure. They found that the suspended nanoparticles substantially increase the heat transfer rate at any given Grashof number. Kumar et al. [2] used a single phase thermal dispersion model to simulate the flow and heat transfer. Their results showed that boundary surface of nanoparticles and their chaotic movement greatly enhances the fluid heat conduction contribution. Heat transfer of a nanofluid flow which is squeezed

* Corresponding author at: Department of Mechanical Engineering, Babol University of Technology, Babol, Iran. Tel.: +98 911 3968030; fax: +98 911 3968030. E-mail addresses: [email protected] (M. Sheikholeslami), [email protected] (D.D. Ganji).

between parallel plates was investigated by Sheikholeslami and Ganji [3].They reported that Nusselt number has direct relationship with nanoparticle volume fraction, the squeeze number and Eckert number when two plates are separated. The transient natural convection heat transfer of aqueous nanofluids in a horizontal annulus between two coaxial cylinders has been studied by Zi-Tao et al. [4]. They found that at constant Rayleigh numbers, the time-averaged Nusselt number is gradually lowered as the volume fraction of nanoparticles is increased. Abu-Nada et al. [5] investigated natural convection heat transfer enhancement in horizontal concentric annuli field by nanofluid. They showed that for low Rayleigh numbers, nanoparticles with higher thermal conductivity cause more enhancement in heat transfer. Khan and Pop [6] published a paper on boundary-layer flow of a nanofluid past a stretching sheet. Their model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. They have taken into account the Prandtl number, Lewis number, Brownian motion numbers and thermophores number. MHD effect on natural convection heat transfer in an inclined L-shape enclosure filled with nanofluid was studied by Sheikholeslami et al. [7]. They found that enhancement in heat transfer has direct relationship with Hartmann number. Sheikholeslami et al. [8] used heatline analysis to simulate two phase simulation of nanofluid flow and heat transfer. Their results indicated that the average Nusselt number decreases as buoyancy ratio number increases until it reaches a minimum value and then

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Please cite this article in press as: Sheikholeslami M, et al. MHD free convection in an eccentric semi-annulus filled with nanofluid. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.03.010

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Nomenclature B0 c ci Cp F f eq f g g eq gy Ha k L Nu Nu p Pr Ra r T ðu; vÞ ðx; yÞ ðX; YÞ

magnetic field lattice speed discrete particle speeds specific heat at constant pressure external forces density distribution functions equilibrium density distribution functions internal energy distribution functions equilibrium internal energy distribution functions gravitational acceleration [m/s2] pffiffiffiffiffiffiffiffiffiffi Hartmann number ð¼ LB0 s =mÞ thermal conductivity height or width of the enclosure average Nusselt number local Nusselt number pressure [Pa] Prandtl number ð¼  y=aÞ  Rayleigh number ¼ g bDTL3 =ay radius of circle fluid temperature velocity components in (x,y) directions, respectively Cartesian coordinates dimensionless coordinates

Greek symbols a thermal diffusivity [m2/s] f volume fraction u dimensionless temperature m dynamic viscosity [Pa/s] y kinematic viscosity [m2/s] l aspect ratio ð¼ L=2rÞ r fluid density [kg/m3] tc relaxation time for temperature s electrical conductivity tv relaxation time for flow b thermal expansion coefficient [K1] c stream function d position of inner cylinder uM direction of the magnetic field Subscripts cold c hot h nanofluid nf base fluid f solid particles s

starts increasing. Recently, several numerical studies have been carried out to simulate nanofluid flow and heat transfer [9–24]. The flow of an electrically conducting fluid in a magnetic field is influenced by magnetohydrodynamic (MHD) forces resulting from the interaction of induced electric currents with the applied magnetic field. An externally imposed magnetic field is also a widely used tool for control of melt flows in the bulk crystal growth of semiconductors. One of the main purposes of electromagnetic control is to stabilize the flow and suppress oscillatory instabilities,

which degrades the resulting crystal. Rudraiah et al. [25] investigated numerically the effect of magnetic field on natural convection in a rectangular enclosure. They found that the magnetic field decreases the rate of heat transfer. Cross-diffusion effects on convection from a vertically spinning cone under the influence of an external magnetic field was considered by Narayana et al. [26].They showed that larger Schmidt numbers lead to a considerable thinning of the concentration boundary layer, and hence a reduction in the solute concentration. Rashidi et al. [27] considered the analysis of the second law of thermodynamics applied to an electrically conducting incompressible nanofluid fluid flowing over a porous rotating disk. They concluded that using magnetic rotating disk drives has important applications in heat transfer enhancement in renewable energy systems and industrial thermal management. Ellahi [28] studied the magnetohydrodynamic flow of non-Newtonian nanofluid in a pipe. He observed that the MHD parameter decreases the fluid motion and the velocity profile is larger than that of temperature profile even in the presence of variable viscosities. Several papers were published about this field of science in recent decade [29–39]. The lattice Boltzmann method (LBM) is a powerful numerical technique based on kinetic theory for simulating fluid flows and modeling the physics in fluids [40,41]. In comparison with the conventional CFD methods, the advantages of LBM include simple calculation procedure, simple and efficient implementation for parallel computation, easy and robust handling of complex geometries, and others. Kefayati et al. [42] studied MHD flow in a lid-driven cavity by a linearly heated wall. They found that heat transfer on the heated wall at the bottom of the cavity behaves like the linearly heated wall regarding the effect of the magnetic field. The lattice Boltzmann method has been applied to investigate the nanofluid flow and heat transfer characteristic. Free convection heat transfer in a concentric annulus between a cold square and heated elliptic cylinders in presence of magnetic field was investigated by Sheikholeslami et al. [43]. They found that the enhancement in heat transfer increases as Hartmann number increases but it decreases with increase of Rayleigh number. This method was successfully applied for different applications [44–47]. Free convection in an enclosure is appropriate to many environmental and industrial applications such as heat exchangers, nuclear and chemical reactors, cooling of electronic equipment, and stratified atmospheric boundary layers. In engineering applications, the geometries that arise, however, are more complicated than a simple enclosure filled with a convective fluid. The geometric shape of interest is with the presence of bodies embedded within the enclosure. However, there is little information about free convection processes when a heated circular cylinder exists within a cooled square enclosure and the location of the inner heated circular cylinder is changed along the vertical centerline of the square enclosure. In this situation, the flow and heat transfer in the enclosure are largely affected by the location of the inner circular cylinder for different Rayleigh numbers. The purpose of the present geometry is to examine how the position of the inner circular cylinder relative to the outer square cylinder affects the natural convection phenomena for different active parameters when a hot inner circular cylinder is located at different positions along the vertical centerline of the outer square cylinder. The aim of the present work is to study MHD natural convection of Cu–water nanofluid in a concentric annulus using lattice Boltzmann method. Effects of Hartmann number, nanoparticle volume fraction, Rayleigh numbers and position of the inner circular cylinder on the flow and heat transfer characteristics have been examined.

Please cite this article in press as: Sheikholeslami M, et al. MHD free convection in an eccentric semi-annulus filled with nanofluid. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.03.010

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JTICE-871; No. of Pages 13 M. Sheikholeslami et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx

3

2.2. The lattice Boltzmann method The LB model used here is the same as that employed in [17]. The thermal LB model utilizes two distribution functions, f and g, for the flow and temperature fields, respectively. It uses modeling of movement of fluid particles to capture macroscopic fluid quantities such as velocity, pressure and temperature. 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 these discrete directions. The D2Q9 model was used and values of w0 ¼ 4=9 for jc0 j ¼ 0 (for the static particle), w14 ¼ 1=9 for jc14 j ¼ 1 and pffiffiffi w59 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 and g, 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 as follows: Fig. 1. Geometry of the problem.

2. Problem definition and mathematical model 2.1. Problem statement The physical model used in this work is shown in Fig. 1. In this figure, l ¼ L=2r denotes aspect ratios and g is measured counterclockwise from the upward vertical plane through the center of outer cylinders. The system consists of a square enclosure with sides of length L, within which a circular cylinder with l ¼ 3:5 is located and moves along the vertical centerline in the range from d/L = 0.2 to 0:8. The walls of the square enclosure was kept at a constant low temperature of T c , whereas the cylinder was kept at a constant high temperature of T h , ðT h > T c Þ. Also, it is also assumed ~ ¼ Bx e!x þ By e!y ) of constant that the uniform magnetic field (B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! magnitude B ¼ B2x þ B2y is applied, where ex and ey are unit vectors in the Cartesian coordinate system. The orientation of the magnetic field form an angle u M with horizontal axis such that uM ¼ cot1 ðBx =By Þ. The electric current J and the electromagnetic ~  BÞ ~ and F ¼ s ðV ~  BÞ ~ ~ B, respecforce F are defined by J ¼ s ðV tively. In this study we considered u M ¼ 0.

For the flow field:    Dt  eq fi ðx; t Þ  f i ðx; t Þ f i x þ ci Dt; t þ Dt ¼ f i ðx; t Þ þ

tv

þ Dtci F k :

(1)

For the temperature field:    Dt  eq gi ðx; t Þ  g i ðx; t Þ g i x þ ci Dt; t þ Dt ¼ g i ðx; t Þ þ

tc

(2)

where Dt denotes lattice time step, ci is the discrete lattice velocity in direction i, F k is the external force in direction of lattice velocity, t v and t c denotes the lattice relaxation time for the flow and temperature fields. The kinetic viscosity y and the thermal diffusivity a, are defined in terms of their respective relaxation times, i.e. y ¼ cs2 ðt v  1=2Þ and a ¼ cs2 ðt c  1=2Þ, respectively. Note that the limitation 0:5 < t should be satisfied for both relaxation times to ensure that viscosity and thermal diffusivity are positive. Furthermore, the local equilibrium distribution function which determines the type of problem that needs to be solved. It also models the equilibrium distribution functions, which are calculated with Eqs. (3) and (4) for flow and

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

Please cite this article in press as: Sheikholeslami M, et al. MHD free convection in an eccentric semi-annulus filled with nanofluid. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.03.010

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temperature fields, respectively. " fieq ¼ wi r 1 þ

ci  u 1 ðci  uÞ2 1 u2 þ    2 2 2 cs cs2 cs4

#



c u gieq ¼ wi T 1 þ i 2 cs

(3)

neq g a¯ ðxb ; t Þ ¼ gaeq ¯ ðxb ; t Þ þ ga ¯ ðxb ; t Þ:

(4)

where wi is a weighting factor and r is the lattice fluid density. In order to incorporate buoyancy forces and magnetic forces in the model, the force term in the Eq. (1) needs to be calculated as below [43]: F ¼ F x þ F hy

i  F x ¼ 3wi r Aðv sinðuM ÞcosðuM ÞÞ  u sin2 ðu M Þ ; h  i F y ¼ 3wi r g y bðT  T m Þ þ Aðu sinðuM ÞcosðuM ÞÞ  v cos2 ðuM Þ (5) p ffiffiffiffiffiffiffiffiffiffi where A is A ¼ Ha2 y=L , Ha ¼ LB0 s =m is Hartmann number and uM is the direction of the magnetic field. For natural convection, the Boussinesq approximation is applied and radiation heat transfer is negligible. To ensure that the code works in near incompressible regime, the characteristic velocity of the flow for natural qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðV natural  bg y DTHÞ regime must be small compared with the fluid speed of sound. In the present study, the characteristic velocity selected as 0.1 of sound speed. Finally, macroscopic variables are calculated with the following formula: 2

Flow density : Momentum : Temperature :

r¼ ru ¼ T¼

X i X i X

f i; ci f i ;

(9)

Substituting Eq. (9) into Eq. (2) leads to:

  1 ð x ; t Þ þ 1  ganeq g˜a¯ xb ; t þ Dt ¼ gaeq b ¯ ¯ ðxb ; t Þ:

(10)

ts

neq Obviously, both gaeq ¯ ðxb ; tÞ and ga ¯ ðxb ; tÞ are needed to calculate the value of g˜a¯ ðxb ; t þ DtÞ. In Eq. (10) the equilibrium part is defined as:

3   ð x ; t Þ ¼ w T 1 þ e  u (11) gaeq a¯ b a¯ b b ¯ c2

where T b  is defined as a function of T b1 ¼ ½T w þ ðD  1ÞT f =D and T b2 ¼ ½2T w þ ðD  1ÞT f f =ð1 þ DÞ. Tb ¼ T b1 ; if D  0:75 Tb ¼ T b1 þ ð1  DÞT b2 ; if D  0:75

(12)

and ub is defined as a function of ub1 ¼ ½uw þ ðD  1Þu f =D and ub2 ¼ ½2uw þ ðD  1Þu f f =ð1 þ DÞ. ub ¼ ub1 ; if D 0:75 ub ¼ ub1 þ 1  D ub2 ;

if D  0:75

(13)

:

The non equilibrium part in Eq. (14) is defined as:       ga ðxb ; t Þ ¼ Dganeq x f ; t þ 1  D ganeq x f f ; t : neq

(14)

2.4. The lattice Boltzmann model for nanofluid (6)

gi:

i

2.3. Boundary conditions 2.3.1. Curved boundary treatment for velocity For treating velocity and temperature fields with curved boundaries, the method proposed in [48] has been used. An arbitrary curved wall separating solid region from fluid is shown in Fig. 2(b). The link between the fluid node x f and the wall node xw intersects the physical boundary at xb .  The fraction of  the   intersected link in the fluid region is D ¼ x f  xw =x f  xb . To calculate the post-collision distribution function f˜a¯ ðxb ; tÞ based upon the surrounding nodes information, a Chapman–Enskog expansion for the post-collision distribution function on the righthand side of Eq. (1) is conducted as:   3 f˜a¯ ðxb ; t Þ ¼ ð1  xÞ f˜a x f ; t þ x fa ðxb ; t Þ þ 2wa r 2 ea¯  uw c

2.3.2. Curved boundary treatment for temperature Following the work of Yan and Zu [48] the non-equilibrium parts of temperature distribution function can be defined as:

(7)

where,    3   fa ðxb ; t Þ ¼ faeq x f ; t þ wa r x f ; t 2 ea  ub f  u f ; c     2D  1 1 ; if 0  D  : ub f ¼ u f f ¼ u x f f ; t ; x ¼ 2 t2    2D  1 1  3 1 D1 ; if ub f ¼ 2D  3 u f þ uw ; x ¼ 2 t  1=2 2D 2D (8) In the above, ea¯   ea ; u f is the fluid velocity near the wall; uw is the velocity of solid wall and ub f is an imaginary velocity for interpolations.

In order to simulate the nanofluid by the lattice Boltzmann method, because of the interparticle potentials and other forces on the nanoparticles, the nanofluid behaves differently from the pure liquid from the mesoscopic point of view and is of higher efficiency in energy transport as well as better stabilization than the common solid–liquid mixture. For pure fluid in absence of nanoparticles in the enclosures, the governed equations are Eqs. (1)–(14). However, for modeling the nanofluid because of changing in the fluid thermal conductivity, density, heat capacitance and thermal expansion, some of the governed equations should change. The nanofluid is a two component mixture with the following assumptions: incompressible; no-chemical reaction; negligible viscous dissipation; negligible radiative heat transfer; nano-solid-particles and the base fluid are in thermal equilibrium and no slip occurs between them. The thermo physical properties of the nanofluid are given in Table 1 [1]. The effective density rn f , the effective heat capacity ðr C p Þnf and thermal expansion ðrbÞnf of nanofluid are defined as [3]:

rnf ¼ rf ð1  fÞ þ rs f 

rC p

 nf





(15) 



¼ rC p f ð1  fÞ þ rC p s f

(16)

ðrbÞnf ¼ ðrbÞf ð1  fÞ þ ðrbÞs f

(17)

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

r ðkg=m3 Þ C p ðJ=kg KÞ k ðW=m KÞ b ðK1 Þ Pure water Copper (Cu) Silver (Ag) Alumina (Al2O3) Titanium oxide (TiO2)

997.1 8933 10 500 3970 4250

4179 385 235 765 686.2

0.613 401 429 40 8.9538

5

21  10 1.67  105 1.89  105 0.85  105 0.9  105



1

s Vm

0:05 5.96  107 3.60  107 1  1010 1  1012

Please cite this article in press as: Sheikholeslami M, et al. MHD free convection in an eccentric semi-annulus filled with nanofluid. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.03.010

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where f is the solid volume fraction of the nanoparticles and subscripts f, nf and s stand for base fluid, nanofluid and solid, respectively. The viscosity of the nanofluid containing a dilute suspension of small rigid spherical particles is (Brinkman model [49]):

mnf ¼

mf : 2:5 ð1  fÞ

knf ks þ 2kf  2fðkf  ks Þ : ¼ ks þ 2kf þ fðkf  ks Þ kf

(19)

Electrical conductivity ðs Þnf of the nanofluid is defined as [7]:

s nf 3ðs s =s f  1Þf ¼1þ : sf ðs s =s f þ 2Þ  ðs s =s f  1Þf

(20)

In order to compare total heat transfer rate, Nusselt number is used. The local and average Nusselt numbers are defined as follows: knf @T 1 and Nu ¼  L kf @n

Table 2 Comparison of the average Nusselt number along the surface of the inner cylinder Nuh for different grid resolution at Ra ¼ 106 , l ¼ 3:5; d=L ¼ 0:8; Ha ¼ 40 and f ¼ 0:06 (Cu–water nanofluid). Mesh size

190  190

200  200

210  210

220  220

Nuh

12.7358

12.7597

12.7602

12.7643

(18)

The effective thermal conductivity of the nanofluid can be approximated by the Maxwell–Garnetts (MG) model as [3]:

Nu ¼

5

ZL

Nu dS:

(21)

0

3. Grid testing and code validation To verify the grid independence of the solution scheme, numerical experiments are performed as shown in Table 2. Different mesh sizes were used for the case of Ra = 106, l = 3.5, Ha = 100, d/L = 0.8 and f = 0.06 (Cu–water). The present code is tested for grid independence by calculating the average Nusselt number on the inner wall. It is found that a grid size of 200  200 ensure the grid independent solution for the present case. The convergence criterion for the termination of all computations is:    nþ1 n maxgrid G  G   107 (22) where n is the iteration number and G stands for the independent variables (U; V; T). In this study numerical solution is validated by comparing the present code results against the results of Kuehn and Goldstein [50], Laboniai and Guj [51] for viscous flow ðf ¼ 0Þ (see Fig. 3). Furthermore, another validation test was carried for natural convection in an enclosure filled with Cu–water for different Grashof numbers with the results of Khanafer et al. [1] in Fig. 4.

Another test for validation of the current code was performed for the case of natural convection in a square enclosure in the presence of magnetic field. In this test case, the average Nusselt number using different Gr and Ha number have been compared with those obtained by Rudraiah et al. [25] as shown in Table 3. All of the previous comparisons indicate the accuracy of the present LBM code. 4. Results and discussion LBM is applied in order to solve the problem of MHD effect on free convection in an eccentric annulus filled with nanofluid. Calculations are made for various values of Hartmann number (Ha ¼ 0  40), volume fraction of nanoparticle (f = 0, 0.02, 0.04 and 0.06), Rayleigh number (Ra = 104, 105 and 106) and position of the inner circular cylinder (d/L = 0.2, 0.4, 0.6 to 0.8) when Prandtl number ðPr ¼ 6:8Þ and aspect ratio ðl ¼ 3:5Þ are fixed. Effects of Cu nanoparticles on the streamlines and isotherms in presence and absence of magnetic field are shown in Fig. 5. The flow strength (absolute values of stream function) increases when nanoparticle added in to the base fluid. Thermal boundary layer thickness increases as volume fraction of nanofluid increases. It is interesting to note from this figure that effect of nanofluid volume fraction is more pronounced when the hot cylinder located at upper location in presence of magnetic field. This is owing to domination of conduction heat transfer mechanism below the hot cylinder. Also presence of magnetic field helps conduction mechanism to be dominated. Fig. 6 shows the effects of Ha, Ra and d=L on isotherms and streamlines contours. At 104 , increasing d=L up to 0.6 leads to a decrease in the maximum value of stream function while as d=L   enhances furthermore, cmax nf increases.   At high Raleigh number c  continuously decreases (e.g.Ra = 105 and 106) the values of max nf   by augment of d=L. As seen, cmax nf has inverse relationship with Hartmann number. At Ra ¼ 104 the distribution of the flow and thermal fields for d=L ¼ 0:2 and 0:4 shows the symmetric shapes about the

Fig. 3. Comparison of isotherms between the present work and experimental study of (a) Kuehn and Goldstein [50]; and (b) Laboni and Guj [51] for viscous flow ðf ¼ 0; Pr ¼ 0:71Þ when (a)R0/Ri = 2.6, Ra = 4.7  104; (b) R0 =Ri ¼ 2:36; Ra ¼ 0:9  105 .

Please cite this article in press as: Sheikholeslami M, et al. MHD free convection in an eccentric semi-annulus filled with nanofluid. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.03.010

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M. Sheikholeslami et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx Table 3 Average Nusselt number versus at different Grashof number under various strengths of the magnetic field at Pr = 0.733. Ha

0 10 50 100

Fig. 4. Comparison of the temperature on axial midline between the present results and numerical results by Khanafer et al. [1] f ¼ 0:1 and Pr = 6.2 (Cu–water).

Gr = 2  104

Gr = 2  105

Present

Rudraiah et al. [25]

Rudraiah

Rudraiah et al. [25]

2.5188 2.2234 1.0856 1.0110

2.566894 2.261644 1.083047 1.008833

4.9198 4.8053 2.8442 1.4317

5.151789 4.952211 2.689937 1.456019

horizontal center line, compared with that of corresponding d=L ¼ 0:8 and 0:6, respectively, because conduction is the dominant mode of heat transfer at this low Rayleigh number. The circulation of flow shows one rotating eddy for the streamlines. When d=L < 0:5 the core of the main vortex is located in the upper half of enclosure but when d=L > 0:5 the main vortex moves downward. Also a thermal plume starts to appear on the top of the inner cylinder and as a result the isotherms move upward. The thermal plume diminishes as the inner cylinder moves upward. As the Rayleigh number increases up to 105 , the role of convection in heat transfer becomes more significant and consequently the thermal boundary layer thicknesses on the surface of the inner cylinder become thinner. It is an interesting to notice that at Ra ¼ 105 the core of main vortex is located in upper

Fig. 5. Comparison of the isotherms (left) and streamlines (right) contours between nanofluid (f ¼ 0:06) (––) and pure fluid (f ¼ 0) ( ) for different values of Ha and d/L at l ¼ 3:5; Ra ¼ 106 and Pr = 6.8.

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Fig. 6. Effects of Ha, Ra and d=L on isotherms (left) and streamlines (right) contours for Cu–water case (f ¼ 0:06) when l ¼ 3:5 and Pr = 6.8.

half of enclosure for all position of inner cylinder. Also this figure shows that at d=L ¼ 0:8, the thermal plume formed at the top of left cylinder. A similar flow pattern is observed at Ra ¼ 106 but the    value of cmax nf and thermal gradient near the hot inner cylinder is greater at this Raleigh number. When the magnetic field is imposed on the enclosure, the velocity field suppressed owing to the retarding effect of the Lorenz force. So intensity of convection weakens significantly. The braking effect of the magnetic field is observed from the maximum stream function value. Also applying magnetic field makes the isotherms parallel to each other due to domination of conduction mode of heat transfer. Also effect of position of inner cylinder is more obvious in presence of magnetic field because magnetic field boosts the conduction heat transfer mechanism. Distribution of the local Nusselt numbers only over the left half of the enclosure and inner cylinder are shown in Figs. 7 and 8 for different values of Ha, Ra, d=L and f. Nusselt number expected to decrease with increase of volume fraction because of decrement in temperature gradient in but opposite trend is observed in Figs. 7 and 8. This observation is due to this fact that 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. At Ra ¼ 104 , when the center of inner cylinder located in the lower

half of the enclosure minimum value of Nuh is obtained at g ¼ 0

and its maximum value occurs at g ¼ 180 , but opposite trend is observed for d=L > 0:5. At Ra = 105 and 106 the Nuh profile shows two different behaviors at d=L greater and smaller than 0.6. This means that for d=L < 0:6, local Nusselt number increases with augment of g whereas for d=L > 0:6, at first Nuh decreases and then increases. The minimum values of Nuh , for these cases are corresponding to the existence of thermal plume over the heated cylinder. It can be seen that at high Rayleigh number magnetic field effect on the local Nusselt number is more obvious. As inner cylinder situated at upper position, the Nuc profiles become smoother due to type of heat transfer mechanism. When d=L ¼ 0:2, maximum values of local Nusselt number obtain at g ¼ 180

because minimum distant between cold and hot cylinders exists at this region. The corresponding polynomial representations of such model for Nusselt number along the walls of the enclosure ðNuc Þ and along the surface of the inner cylinder ðNuh Þ are as follows: Nuc ¼ b13 þ b23 Y 1 þ b33 Y 2 þ b43 Y12 þ b53 Y22 þ b63 Y 1 Y 2  2  Y 1 ¼ b11 þ b21 Ra þ b31 d þ b41 Ra2 þ b51 d þ b61 Ra d 2  2  Y 2 ¼ b12 þ b22 Ha þ b32 f þ b42 Ha þ b52 f þ b62 Ha f

(23)

Nuh ¼ a13 þ a23 Y 1 þ a33 Y 2 þ a43 Y12 þ a53 Y22 þ a63 Y 1 Y 2  2  Y 1 ¼ a11 þ a21 Ra þ a31 d þ a41 Ra2 þ a51 d þ a61 Ra d 2  2  Y 2 ¼ a12 þ a22 Ha þ a32 f þ a42 Ha þ a52 f þ a62 Ha f

(24)

Please cite this article in press as: Sheikholeslami M, et al. MHD free convection in an eccentric semi-annulus filled with nanofluid. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.03.010

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Fig. 7. Effects of Ha, Ra, d=L and f on local Nusselt number along the walls of the enclosure (Nuc) for Cu–water case when l ¼ 3:5 and Pr = 6.8.

where Ra* = log(Ra), d* = d/L and Ha* = Ha/100. Also ai j and bi j can be found in Tables 4 and 5. Effects of Ha, Ra and d=L on the average Nusselt number along the walls of the enclosure and the surface of the inner cylinder is depicted in Figs. 9 and 10. Variation of the average Nusselt number over the hot surface is similar to that of cold surface. In addition the value of Nusselt number over the inner cylinder is considerably greater than the cold one. Since the

conduction heat transfer mechanism is dominate at Ra ¼ 104 , the profile of average Nusselt number is almost symmetric about d=L ¼ 0:5. It can be found that the minimum value of average Nusselt number is obtained at d=L ¼ 0:6 when Ra ¼ 105 . For Ra ¼ 106 the average Nusselt number decreases continuously with increases of d=L. Increasing Hartmann number causes Lorenz force to increase and leads to a substantial suppression of the

Please cite this article in press as: Sheikholeslami M, et al. MHD free convection in an eccentric semi-annulus filled with nanofluid. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.03.010

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Fig. 8. Effects of Ha, Ra, d=L and f on local Nusselt number along the surface of the inner cylinder (Nuh) for Cu–water case when l ¼ 3:5 and Pr = 6.8.

Table 4 Constant coefficient for using Eq. (23). bi j

i=1

i=2

i=3

Table 5 Constant coefficient for using Eq. (24). i=4

i=5

i=6

j=1 4.983644 1.8922 1.86745 0.355825 5.82276 1.02723 j=2 2.777641 0.49729 7.766133 0.19892 9.476536 1.33354 j = 3 0.85206 0.598477 0.000487 0.09365 0.00053 0.345664

ai j

i=1

i=2

i=3

i=4

i=5

i=6

j = 1 21.5249 8.15581 8.09223 1.553241 25.67044 4.56336 j = 2 12.40464 2.22502 31.3948 0.89001 43.90568 5.46778 j = 3 3.82584 0.602369 0.003993 0.02115 0.00012 0.077697

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Fig. 9. Effects of Ha, Ra and d=L on average Nusselt number along (a) the walls of the enclosure ðNuc Þ; (b) the surface of the inner cylinder ðNuh Þ for Cu–water case when f ¼ 0:06; l ¼ 3:5.

convection. So Nusselt number has reverse relationship with Hartmann number. The enhancement of heat transfer between the case of f ¼ 0:06 and the pure fluid (base fluid) case is defined as:

En ¼

Nuh ðf ¼ 0:06Þ  Nuh ðbasefluidÞ  100: Nuh ðbasefluidÞ

(25)

Fig. 11 depicts the effects of Raleigh number, Hartmann number and position of inner cylinder on the enhancement of heat transfer. As seen, the effect of nanoparticles is more pronounced at low Rayleigh number than at high Rayleigh number because of

greater amount of rate of enhancement and increasing Rayleigh number leads to decrease in ratio of enhancement of heat transfer. This observation can be explained by noting that at low Rayleigh number the heat transfer is dominant by conduction. Therefore, the addition of high thermal conductivity nanoparticles will increase the conduction and therefore make the enhancement more effective. Also it can be seen that maximum value of En is obtained at d=L ¼ 0:4 for Ra ¼ 104 , whereas the minimum value of heat transfer enhancement at Ra ¼ 105 and 106 is obtained at d=L ¼ 0:6 and 0:4, respectively. Also this figure shows that enhancement increases with augment of Hartmann number. Fig. 12 shows the effects of types of nanoparticles on enhancement in heat transfer due to addition of nanoparticles. It can be seen that maximum of enhancement occurs for Cu–water case.

Please cite this article in press as: Sheikholeslami M, et al. MHD free convection in an eccentric semi-annulus filled with nanofluid. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.03.010

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Fig. 10. Variation of average Nusselt number along (a) the walls of the enclosure ðNuc Þ; (b) the surface of the inner cylinder ðNuh Þ for various input parameters.

Please cite this article in press as: Sheikholeslami M, et al. MHD free convection in an eccentric semi-annulus filled with nanofluid. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.03.010

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Fig. 11. Effects of Ha, Ra and d=L on ratio of enhancement in heat transfer due to addition of nanoparticles for Cu–water case when l ¼ 3:5.

References

Fig. 12. Effects of types of nanoparticle on enhancement in heat transfer due to addition of nanoparticles when l ¼ 3:5; Pr ¼ 6:8; Ha ¼ 0.

5. Conclusions In this study, LBM is used for simulating natural convection heat transfer in an eccentric annulus filled with nanofluid in presence and absence of magnetic field. Effects of Hartmann number, nanoparticle volume fraction, Rayleigh number and position of inner cylinder on the flow and heat transfer characteristics have been examined. The results show that Nusselt number increases with increase of nanoparticle volume fraction and Rayleigh number but it decreases with enhance of Hartmann number. Effect of position of inner cylinder on Nusselt number is different for various values of Rayleigh numbers. At Ra ¼ 104 the profile of average Nusselt number is almost symmetric about d=L ¼ 0:5. Minimum value of average Nusselt number occurs at d=L ¼ 0:6 when Ra ¼ 105 . At Ra ¼ 106 Nusselt number decreases with increase of d=L. Also the results indicate that effect of nanoparticles is more pronounced at low Rayleigh number and high Hartmann number due to conduction domination.

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