《高等传热学》课程教学资源（文献资料）Numerical simulation of thermomagnetic convection of air in a porous square enclusure under a magnetic quadrupole field using LTNE models

of Heat and(015)1 Contents lists available at ScenceDirect International Journal of Heat and Mass Transfer ELSEVIER journal homepage:www.elsevier.com/locate/ijhmt 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 ARTICLE INFO ABSTRACT convection of air in a tw 23m March 2015 in this paper. ddle pla of decrease at first and then i 2015 Elsevier Ltd.All rights reserved. 1.Introduction etic field the heattransfer and uid fow has There are electro ic packaging purific al gro porousmedium using the homotopy analysis method. sfe dary con ditions have an important influence on the ngth and field have a strons porous square cavities.Chankim et al.analyzed theoretically porous medium4.Sathiyam 5analWzedtheconveCtw gnetic field.Magnetohydr in E-ma 7oo2aS2aa2&

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 article info Article history: Received 21 August 2014 Received in revised form 3 March 2015 Accepted 23 July 2015 Available online 6 August 2015 Keywords: Thermomagnetic convection Numerical simulation Porous media Magnetic quadrupole field Magnetic force abstract 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 num￾ber 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 por￾ous 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. 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 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 http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.07.103 0017-9310/ 2015 Elsevier Ltd. All rights reserved. ⇑ 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: cw_jiang@163.com (C. Jiang). International Journal of Heat and Mass Transfer 91 (2015) 98–109 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

99 Nomenclature density.bo=Br(T) n coordinates 4BB nanent magnets(T) dimensionless fluid temperature.= pefficient (Wm-2K-) solid temperature. netic nsit wm aticviscos average Nusselt number mass magnenss m e(K) reference value velocity vector solid Wakayama and cow orkers [26-28]studi d the ransfe uation for nagnetic conve ing a m milar to ic d app cubi 13-34 ed Finite Eleme t Method (C M). The results ind ases ile an opposite trend d that the magn e can be us un erical availabe from Neody mium-Iro out hat t f air or they found tha natural magne and depends on the relati rate of pa dients theoretically and experim ntally and found that th on of par s.How wa onal fo ces on the natura cated in n in n. e stability of p uid layer by the action im u diamagnetic field.Gray et al.125]obtained the nu erica number and magnet the esult ifcant effect on the flow field and heat transfer in a paramagnet

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 cir￾cular cylinder in the presence of horizontal magnetic field is inves￾tigated numerically by Sheikholeslami et al. [17] using the Control Volume based Finite Element Method (CVFEM). The results indi￾cate 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 inves￾tigated 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 oxy￾gen 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 buoy￾ancy 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. 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 para￾magnetic, 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 natu￾ral 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 sup￾pression 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 ana￾lyzed 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 med￾ium 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 sig￾nificant effect on the flow field and heat transfer in a paramagnetic Nomenclature b magnetic flux density (T) b0 reference magnetic flux density, b0 ¼ Br (T) B dimensionless magnetic flux Br magnetic flux density of permanent magnets (T) C C ¼ 1 þ 1 T0b cp fluid specific heat at constant pressure (J kg1 K1 ) Da Darcy number, j L2 fm magnetic force g gravitational acceleration (m s2 ) h solid-to-fluid heat transfer coefficient (W m2 K1 ) H dimensionless solid-to-fluid heat transfer coefficient, hL2 ekf H magnetic field intensity kf fluid thermal conductivity (W m1 K1 ) ks solid thermal conductivity (W m1 K1 ) L length of the enclosure, (m) Num average Nusselt number p pressure, Pa P dimensionless pressure p0 pressure at reference temperature, Pa p0 pressure difference due to the perturbed state, Pa Pr Prandtl number, Pr ¼ mf af Ra Rayleigh number, Ra ¼ gbðThTc ÞL3 af mf T0 T0 ¼ ThþTc 2 (K) Tc cold wall temperature (K) Tf fluid temperature (K) Th hot wall temperature (K) Ts solid temperature (K) u,v velocity components (m s1 ) U,V dimensionless velocity components U velocity vector x,y Cartesian coordinates X,Y dimensionless Cartesian coordinates Greek symbols af fluid thermal diffusivity (m s1 ) b thermal expansion coefficient (K1 ) c dimensionless magnetic strength parameter, c ¼ v0b2 0 lmgL hf dimensionless fluid temperature, hf ¼ Tf T0 ThTc hs dimensionless solid temperature, hs ¼ TsT0 ThTc e porosity l0 magnetic permeability of vacuum (H m1 ) lm magnetic permeability (H m1 ) lf fluid kinematic viscosity (kg m1 s 1 ) mf fluid dynamic viscosity (m2 s1 ) qf fluid density (kg m3 ) qs solid density (kg m3 ) v mass magnetic susceptibility (m3 kg1 ) v0 reference mass magnetic susceptibility (m3 kg1 ) vm volume magnetic susceptibility j Permeability (m2 ) K dimensionless thermal conductivity, K ¼ ekf ð1eÞks um scalar magnetic potential Subscripts 0 reference value c cold f fluid h hot s solid C. Jiang et al. / International Journal of Heat and Mass Transfer 91 (2015) 98–109 99

100 amr91(2015)9g-109 :Pyis the fluid density.kgm is the mass earfuture Thus the sty of on nat- ndudes the rce research and Even tho can be pres engineering app ough fluids in a rous cubic ncoimthanceantcalcolharet (2) n which.U is the velocity ye The on natural com en erical ing At the refere convectio ord rynumbedmeolid-t-heat 0=-+2+p (3) 2.Physical model where is the pressure at reference temperature.Pa:is the ility at reference temperature.mkg subtracting ge oesuareencos le the ot tw +22+-咖8 (4 the present study.the size of the rnethand 3.Mathematical formulation 4-4。+(）西-10+ (5) n饮pmeg 4=m+().-+ x-号 (7) x-9。=(器x-p)-+ N =w+)-+ 张(+)-+-8 1.Physical model and

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, combus￾tion. There may be plenty of applications in engineering field in the near future. Thus, the study of effect of magnetic force on nat￾ural convection in porous media is very important for both scien￾tific 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 con￾vection of air in a porous square enclosure under a magnetic quad￾rupole field using LTNE models has yet to appear in the literature. The present paper represents the results of a numerical investi￾gation 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 car￾ried 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 mag￾nets L2 are 0.024 m, 0.02 m and 0.03 m respectively. 3. Mathematical formulation 3.1. Governing equations The assumptions in the model are as follows: the fluid is consid￾ered 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: fm ¼ vm 2lm rb2 ¼ qf v 2lm rb2 ð1Þ where, fm is the magnetic force; vm is the volumetric magnetic sus￾ceptibility; lm is the magnetic permeability, H m1 ; b is the mag￾netic 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: qf e2 U rU ¼ rp lf j U þ lf e r2 U qf e3=2 1:75 ffiffiffiffiffiffiffiffiffi 150 p jUjU ffiffiffi j p þ qf v 2lm rb2 þ qf g ð2Þ 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 acceler￾ation, 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 qf0v0 2lm rb2 þ qf 0g ð3Þ where p0 is the pressure at reference temperature, Pa; qf0 is the fluid density at reference temperature, kg m3 ; v0 is the mass mag￾netic susceptibility at reference temperature, m3 kg1 , subtracting (3) from (2) gives: qf e2 U rU ¼ rp0 lf j U þ lf e r2 U qf e3=2 1:75 ffiffiffiffiffiffiffiffiffi 150 p jUjU ffiffiffi j p þ ðqf v qf 0v0Þ 2lm rb2 þ ðqf qf0Þg 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: qf v ¼ ðqf vÞ 0 þ @ðqf vÞ @Tf  0 ðTf T0Þþ ð5Þ qf ¼ qf 0 þ @qf @Tf  0 ðTf Tf 0Þþ ð6Þ For the paramagnetic fluid air, the mass magnetic susceptibility is inverse proportion to absolute temperature, according to the Curie’s law: v ¼ 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 ðqfvÞ 0 ¼ @qf @Tf v qf v Tf  0 ðTf T0Þþ ¼ qf bv qf v Tf  0 ðTf T0Þþ ¼ qf0v0b 1 þ 1 T0b ðTf T0Þþ ð8Þ The higher order small amount in Eq. (8) is omitted and gener￾ated into Eq. (4), Eq. (4) becomes as follows. 1 e2 U rU ¼ rp0 qf0 lf qf 0j U þ lf qf 0e r2 U 1 e3=2 1:75 ffiffiffiffiffiffiffiffiffi 150 p jUjU ffiffiffi j p v0b 2lm 1 þ 1 T0b ðTf T0Þrb2 þ bðTf T0Þg ð9Þ where b is thermal expansion coefficient, K1 . Fig. 1. Physical model and coordinate system. Accordingly, the governing equations can be written as: 100 C. Jiang et al. / International Journal of Heat and Mass Transfer 91 (2015) 98–109

C.Jiang et al.Int nd Mass Tranfer91(2015)98-109 verage Nusselt numbers with [45]at various values of Br. 25 Continuity equation: 贺+-0 Momentum equation △T 袋》贵货-源品 Present results Relative error +兰快+ (+制 0 02 Yang et al. Present work with 35](left:temperature field and right:velocity field)

Continuity equation: @u @x þ @v @y ¼ 0 ð10Þ Momentum equation: qf e2 u @u @x þ v @u @y  ¼ @p @x lf j u qf 1:75 ffiffiffiffiffiffiffiffiffi 150 p ðu2 þ v2Þ 1=2 ffiffiffi j p u e3=2 þ lf e @2 u @x2 þ @2 u @y2 ! 1 þ 1 T0b  v0bðTf T0Þ 2lm @ðb2 Þ @x ð11Þ 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-gravita￾tional condition. Grid dimension Num 30  30 2.0635 40  40 2.0519 50  50 2.0478 60  60 2.0460 70  70 2.0454 Table 2 Comparison of present results with [35]. DT Num Yang et al. [35] Present results Relative error/% 1 1.003 1.003 0 10 1.214 1.244 2.47 50 2.120 2.166 2.17 Fig. 2. The comparison of present results with [35] (left: temperature field and right: velocity field). Table 3 Comparison of average Nusselt numbers with [45] at various values of Br. Br Num Song et al. [45] Present results 0.0 T 4.520 4.525 0.5 T 4.506 4.524 1.5 T 4.352 4.360 2.5 T 3.898 3.923 C. Jiang et al. / International Journal of Heat and Mass Transfer 91 (2015) 98–109 101

(b) Fig.3.(a)Distribution of gradient of squ ion (VB"L (b)vec (-CRaPrB22)under the non-gravitational condito Ra=1x10 Ra=5×1 yRa=1×10 7Ra=1x10 rnthe(e)m iddle)and solid phase temperature (right)for Da-10-3,H-10 and 2-05 under the non (装+岁需柴-照”品 Fluid phase energy equation VK 偿 w要+-（德-功间 Solid phase energy equation: -1+) 0=1-9k(++- (14) +PygB(T;-To) (12)

qf e2 u @v @x þ v @v @y  ¼ @p @y lf j v qf 1:75 ffiffiffiffiffiffiffiffiffi 150 p ðu2 þ v2Þ 1=2 ffiffiffi j p v e3=2 þ lf e @2 v @x2 þ @2 v @y2 ! 1 þ 1 T0b  v0bðTf T0Þ 2lm @ðb2 Þ @y þ qf gbðTf T0Þ ð12Þ Fluid phase energy equation: ðqcpÞf u @Tf @x þ v @Tf @y  ¼ ekf @2 Tf @x2 þ @2 Tf @y2 ! þ hðTs TfÞ ð13Þ Solid phase energy equation: 0 ¼ ð1 eÞks @2 Ts @x2 þ @2 Ts @y2 ! þ hðTf TsÞ ð14Þ 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 non￾gravitational convection. 102 C. Jiang et al. / International Journal of Heat and Mass Transfer 91 (2015) 98–109

103 s:u.vare 32.Mathematical mogetic field ity.ms the slid-t-u eat T.B=0 (20 又×H=0 心 oleinuiyequaio 贺%0 (15) B-jH (22 Momentum equation: 喂爱-照品 e水mnl +偎别）- (16) 袋+别-器-源心品 Vom-0 (24 +食别- Fluid phase energyeqution: +(微+v器)-股+器+Ha- ✉ Solid phase energy equation: 0-旋祭+Ng- The dimensionless variables and parameters in the above equa tions are defined as x-Y-u-婴v- 1.0 -器严 m-多w-臣'，B-品=所C=1+ 7盛H-器A co wall temperature. 0.0广0102广03广04广0506广07广08广09广10 Chm nm

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 conduc￾tivity, W m1 K1 ; ks is the solid thermal conductivity, W m1 K1 ; h is the solid-to-fluid heat transfer coefficient, Wm2 K1 ; Ts is solid phase temperature, K. The above equations (10)–(14) can be non-dimensionalised as follows: Continuity equation: @U @X þ @V @Y ¼ 0 ð15Þ Momentum equation: 1 e2 U @U @X þ V @U @Y  ¼ @P @X Pr DaU 1:75 ffiffiffiffiffiffiffiffiffi 150 p ðU2 þ V2 Þ 1=2 ffiffiffiffiffiffi Da p U e3=2 þ Pr e @2 U @X2 þ @2 U @Y2 ! cRaPrhf C 2 @ðB2 Þ @X ð16Þ 1 e2 U @V @X þ V @V @Y  ¼ @P @Y Pr Da V 1:75 ffiffiffiffiffiffiffiffiffi 150 p ðU2 þ V2 Þ 1=2 ffiffiffiffiffiffi Da p V e3=2 þ Pr e @2 V @X2 þ @2 V @Y2 ! cRaPrhf C 2 @ðB2 Þ @Y þ RaPrhf ð17Þ Fluid phase energy equation: ð1 þ K1 Þ U @hf @X þ V @hf @Y  ¼ @2 hf @X2 þ @2 hf @Y2 þ Hðhs hfÞ ð18Þ Solid phase energy equation: 0 ¼ @2 hs @X2 þ @2 hs @Y2 þ KHðhf hsÞ ð19Þ The dimensionless variables and parameters in the above equa￾tions are defined as X ¼ x L ; Y ¼ y L ; U ¼ uL af ; V ¼ vL af hf ¼ Tf T0 Th Tc ; hs ¼ Ts T0 Th Tc ; P ¼ pL2 qf a2 f ; Ra ¼ gbðTh TcÞL3 af mf Pr ¼ mf af ; Da ¼ j L2 ; T0 ¼ Th þ Tc 2 ; B ¼ b b0 ; b0 ¼ Br; C ¼ 1 þ 1 T0b c ¼ v0b2 0 lmgL ; H ¼ hL2 ekf ; K ¼ ekf ð1 eÞks where: X, Y are dimensionless Cartesian coordinates; U, V are dimensionless velocity components; P is the dimensionless pres￾sure; hf is the dimensionless fluid temperature; hs is the dimension￾less solid temperature; Th is the hot wall temperature, K; Tc is the cold wall temperature, K; Pr is the Prandtl number; b0 is the refer￾ence magnetic flux density, T; Br is the magnetic flux density of per￾manent 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 ther￾mal conductivity. 3.2. Mathematical formulation of magnetic field calculation The maxwell equations are applied to describe the magnetic quadrupole field. r B ¼ 0 ð20Þ r  H ¼ 0 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 ð22Þ The scalar magnetic potential um is commonly used to calculate the magnetic field and it satisfies: H ¼ rum 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 ð24Þ 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. C. Jiang et al. / International Journal of Heat and Mass Transfer 91 (2015) 98–109 103

104 C.Jiang性al/aem 33.Boundary conditions 65 5.0 4.5 ◆Da=10 25 3.4.Nusselt number 华 aw-微州 (25) 05.00000的000广 For solid: Rax10 uu=-心 26) The average total Nusselt number 44]is given by: 3.5.Numerical procedure and code verification a-瓜（欲ls+微n 27 =10 H=100 (left).fluid phase e(right)for Da-1 and5 under the non

3.3. Boundary conditions The non-slip conditions are imposed for the two velocity com￾ponents 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 @X     X¼0:5 dY ð25Þ For solid: Nums ¼ Z 0:5 0:5 @hs @X     X¼0:5 dY ð26Þ The average total Nusselt number [44] is given by: Num ¼ 1 ð1 þ KÞ Z 0:5 0:5 K@hf @X     X¼0:5 þ @hs @X     X¼0:5 dY ð27Þ 3.5. Numerical procedure and code verification 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 non￾gravitational condition. Fig. 7. Effect of Darcy number on the average Nusselt number for e = 0.5 under the non-gravitational condition. 104 C. Jiang et al. / International Journal of Heat and Mass Transfer 91 (2015) 98–109

of Heat and Mass Transfer 91(2015)9-109 10e third-ord results for Da=107 s=09999 H=0g= for the aver Nusselt numbe atur ine procedure. combining the tridiagona he present results for Da=.H=0under gravita The t is se that the d by previous authors. of the mat atical mode code must be chec 4.Results and discussio ked before To al grid indepen H.10. A=10 Ra-1×105 arar Da,H and Da=1×10- resent stuc nde on-gra the pre de er on th .71.A-10.and Re a grid (y)dimensionles solid-to-fluid h eat transfer coefficien alieendwasemplog haracteristics in the p 4.1.Numerical results under the non-gravitational condition in a cdgrae drupole field.The B2)h 10 =100 on the erature (middle)and solid pbase emperature (right)for Da=10-Ra-1

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 pres￾sure 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 indepen￾dent 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 com￾parison. 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 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 gravita￾tional 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 convec￾tion 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 gra￾dient 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. C. Jiang et al. / International Journal of Heat and Mass Transfer 91 (2015) 98–109 105

106 amr91(2015)98-109 tem hand side hot air izing force is irec to the square ent nea of the a Nusselt pumbe (Nu, of various Darc gnetizing force s symmetrical in terms of the as 7Ra increases for each Darcy numb r case.On the then th Darcy number incre s from 10-1 to 10- at vari YRa.At low the d b that the low in t enc sure is ve weak and hence the heat tr ith b ransfer in the porous 4.1.1.Effect of yRa increased 42.N cal results under gravitational conditio 0.5.In each graphic nally from opposing wa the the h the i streamline sug s that e flow in the enclos ure is that the ninate from the wall to the s severe def the s are po the ber 0.1020.30.40.50.60.70.809 I th 0-10 hat the cal Nusselt numbe netric about hor which is con stent ddle pos of the hot H the treamlines.fluid phase tempe ondition 0.102030.405060.7广0.809 1. 0.It may be note that the inreased inter-phase heat transfe ases.The incre heat trans se

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 mid￾dle 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 enclo￾sure 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 isother￾mally 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 cel￾lular structure with horizontal symmetry about the middle of the enclosure. The distribution of isotherms suggests that the iso￾therms 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 rela￾tively small, such as cRa = 1  105 , the convection of air in the por￾ous 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 sym￾metric about horizontal centerline, which is consistent with sym￾metric 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 bot￾tom of the hot wall. For the cold wall, the local Nusselt number is maximum in the middle position and gradually presents mono￾tonic 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 magni￾tude of cRa has little effect on the heat transfer rate. With the increase of cRa, the convection in the porous enclosure is strength￾ened 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. 106 C. Jiang et al. / International Journal of Heat and Mass Transfer 91 (2015) 98–109

C.Jiang et al.Int onal jo Het and Mass Transfer 91 (2015)9-109 Da=10-3 Ra=105 and =05 unde the middle o to the case of the pure and temperature feld are very comple and bottom n the ho wall be om nthelowerpirtoi sorce nmDer increasem namely msare approximately the gravity buoyancy force and respectsa clockwi e flo th iatio of the loc Nusselt numbe ncrease of the mag ic force number .so there is a numbers when.From the variatio andgTaCAC the tl ed to be m h f th otati corn r of the po te on the of the left wa ases at fir of th ortex unde the vancy force the number is l ate of the air flo in the right of th on the top of right wall dece H-10 H=100 H=100 the sre (eft).fluid phas ture (middle)and solid phase re (right)for Da=10-7=10.Ra=10 and =05 under the

Da = 103 , Ra = 105 and e = 0.5 under the gravitational environ￾ment. Now the air in the porous enclosure is acted by both mag￾netic 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 mag￾netic field, the counteraction of magnetic buoyancy force and grav￾itational 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 enclo￾sure gradually enlarges and shrinks respectively. If the magnetic force number is larger, the state of the air flow in the porous enclo￾sure is mainly determined by the magnetic buoyancy force, the flow in the enclosure is of two cellular structures with approximately horizontal symmetry about the middle of the enclo￾sure. 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 iso￾therms in the porous enclosure is similar to the case of the pure magnetic convection, namely isotherms are approximately hori￾zontal 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 syn￾ergy 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 bot￾tom 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. C. Jiang et al. / International Journal of Heat and Mass Transfer 91 (2015) 98–109 107