Numerical simulation of thermomagnetic convection of air in a porous square enclosure under a magnetic quadrupole field using LTNE models

International Journal of Heat and Mass Transfer 91 (2015) 98–109

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Numerical simulation of thermomagnetic convection of air in a porous square enclosure under a magnetic quadrupole field using LTNE models Changwei Jiang ⇑, Er Shi, Zhangmao Hu, Xianfeng Zhu, Nan Xie School of Energy and Power Engineering, Changsha University of Science and Technology, Changsha 410114, China Key Laboratory of Efficient and Clean Energy Utilization, College of Hunan Province, Changsha 410114, China

a r t i c l e

i n f o

Article history: Received 21 August 2014 Received in revised form 3 March 2015 Accepted 23 July 2015

Keywords: Thermomagnetic convection Numerical simulation Porous media Magnetic quadrupole field Magnetic force

a b s t r a c t In this paper, thermomagnetic convection of air in a two-dimensional porous square enclosure under a magnetic quadrupole field has been numerically investigated. The scalar magnetic potential method is used to calculate the magnetic field. A generalized model, which includes a Brinkman term, a Forcheimmer term and a nonlinear convective term, is used to solve the momentum equations and the energy for fluid and solid are solved with the local thermal non-equilibrium (LTNE) models. The results are presented in the form of streamlines and isotherms and local and average Nusselt numbers. Numerical results are obtained for a range of the magnetic force parameter from 0 to 100, the Darcy number from 105 to 101 and dimensionless solid-to-fluid heat transfer coefficient from 1 to 1000. The results show that the magnetic force number, Darcy number, Rayleigh number and dimensionless solid-to-fluid heat transfer coefficient have significant effect on the flow field and heat transfer in a porous square enclosure. The flow characteristics presents two cellular structures with horizontal symmetry about the middle plane of the enclosure and the Nusselt numbers are increased as the magnetic force number increases under the non-gravitational condition. The average Nusselt number respects the trend of decrease at first and then increases when the magnetic force number increases under gravitational condition. The non-equilibrium effect on fluid phase temperature and solid phase temperature gradually reduces with the increase of value of H. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Natural convection heat transfer in porous enclosure is widely used in many industrial applications such as cooling of electronic devices, solar collectors, heat exchangers and so on. There are many open literature related to natural convection in porous enclosures [1–3]. Ellahi et al. [4,5] have analyzed the influence of variable viscosity and viscous dissipation on the non-Newtonian flow in porous medium using the homotopy analysis method. Numerical investigation of natural convection within porous square enclosures for various thermal boundary conditions has been done by Ramakrishna et al. [6]. It is found that thermal boundary conditions have an important influence on the flow and heat transfer characteristics during natural convection within porous square cavities. Chankim et al. [7] analyzed theoretically

⇑ Corresponding author at: School of Energy and Power Engineering, Changsha University of Science and Technology, Changsha 410114, China. Tel./fax: +86 731 85258409. E-mail address: [email protected] (C. Jiang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.07.103 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

the onset of convection motion in an initially quiescent, horizontal isotropic porous layer. Magnetic field effect on the heat transfer and fluid flow has received much attention in recent years due to its importance in electronic packaging, purification of molten metals, crystal growth in liquids and many others [8–11]. Saleh et al. [12] analyzed the effect of a magnetic field on steady convection in a trapezoidal enclosure filled with a fluid-saturated porous medium by the finite difference method. Grosan et al. [13] examined the effects of a magnetic field and internal heat generation on natural convection heat transfer in an inclined square enclosure filled with a fluid-saturated porous medium. It was shown that both the strength and inclination angle of the magnetic field have a strong influence on convection modes. Nield studied MHD convection in porous medium [14]. Sathiyamoorthy [15] analyzed the convective heat transfer in a square cavity filled with porous medium under a magnetic field. Magnetohydrodynamic natural convection in a rectangular cavity under a uniform magnetic field at different angles with respect to horizontal plane has been investigated by Yu et al. [16]. They concluded that the heat transfer is not only determined by the strength of the magnetic field, but also

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Nomenclature b b0 B Br C cp Da fm g h H H kf ks L Num p P p0 p0 Pr Ra T0 Tc Tf Th Ts u, v U, V U

magnetic flux density (T) reference magnetic flux density, b0 ¼ Br (T) dimensionless magnetic flux magnetic flux density of permanent magnets (T) C ¼ 1 þ T10 b fluid specific heat at constant pressure (J kg1 K1) Darcy number, Lj2 magnetic force gravitational acceleration (m s2) solid-to-fluid heat transfer coefficient (W m2 K1) 2 dimensionless solid-to-fluid heat transfer coefficient, hL ekf magnetic field intensity fluid thermal conductivity (W m1 K1) solid thermal conductivity (W m1 K1) length of the enclosure, (m) average Nusselt number pressure, Pa dimensionless pressure pressure at reference temperature, Pa pressure difference due to the perturbed state, Pa m Prandtl number, Pr ¼ aff 3 c ÞL Rayleigh number, Ra ¼ gbðTahfT mf T h þT c T 0 ¼ 2 (K) cold wall temperature (K) fluid temperature (K) hot wall temperature (K) solid temperature (K) velocity components (m s1) dimensionless velocity components velocity vector

influenced by the inclination angle. Especially, when the aspect ratio is less or more than 1, it is found that the inclination angle plays a great role on flow and heat transfer. The natural convection heat transfer of Cu–water nanofluid in a cold outer circular enclosure containing a hot inner sinusoidal circular cylinder in the presence of horizontal magnetic field is investigated numerically by Sheikholeslami et al. [17] using the Control Volume based Finite Element Method (CVFEM). The results indicate that in the absence of magnetic field, enhancement ratio decreases as Rayleigh number increases while an opposite trend is observed in the presence of magnetic field. In recent years, with the development of a superconducting magnet providing strong magnetic induction of 10 T or more, the natural convection of paramagnetic fluid like oxygen gas and air under magnetic field has been an interesting research topic investigated by many researchers [18,19]. The effect of the magnetic buoyancy force on the convection of paramagnetic fluids was first reported by Braithwaite et al. [20], they found that the effect of magnetic buoyancy force on convection depends on the relative orientation of the magnetic force and the temperature gradient. Carruthers and Wolfe [21] studied the thermal convection of oxygen gas in a rectangular container with thermal and magnetic field gradients theoretically and experimentally, and found that the magnetic buoyancy force cancel out the influence of gravity buoyancy force when rectangle enclosure heated from one vertical wall and cooled from opposing wall was located in horizontal magnetic field with vertical magnetic field gradient. Huang et al. [22–24] studied the stability of paramagnetic fluid layer by the action of a non-uniform magnetic field and the thermomagnetic convection of the diamagnetic field. Gray et al. [25] obtained the numerical similarity solutions for the two-dimensional plums driven by the interaction of a line heat source and a non-uniform magnetic field.

x, y X, Y

Cartesian coordinates dimensionless Cartesian coordinates

Greek symbols af fluid thermal diffusivity (m s1) b thermal expansion coefficient (K1) v b2 c dimensionless magnetic strength parameter, c ¼ l0 gL0 hf hs

e l0 lm lf mf qf qs v v0 vm j K

um

T T

m

dimensionless fluid temperature, hf ¼ T fh T0c 0 dimensionless solid temperature, hs ¼ TT hs T T c porosity magnetic permeability of vacuum (H m1) magnetic permeability (H m1) fluid kinematic viscosity (kg m1 s1) fluid dynamic viscosity (m2 s1) fluid density (kg m3) solid density (kg m3) mass magnetic susceptibility (m3 kg1) reference mass magnetic susceptibility (m3 kg1) volume magnetic susceptibility Permeability (m2) ek dimensionless thermal conductivity, K ¼ ð1efÞks scalar magnetic potential

Subscripts 0 reference value c cold f fluid h hot s solid

Wakayama and coworkers [26–28] studied the magnetic control of thermal convection. Tagawa’s group [29–31] derived a model equation for magnetic convection using a method similar to the Boussinesq approximation and studied natural convection of paramagnetic, diamagnetic and electrically conducting fluids in a cubic enclosure with thermal and magnetic field gradients at different thermal boundary. Ozoe and co-workers [18,32–34] studied natural convection of paramagnetic and diamagnetic fluids in a cylinder under gradient magnetic field at different thermal boundary and found that the magnetic force can be used to control heat transfer rate of paramagnetic and diamagnetic fluids. Yang et al. [35,36] numerically and experimentally investigated magnetothermal convection of air or oxygen gas in a enclosure by using the gradient magnetic field available from Neodymium–Iron–Boron permanent magnet systems, and pointed out that the enhancement or suppression of magnetothermal convection of air or oxygen gas can be achieved by gradient magnetic field. Tomasz and co-workers [37–40] studied natural convection of paramagnetic fluids in a cubic enclosure under magnetic field by an electric coil and analyzed that the effect of inclined angle of electric coil, location of electric coil, Ra number on heat transfer rate of paramagnetic fluid. Above studies are concerned with the effect of magnetic force on natural convection of paramagnetic fluids. However, only a few studies are paid on the combined effects of both magnetic and gravitational forces on the natural convection of paramagnetic fluids in porous medium. Natural convection in an enclosure filled with a paramagnetic or diamagnetic fluid-saturated porous medium under strong magnetic field was numerically investigated by Wang et al. [41–43]. Considering the effect of Darcy number, Rayleigh number and magnetic force number, the results of numerical investigation showed that the magnetic force had a significant effect on the flow field and heat transfer in a paramagnetic

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or diamagnetic fluid-saturated porous medium. The application of strong magnetic field for porous medium may be found in the field of medical treatment, metallurgy, materials processing, combustion. There may be plenty of applications in engineering field in the near future. Thus, the study of effect of magnetic force on natural convection in porous media is very important for both scientific research and engineering application. Even though earlier studies on the natural convection of paramagnetic or diamagnetic fluids in a porous cubic enclosure with an electrical coil have been carried out by wang et al., a detailed investigation of natural convection of air in a porous square enclosure under a magnetic quadrupole field using LTNE models has yet to appear in the literature. The present paper represents the results of a numerical investigation on natural convection of air in a porous square enclosure under a magnetic quadrupole field using local thermal no-equilibrium (LTNE) models. The numerical investigation is carried out for different governing parameters such as the magnetic force number, Darcy number, and dimensionless solid-to-fluid heat transfer coefficient etc. 2. Physical model The schematic of the system under consideration is shown in Fig. 1. The system consists of a porous square enclosure which is kept in a horizontal position and four permanent magnets which generate a magnetic field. The porous square enclosure filled with air is heated isothermally from left-hand side vertical wall and cooled isothermally from opposing wall while the other two walls are thermally insulated. The gravitational force acts in the Y direction. In the present study, the size of the enclosure L, the size of the permanent magnet L1 and the distance of permanent magnets L2 are 0.024 m, 0.02 m and 0.03 m respectively. 3. Mathematical formulation

where, f m is the magnetic force; vm is the volumetric magnetic susceptibility; lm is the magnetic permeability, H m1; b is the magnetic flux density, T; qf is the fluid density, kg m3; v is the mass magnetic susceptibility, m3 kg1. The Navier–Stokes equation which includes the magnetic force can be presented as:

lf lf qf qf 1:75 jUjU U  rU ¼ rp  U þ r2 U  3=2 pffiffiffiffiffiffiffiffiffi pffiffiffiffi e2 j e e 150 j qf v 2 rb þ qf g þ 2l m

In which, U is the velocity vector; p is the pressure, Pa; lf is the fluid kinematic viscosity, kg m1 s1; g is the gravitational acceleration, m s2; j is the permeability, m2. At the reference state of the isothermal state, there will be no convection. Therefore, Eq. (2) becomes as follows.

0 ¼ rp0 þ

qf 0 v0 2 rb þ q f 0 g 2lm

lf lf qf qf 1:75 jUjU U  rU ¼ rp0  U þ r2 U  3=2 pffiffiffiffiffiffiffiffiffi pffiffiffiffi e2 j e e 150 j ðqf v  qf 0 v0 Þ 2 þ rb þ ðqf  qf 0 Þg 2lm

ð4Þ

where: p ¼ p0 þ p0 , p0 is the pressure difference due to the perturbed state, Pa. Because qf and v are function of temperature, according to Taylor expansion method, qfv and qf can be respectively indicated as:

3.1. Governing equations

qf v 2 v 2 f m ¼ m rb ¼ rb 2lm 2l m

ð3Þ

where p0 is the pressure at reference temperature, Pa; qf0 is the fluid density at reference temperature, kg m3; v0 is the mass magnetic susceptibility at reference temperature, m3 kg1, subtracting (3) from (2) gives:

qf v ¼ ðqf vÞ0 þ

The assumptions in the model are as follows: the fluid is considered to be steady, incompressible, and Newtonian fluid. Both the viscous heat dissipation and magnetic dissipation are assumed to be negligible. According to Braithwaite et al. [20], the magnetic force can be given as follows:

ð2Þ

qf ¼ qf 0 þ

  @ðqf vÞ ðT f  T 0 Þ þ    @T f 0

  @ qf ðT f  T f 0 Þ þ    @T f 0

ð5Þ

ð6Þ

For the paramagnetic fluid air, the mass magnetic susceptibility is inverse proportion to absolute temperature, according to the Curie’s law:

v¼ ð1Þ

m Tf

ð7Þ

where m is the constant value; Tf is the fluid phase temperature, K; T0 = (Th + Tc)/2, K; Subscripts 0, h, c represent reference value, hot and cold respectively. So Eq. (5) can be written as:

  @ qf v v  qf ðT f  T 0 Þ þ    @T f Tf 0   v ¼ qf bv  qf ðT f  T 0 Þ þ    Tf 0   1 ¼ qf 0 v0 b 1 þ ðT f  T 0 Þ þ    T0b

qf v  ðqf vÞ0 ¼

ð8Þ

The higher order small amount in Eq. (8) is omitted and generated into Eq. (4), Eq. (4) becomes as follows.

1

e

2

Fig. 1. Physical model and coordinate system.

lf lf 2 1 1:75 jUjU  Uþ r U  3=2 pffiffiffiffiffiffiffiffiffi pffiffiffiffi qf 0 qf 0 j qf 0 e e 150 j   v0 b 1 2 ðT f  T 0 Þrb þ bðT f  T 0 Þg  1þ 2lm T0b

U  rU ¼ 

rp0

where b is thermal expansion coefficient, K1. Accordingly, the governing equations can be written as:

ð9Þ

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C. Jiang et al. / International Journal of Heat and Mass Transfer 91 (2015) 98–109 Table 1 Comparison of the average Nusselt number Num for different grid resolution at e = 0.5, Pr = 0.71, H = 10, K = 10, cRa = 1  106, Da = 1  103 under non-gravitational condition. Grid dimension

Num

30  30 40  40 50  50 60  60 70  70

2.0635 2.0519 2.0478 2.0460 2.0454

Table 3 Comparison of average Nusselt numbers with [45] at various values of Br. Br

Num Song et al.

0.0 T 0.5 T 1.5 T 2.5 T

[45]

4.520 4.506 4.352 3.898

Present results 4.525 4.524 4.360 3.923

Continuity equation:

@u @ v þ ¼0 @x @y Momentum equation:

Table 2 Comparison of present results with [35].

DT

Num Yang et al.

1 10 50

1.003 1.214 2.120

[35]

ð10Þ

Present results

Relative error/%

1.003 1.244 2.166

0 2.47 2.17

  1=2 qf @u @u @p lf 1:75 ðu2 þ v 2 Þ u pffiffiffiffi ¼   u  qf pffiffiffiffiffiffiffiffiffi u þv 2 @x @y @x j e e3=2 j 150 ! lf @ 2 u @ 2 u þ þ e @x2 @y2   2 1 v0 bðT f  T 0 Þ @ðb Þ  1þ T 0b @x 2lm

Fig. 2. The comparison of present results with [35] (left: temperature field and right: velocity field).

ð11Þ

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Fig. 3. (a) Distribution of gradient of square magnetic induction (rB2), (b) vectors of the magnetizing force (CcRaPrhfrB2/2) under the non-gravitational condition.

Fig. 4. Effect of cRa number on the streamlines (left), fluid phase temperature (middle) and solid phase temperature (right) for Da = 103, H = 10 and e = 0.5 under the nongravitational convection.





1=2 qf @ v @v @p lf 1:75 ðu2 þ v 2 Þ v p p ffiffiffi ffi ffiffiffiffiffiffiffiffi ffi ¼  u þ v  v  q f @y j e2 @x @y e3=2 j 150 ! lf @ 2 v @ 2 v þ þ e @x2 @y2

Fluid phase energy equation:

!   @T f @T f @2T f @2T f þ hðT s  T f Þ ¼ ekf ðqcp Þf u þv þ @x @y @x2 @y2 Solid phase energy equation:

  2 1 v0 bðT f  T 0 Þ @ðb Þ  1þ T0b @y 2lm þ qf gbðT f  T 0 Þ

ð13Þ

0 ¼ ð1  eÞks ð12Þ

@2T s @2T s þ 2 @x2 @y

! þ hðT f  T s Þ

ð14Þ

C. Jiang et al. / International Journal of Heat and Mass Transfer 91 (2015) 98–109

where: x, y are Cartesian coordinates; u, v are velocity components, m s1; af is the fluid thermal diffusivity, m s1; mf is the fluid dynamic viscosity, m2 s1; cpf is the fluid specific heat at constant pressure, J kg1 K1; e is the porosity; kf is the fluid thermal conductivity, W m1 K1; ks is the solid thermal conductivity, W m1 K1; h is the solid-to-fluid heat transfer coefficient, Wm2K1; Ts is solid phase temperature, K. The above equations (10)–(14) can be non-dimensionalised as follows: Continuity equation:

@U @V þ ¼0 @X @Y

ð15Þ

3.2. Mathematical formulation of magnetic field calculation The maxwell equations are applied to describe the magnetic quadrupole field.

rB¼0

ð20Þ

rH ¼0

ð21Þ

where, B is the magnetic flux density and H is the magnetic field intensity. The constitutive relation that describes the behavior of the magnetic material is:

B ¼ lH

Momentum equation:

  1=2 1 @U @U @P Pr 1:75 ðU 2 þ V 2 Þ U pffiffiffiffiffiffi ¼ U  pffiffiffiffiffiffiffiffiffi U þV  2 @X @Y @X Da e e3=2 Da 150 ! Pr @ 2 U @ 2 U C @ðB2 Þ  cRaPrhf þ þ 2 2 2 @X e @X @Y

ð22Þ

The scalar magnetic potential um is commonly used to calculate the magnetic field and it satisfies:

H ¼ rum ð16Þ

ð23Þ

In homogeneous magnetic medium, the permeability is assumed to be constant. Combining Eqs. (22), (23) and (20), the scalar magnetic potential satisfies the Laplace’s Equation:

rum ¼ 0

  1=2 @V @V @P Pr 1:75 ðU 2 þ V 2 Þ V p p ffiffiffiffiffiffi ffiffiffiffiffiffiffiffi ffi ¼  V  U þ V  3=2 @X @Y @Y Da e2 e Da 150 ! 2 2 2 Pr @ V @ V C @ðB Þ  cRaPrhf þ þ 2 @Y e @X 2 @Y 2 1

þ RaPrhf

103

ð24Þ

ð17Þ

Fluid phase energy equation:

  @hf @hf @ 2 hf @ 2 hf ¼ ð1 þ K1 Þ U þV þ þ Hðhs  hf Þ @X @Y @X 2 @Y 2

ð18Þ

Solid phase energy equation:

@ 2 hs

0¼

@X 2

þ

@ 2 hs @Y 2

þ KHðhf  hs Þ

ð19Þ

The dimensionless variables and parameters in the above equations are defined as

x X¼ ; L

y Y¼ ; L

U¼

uL

af

;

V¼

vL af pL2

gbðT h  T c ÞL3

hf ¼

Tf  T0 ; Th  Tc

Pr ¼

mf j Th þ Tc b 1 ; Da ¼ 2 ; T 0 ¼ ; B ¼ ; b0 ¼ Br; C ¼ 1 þ b0 T0b af 2 L

c¼

hs ¼

Ts  T0 ; Th  Tc

P¼

qf a

2 f

;

Ra ¼

a f mf

2 ekf v0 b20 hL ; H¼ ; K¼ lm gL ekf ð1  eÞks

where: X, Y are dimensionless Cartesian coordinates; U, V are dimensionless velocity components; P is the dimensionless pressure; hf is the dimensionless fluid temperature; hs is the dimensionless solid temperature; Th is the hot wall temperature, K; Tc is the cold wall temperature, K; Pr is the Prandtl number; b0 is the reference magnetic flux density, T; Br is the magnetic flux density of permanent magnets, T; Ra is the Rayleigh number; c is the dimensionless magnetic strength parameter; B is the dimensionless magnetic flux; Da is the Darcy number; H is the dimensionless solid-to-fluid heat transfer coefficient; K is the dimensionless thermal conductivity.

Fig. 5. Effect of cRa number on the local Nusselt numbers of the left (top) and right (bottom) side walls for Da = 103 and e = 0.5 under the non-gravitational condition.

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3.3. Boundary conditions The non-slip conditions are imposed for the two velocity components on the solid walls. The temperature boundary conditions are as follows: (1) left vertical wall (X = 0.5): hf = hs = 0.5; (2) right vertical wall (X = 0.5): hf = hs = 0.5; (3) top and bottom horizontal adiabatic walls (Y = 0.5, 0.5): @hf/@ Y = @hs/@ Y = 0 3.4. Nusselt number calculation In order to compare total heat transfer rate, the Nusselt number is used. The average Nusselt number at the hot wall is defined as follows: For fluid:

Numf ¼ 

Z

0:5

0:5

 @hf  dY @X X¼0:5

ð25Þ

 @hs  dY @X X¼0:5

ð26Þ

For solid:

Nums ¼ 

Z

0:5

0:5

The average total Nusselt number [44] is given by:

    @h  1 @hs  dY Num ¼ K f  þ  ð1 þ KÞ 0:5 @X X¼0:5 @X X¼0:5 Z

Fig. 7. Effect of Darcy number on the average Nusselt number for e = 0.5 under the non-gravitational condition.

3.5. Numerical procedure and code verification

0:5

ð27Þ

The governing equations, Eqs. (15)–(19) are discretized by the finite-volume method (FVM) on a uniform grid system. The

Fig. 6. Effect of H on the streamlines (left), fluid phase temperature (middle) and solid phase temperature (right) for Da = 103, cRa = 106 and e = 0.5 under the nongravitational condition.

C. Jiang et al. / International Journal of Heat and Mass Transfer 91 (2015) 98–109

third-order Quick scheme and the second-order central difference scheme are implemented for the convection and diffusion terms, respectively. The set of discretized equations for each variable is solved by a line-by-line procedure, combining the tridiagonal matrix algorithm (TDMA) with the successive overrelaxation (SOR) iteration method. The coupling between velocity and pressure is solved by the SIMPLE algorithm. The convergence criterion is that the maximal residual of all the governing equations is less than 106. The reliability and accuracy of the mathematical model and code must be checked before calculation. To allow grid independent examination, the numerical procedure has been conducted for different grid resolutions. Table 1 demonstrates the influence of number of grid points for the case of at e = 0.5, Pr = 0.71, H = 10, K = 10, cRa = 1106, Da = 1  103 under non-gravitational condition. The present code is tested for grid independence by calculating the average Nusselt number on the hot wall. It was found that a grid size of 60  60 ensures a grid-independent solution. Therefore, for all computations in this article, a 60  60 uniform grid was employed. In order to validate the numerical methods and codes of the present work, the recent, similar works by Yang et al. [35] and Song et al. [45] were selected as the benchmark solution for comparison. Yang considered the thermomagnetic convection in an air-filled 2-D square enclosure confined to a magnetic quadrupole field under zero-gravity. Table 2 and Fig. 2 present comparisons

105

between the present results for Da = 107, e = 0.9999, H = 0, g = 0 and those of Yang for the average Nusselt number, temperature field and velocity field [35]. Table 3 presents comparisons between the present results for Da = 107, e = 0.9999, H = 0 under gravitational condition and those of Song for the average Nusselt number. It is seen that the present results are in very good agreement with those obtained by previous authors, which validates the present numerical code. 4. Results and discussion As indicated by above mathematic model, the natural convection under consideration is governed by seven nondimensional parameters: e, c, Ra, Pr, Da, H and K. In the present study, the porosity (e), the Prandtl number, K, the Rayleigh number are kept constant at e = 0.5, Pr = 0.71, K = 10, and Ra = 1  105 respectively, and therefore main attention is paid to the effect of magnetic force number (c), dimensionless solid-to-fluid heat transfer coefficient (H) and Darcy number (Da) on the fluid flow and heat transfer characteristics in the porous enclosure. 4.1. Numerical results under the non-gravitational condition Fig. 3(a) shows the gradient of square of magnetic induction (rB2) that is produced under a magnetic quadrupole field. The gradient of square of magnetic induction (rB2) has a centrifugal

Fig. 8. Effect of magnetic force number on the streamlines (left), fluid phase temperature (middle) and solid phase temperature (right) for Da = 103, Ra = 105 and e = 0.5 under the gravitational condition.

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character. Fig. 3(b) shows the vector of the magnetizing force (CcRaPrhfrB2/2) under the non-gravitational condition. The model equation includes the effect of magnetic force in which the magnetic susceptibility of air is inversely proportional to its temperature. The magnetic force repels the left-hand side hot air and the right-hand side cold air is attracted to the strong magnetic field. So the magnetizing force is direct to the square center near the hot wall and is deviated from the square center near the cold wall, and the magnetizing force is symmetrical in terms of the middle plane of the porous square. The magnetic buoyancy force drives the air moving from the left hot wall to the right cold wall along the horizontal middle line of the porous enclosure, then the air is going downward and upward to the top and bottom walls and returns to the middle of the hot wall, so that the flow in the enclosure is of two cellular structures with horizontal symmetry about the middle of the enclosure. 4.1.1. Effect of cRa Since Ra becomes zero and c becomes infinity when g = 0, finite product cRa has to be used to describe non-gravity cases. Fig. 4 shows the streamlines, the fluid phase isotherms and the solid phase isotherms at various cRa number when Da = 103 and e = 0.5. In each graphic shown, the porous enclosure is heated isothermally from left-hand side vertical wall and cooled isothermally from opposing wall. Obviously, the convection in the porous enclosure is strengthened with the increase of cRa. The distribution of streamline suggests that the flow in the enclosure is of two cellular structure with horizontal symmetry about the middle of the enclosure. The distribution of isotherms suggests that the isotherms in the porous enclosure are horizontally symmetry about the middle of the porous enclosure; isotherms are dense at top and bottom of hot wall and middle of cold wall. When cRa is relatively small, such as cRa = 1  105, the convection of air in the porous enclosure is very weak so that the heat transfer is dominated by the conduction mechanism, where isotherms approximately exhibit a linear trend from the hot wall to the cold wall. With the increase of cRa, the convection in the porous enclosure is strengthened and isotherms occurs severe deformation. For the temperature field, with increase of the magnetic force, the temperature difference between the solid and liquid is greatly increased, so the LTNE models are important in simulating the heat transfer in porous medium when there is a strong magnetic field. Fig. 5 illustrates the variations of the local Nusselt numbers along the left hot wall and the right cold wall at various cRa when Da = 103, e = 0.5. It can be seen that the local Nusselt numbers are increased as the cRa increases; the local Nusselt numbers are symmetric about horizontal centerline, which is consistent with symmetric of isotherms along horizontal centerline. For the hot wall, the minimum local Nusselt number appears in the middle position and the maximal local Nusselt number appears in the top and bottom of the hot wall. For the cold wall, the local Nusselt number is maximum in the middle position and gradually presents monotonic decrease from middle of cold wall to ends of the cold wall. 4.1.2. Effect of dimensionless solid-to-fluid heat transfer coefficient (H) Fig. 6 shows the streamlines, fluid phase temperature and solid phase temperature for various values of H at Da = 103, cRa = 106 and e = 0.5 under the non-gravitational condition. It is obvious from this figure that the non-equilibrium effect is very strong at low values of H. The isotherms of fluid phase are not much affected due to change in the value of H whereas the isotherms of solid phase deviate to larger extent when H is increased from 1 to 1000. It may be noted that the increased inter-phase heat transfer coefficient is associated with increased heat transfer between solid and fluid phases. The increased heat transfer between two phases brings their temperature close to each other thus the fluid phase

and solid phase temperature field look similar at higher values of H, which reveals that the solid and fluid phases are in a state of thermal equilibrium. The streamlines show that the flow pattern is nearly unchanged when H is increased. 4.1.3. Effect of Darcy number (Da) Fig. 7 presents the variations of the average Nusselt number (Num) of the porous enclosure in terms of cRa at various Darcy number (Da) when e = 0.5. Obviously, the average Nusselt number increases as cRa increases for each Darcy number case. On the other hand, the average Nusselt number is also increased as the Darcy number increases from 101 to 105 at various cRa. At low Da number, such as Da = 105, the convection of air in the porous enclosure is very weak and hence the heat transfer in the porous enclosure is mainly governed by the conduction mode, the magnitude of cRa has little effect on the heat transfer rate. With the increase of cRa, the convection in the porous enclosure is strengthened and the effect of cRa number on heat transfer rate is increased. 4.2. Numerical results under gravitational condition 4.2.1. Effect of magnetic force number (c) Fig. 8 depicts the streamlines, the fluid phase isotherms and the solid phase isotherms at various magnetic force number when

Fig. 9. Effect of Magnetic force number on the local Nusselt numbers of the left (top) and right (bottom) side walls for Da = 103, Ra = 105 and e = 0.5 under the gravitational condition.

C. Jiang et al. / International Journal of Heat and Mass Transfer 91 (2015) 98–109

Da = 103, Ra = 105 and e = 0.5 under the gravitational environment. Now the air in the porous enclosure is acted by both magnetic buoyancy force and gravity buoyancy force. The results show that the flow and temperature field are very complex. When c = 0, the air flow in the porous enclosure is pure gravity convection and respects a clockwise vortex. When there is a magnetic field, the counteraction of magnetic buoyancy force and gravitational buoyancy force repels clockwise flow of air in the upper part of the porous enclosure, however, the synergy of magnetic buoyancy force and gravity buoyancy force enhances the air flow in the lower part of the porous enclosure. When c = 1, the magnetic buoyancy force is weak, and the air flow is completely succumb to the gravity buoyancy force and respects a clockwise flow structure. The convection in the porous enclosure is strengthened with the increase of the magnetic force number c, so there is a competition between magnetic and gravity buoyancy forces, which results in the stratified phenomenon of the air flow. At c = 5, the flow pattern is observed to be multi-cellular with a large clockwise rotating cell at the lower part of the porous enclosure and a small counter-clockwise rotating cell at the top right corner of the porous enclosure. As the magnetic force number continues to increase, the vortex under the top part and the bottom part in the porous enclosure gradually enlarges and shrinks respectively. If the magnetic force number is larger, the state of the air flow in the porous enclosure is mainly determined by the magnetic buoyancy force, the flow in the enclosure is of two cellular structures with

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approximately horizontal symmetry about the middle of the enclosure. The distribution of isotherms is similar to the case of the pure gravity convection when the magnetic buoyancy force is very small, that is isotherms on the bottom of the hot wall and the top of cold wall are dense. With increasing of the magnetic force number, isotherms on the top and bottom on the hot wall become dense and the area with dense isotherms on the cold wall moves from the top to the middle of the cold wall gradually. When the magnetic force number increases up to 100, the distribution of isotherms in the porous enclosure is similar to the case of the pure magnetic convection, namely isotherms are approximately horizontal symmetry about the middle of the porous enclosure. Fig. 9 illustrates the variations of the local Nusselt number along the left hot wall and the right cold wall at various magnetic force numbers when Da = 103, Ra = 105, e = 0.5. From the variation of the average Nusselt number on the left wall, it can found that the heat transfer rate on the bottom of left wall increases gradually with increasing of the magnetic force number because of the synergy of the magnetic and gravity buoyancy forces. However, the heat transfer rate on the top of the left wall decreases at first and then increases because of the counteraction of the magnetic and gravity buoyancy forces. From the variation of the average Nusselt number on the right wall, the heat transfer rate on the bottom of right wall also increases gradually with increasing of the magnetic force number, and on the contrary, the heat transfer rate on the top of right wall decreases gradually. When the magnetic

Fig. 10. Effect of H on the streamlines (left), fluid phase temperature (middle) and solid phase temperature (right) for Da = 103, c = 10, Ra = 105 and e = 0.5 under the gravitational condition.

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magnetic field. A generalized model, which includes a Brinkman term, a Forcheimmer term and a nonlinear convective term, was used to solve the momentum equations and the energy for fluid and solid were solved with the local thermal non-equilibrium (LTNE) models. The main conclusions of the present analysis are as follows:

Fig. 11. Effect of Darcy number on the average Nusselt number for Ra = 105 and e = 0.5 under the gravitational condition.

force number is large than 50, the local Nusselt numbers are approximately symmetric about the horizontal centerline of the porous enclosure. 4.2.2. Effect of dimensionless solid-to-fluid heat transfer coefficient (H) Fig. 10 depicts the streamlines and isotherms for various values of H at Da = 103, c = 10, Ra = 105 and e = 0.5 under the gravitational condition. It is evident from this figure that the solid phase temperature is affected to greater degree as compared to the fluid phase temperature when H is increased. At large value of H, the isotherms for both the phases become similar. This happens due to fact that non-equilibrium forces are very weak and thus equilibrium prevails at high values of H. 4.2.3. Effect of Darcy number (Da) Fig. 11 presents the variations of the average Nusselt number (Num) in terms of the magnetic force number (c) at various Darcy number (Da) when Ra = 105 and e = 0.5. Obviously, the average Nusselt number is increased as the Darcy number increases for each magnetic force number case. At low Da number, such as Da = 105, the convection of air in the porous enclosure is very weak so that the heat transfer in the porous enclosure is mainly governed by the conduction mode, the magnitude of magnetic force has little effect on the heat transfer rate. With the increase of the Darcy number, the convection in the porous enclosure is strengthened gradually and the effect of the magnetic force number on heat transfer rate is also increased gradually. At the same time, the average Nusselt number respects the trend of decrease at first and then increases when the magnetic force number increases, which is because of the counteraction of the magnetic buoyancy force and the gravity buoyancy force. When c < 5, the strength of air convection is diminished because of the counteraction of the magnetic buoyancy force and the gravity buoyancy force. When c > 5, the convection of the air is mainly determined by the magnetic buoyancy force, the heat transfer rate in the porous enclosure increases with the increase of the magnetic force number. 5. Conclusion Thermomagnetic convection of air was investigated numerically in a two-dimensional porous square enclosure under a magnetic quadrupole field in the presence or absence of a gravity field. The scalar magnetic potential method was used to calculate the

(1) For the non-gravity condition, the flow of the air in the porous enclosure is of two cellular structures with horizontal symmetry about the middle of the enclosure. The local Nusselt numbers along the left hot wall and the right cold wall are increased as the magnetic force number increases and are symmetric about the horizontal centerline. (2) In the gravity condition, with the increase of the magnetic force number, the air flow in the porous enclosure presents a clockwise vortex at first and then appears the stratified phenomenon, which results in the formation of two vortexes with different rotational directions and intensities. Finally, the air flow in the enclosure is of two cellular structures with approximately horizontal symmetry about the middle of the enclosure. (3) The solid phase temperature is affected to greater degree as compared to the fluid phase temperature when H is increased and the isotherms of fluid phase and solid phase look similar at higher values of H.

Conflict of interest None declared.

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