A.M. Al-Amiri

D. Braithwaite

D. Gobin

Feng Wei

J.R. Carruthers

Jiang Changwei

L.J. Yang

M. Zeng

Q.W. Wang

R. Shigemitsu

T. Bednarz

T. Tagawa

Zen Junyong

Zhong Hui

Zhu Qiangming

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J Supercond Nov Magn (2014) 27:519–525DOI 10.1007/s10948-013-2298-xO R I G I NA L PA P E RNumerical Simulation of Thermomagnetic Convection of Airin a Porous Square Enclosure Under a Magnetic Quadrupole FieldJiang Changwei · Zhong Hui · Feng Wei · Zen Junyong ·Zhu QiangmingReceived: 4 June 2013 / Accepted: 28 June 2013 / Published online: 28 July 2013© The Author(s) 2013. This article is published with open access at Springerlink.comAbstract The thermomagnetic convection of air in a two-dimensional porous square enclosure under a magneticquadrupole field is numerically investigated. The ScalarMagnetic Potential Method is used to calculate the mag-netic field. A generalized model, which includes a Brinkmanterm, a Forcheimmer term, and a nonlinear convective term,is used to solve the momentum. The flow and temperaturefields for the air thermomagnetic convection are presentedand the local and average Nusselt numbers on the walls arecalculated and compared. The results show that the magneticfield intensity, the Darcy number and the Rayleigh numberhave a significant effect on the flow field and heat transfer ina porous square enclosure.Keywords Thermomagnetic convection · Numericalsimulation · Porous media · Magnetic quadrupole field ·Magnetic force1 IntroductionEnhancements or suppressions of the convection phenom-ena and improvement of heat and mass transfer continue tobe an active research area, due to their significance for bothfundamental interests and engineering applications, such assolar receivers, cooling of electronic devices, solidificationJ. Changwei (B) · Z. Hui · F. Wei · Z. Junyong · Z. QiangmingSchool of Energy and Power Engineering, Changsha Universityof Science and Technology, Changsha 410114, Chinae-mail: jiangcw730525@163.comJ. Changwei · Z. Hui · F. Wei · Z. Junyong · Z. QiangmingKey Laboratory of Efficient and Clean Energy Utilization,College of Hunan Province, Changsha 410114, Chinaof materials and so on. There are many methods of enhance-ments or suppressions of the convection phenomena, for ex-ample by placing fins on the heated wall, exerting electricand magnetic fields, etc. [1, 2].Recently, magnetic force has received more attention inthe field of metallic materials, and less in the field of non-metallic materials. With the development of a superconduct-ing magnet providing strong magnetic induction of 10 Teslaor more in recent years, the suppression or enhancementof the natural convection of the paramagnetic fluids likeoxygen gas and air by a magnetic field has become an in-teresting research topic investigated by many researchers.Many research works about magnetically induced naturalconvection have followed [3]. The effect of the magneticbuoyancy force on the convection of the paramagnetic flu-ids was first reported by Braithwaite et al. [4]. They used themagnetic field both to enhance and suppress the Rayleigh–Benard convection in a solution of gadolinium-nitrate in ashallow layer heated from below and cooled from above andshowed that the effect depends on the relative orientationof the magnetic force and the temperature gradient. Car-ruthers and Wolfe [5] studied the thermomagnetic convec-tion of oxygen gas in a rectangular container with thermaland magnetic field gradients theoretically and experimen-tally, and found that magnetic buoyancy force canceled outthe influence of gravitational buoyancy force when the rect-angular enclosure heated from one vertical wall and cooledfrom opposing wall was located in horizontal magnetic fieldwith vertical magnetic field gradient, and horizontal mag-netic field could enhance and suppress the Rayleigh–Benardconvection when the rectangular enclosure heated from be-low and cooled from above was located with the same mode.Shigemitsu’s group [6] derived a model equation for mag-netic convection using a method similar to the Boussinesqapproximation and studied natural convection of paramag-520 J Supercond Nov Magn (2014) 27:519–525netic, diamagnetic and electrically conducting fluids in acubic enclosure with thermal and magnetic field gradientsat different thermal boundaries numerically and experimen-tally. Tagawa and co-workers [7] studied natural convectionof paramagnetic and diamagnetic fluids in a cylinder undergradient magnetic field at different thermal boundaries nu-merically and experimentally and found that the magneticbody due to gradient magnetic field could be used to con-trol heat transfer rate of paramagnetic and diamagnetic flu-ids. Bednarz and co-workers [8] studied natural convectionof paramagnetic fluids in a cubic enclosure under magneticfield by an electric coil numerically and experimentally andanalyzed the effect of inclined angle of electric coil, locationof electric coil, Ra number, γ number on heat transfer rateof paramagnetic fluids.The above studies are concerned with the effect of mag-netic force on natural convection of paramagnetic fluids.However, almost no attention has been paid to the combinedeffects of magnetic and gravitational forces on the naturalconvection of paramagnetic fluids in porous medium. Nat-ural convection in an enclosure filled with a paramagneticor diamagnetic fluid-saturated porous medium under strongmagnetic field was numerically investigated by Wang et al.[9, 10]. Considering the effect of Darcy number, Rayleighnumber and γ number, the results of numerical investiga-tion showed that the magnetic force has a significant effecton the flow field and heat transfer in a paramagnetic or dia-magnetic fluid-saturated porous medium. The application ofstrong magnetic field on porous medium may be found inthe field of medical treatment, metallurgy, materials process-ing, combustion. There may be plenty of applications in thenear future in the field of engineering operations. Thus, thestudy of the effect of magnetic force on natural convectionin porous media is important for both scientific research andengineering application.2 Physical ModelThe schematic of the system under consideration is shownin Fig. 1. The system consists of a porous cubic enclosurewhich is kept in a horizontal position and four permanentmagnets which generate a magnetic field. The porous cubicenclosure filled with air is heated isothermally from the left-hand side vertical wall and cooled isothermally from oppos-ing wall while the other four walls are thermally insulated.The gravitational force acts in the minus Y direction. In thepresent study, the length of the enclosure L, the length of thepermanent magnet L1 and the distance of permanent mag-nets L2 are 0.024, 0.02 and 0.03 m, respectively.Fig. 1 Physical model and coordinate system3 Mathematical Formulation3.1 Governing EquationsThe assumptions in the model are the following. The fluidis considered to be steady, incompressible, Newtonian. Boththe viscous heat dissipation and magnetic dissipation are as-sumed to be negligible.According to Braithwaite et al. [4], the magnetic forcecan be given asfm = χm2μm ∇b2 = ρχ2μm∇b2 (1)where fm is the magnetic force; χm is the volumetricmagnetic susceptibility; μm is the magnetic permeability,H m−1; b is the magnetic flux density, T; ρ is the fluid den-sity, kg m−3; χ is the mass magnetic susceptibility, m3 kg−1.The Navier–Stokes equation which includes the magneticforce can be presented asρε2U · ∇U = −∇p − μκU + με∇2U − ρε3/21.75√150|U|U√κ+ ρχ2μm∇b2 + ρg (2)in which U is the velocity vector; p is the pressure, Pa; μ isthe fluid kinematic viscosity, kg m−1 s−1;g is the gravita-tional acceleration, m s−2; ε is the porosity; κ is the perme-ability, m2.At the reference state of the isothermal state, there willbe no convection. Therefore Eq. (2) becomes0 = −∇p0 + ρ0χ02μm ∇b2 + ρ0g (3)where p0 is the pressure at reference temperature, Pa; ρ0is the fluid density at reference temperature, kg m−3;χ0 isJ Supercond Nov Magn (2014) 27:519–525 521the mass magnetic susceptibility at reference temperature,m3 kg−1. Subtracting (3) from (2) givesρε2U · ∇U = −∇p′ − μκU + με∇2U − ρε3/21.75√150|U|U√κ+ (ρχ − ρ0χ0)2μm∇b2 + (ρ − ρ0)g (4)where p = p0 + p′, p′ is the pressure difference due to theperturbed state, Pa. Since ρ and χ are functions of temper-ature, according to Taylor expansion method, ρχ and ρ canbe respectively indicated asρχ = (ρχ)0 +(∂(ρχ)∂T)0(T − T0) + · · · (5)ρ = ρ0 +(∂ρ∂T)0(T − T0) + · · · (6)For air as a paramagnetic fluid, mass magnetic suscep-tibility is in inverse proportion to absolute temperature; ac-cording to the Curie law:χ = mT(7)where m is the constant value; T is the fluid temperature,K; T0 = (Th + Tc)/2, K; subscripts 0, h, c represent the ref-erence value, hot and cold, respectively. So Eq. (5) can bewritten asρχ − (ρχ)0 =(∂ρ∂Tχ − ρ χT)0(T − T0) + · · ·=(−ρβχ − ρ χT)0(T − T0) + · · ·= −ρ0χ0β(1 + 1T0β)(T − T0) + · · · (8)The higher order small amount in Eq. (8) is omitted and gen-erated into Eq. (4), which becomes1ε2U · ∇U = −∇p′ρ0− μρ0κU + μρ0ε∇2U− 1ε3/21.75√150|U|U√κ− χ0β2μm(1 + 1T0β)(T − T0)∇b2+ β(T − T0)g (9)where β is the thermal expansion coefficient, K−1.For simplicity, subscripts of physical parameters and thesuperscript of pressure are omitted, so the governing equa-tions can be written as:Continuity equation:∂u∂x+ ∂v∂y= 0 (10)Momentum equation:ρε2(u∂u∂x+ v ∂u∂y)= −∂p∂x− μκu − ρ 1.75√150(u2 + v2)1/2√κuε3/2+ με(∂2u∂x2+ ∂2u∂y2)−(1 + 1T0β)χ0β(T − T0)2μm∂(b2)∂x(11)ρε2(u∂v∂x+ v ∂v∂y)= −∂p∂y− μκv − ρ 1.75√150(u2 + v2)1/2√κvε3/2+ με(∂2v∂x2+ ∂2v∂y2)−(1 + 1T0β)χ0β(T − T0)2μm∂(b2)∂y+ ρgβ(T − T0) (12)Energy equation:u∂T∂x+ v ∂T∂y= α(∂2T∂x2+ ∂2T∂y2)(13)where: x, y are Cartesian coordinates; u, v are velocity com-ponents, m s−1;α is the fluid thermal diffusivity, m s−1;ν isthe fluid dynamic viscosity, m2 s−1.The above Eqs. (10)–(13) can be non-dimensionalized asfollows:Continuity equation:∂U∂X+ ∂V∂Y= 0 (14)Momentum equation:1ε2(U∂U∂X+ V ∂U∂Y)= −∂P∂X− PrDaU − 1.75√150(U2 + V 2)1/2√DaUε3/2+ Prε(∂2U∂X2+ ∂2U∂Y 2)− γ RaPrθ C2∂(B2)∂X(15)522 J Supercond Nov Magn (2014) 27:519–5251ε2(U∂V∂X+ V ∂V∂Y)= −∂P∂Y− PrDaV − 1.75√150(U2 + V 2)1/2√DaVε3/2+ Prε(∂2V∂X2+ ∂2V∂Y 2)− γ RaPrθ C2∂(B2)∂Y+ RaPrθ(16)Energy equation:U∂θ∂X+ V ∂θ∂Y= ∂2θ∂X2+ ∂2θ∂Y 2(17)The dimensionless variables and parameters in the aboveequations are defined asX = xLY = yLU = uα/LV = vα/Lθ = T − T0Th − Tc P =pρα2/L2Ra = gβ(Th − Tc)L3ναPr = ναDa = κL2T0 = Th + Tc2B = bb0b0 = Br C = 1 + 1T0βγ = χ0b20μmρgLwhere: X, Y are dimensionless Cartesian coordinates; U , Vare dimensionless velocity components; P is the dimension-less pressure; θ is the dimensionless temperature; Th is thetemperature of the hot wall, K; Tc is the temperature of thecold wall, K; Pr is the Prandtl number; b0 is the referencemagnetic flux density, T; L is the length of the enclosure,m; L1 is the length of the permanent magnet, m; L2 is thedistance of permanent magnets, m; Ra is the Rayleigh num-ber; γ is the dimensionless magnetic strength parameter; Bis the dimensionless magnetic flux density; and Da is theDarcy number.3.2 Mathematical Formulation of Magnetic FieldCalculationMaxwell’s equations are applied to describe the magneticquadrupole field:∇ · B = 0 (18)∇ × H = 0 (19)where B is the magnetic flux density and H is the magneticfield intensity.The constitutive relation that describes the behavior ofthe magnetic material isB = μH (20)The scalar magnetic potential ϕm is commonly used to cal-culate the magnetic field and it satisfiesH = −∇ϕm (21)In homogeneous magnetic medium, the permeability is as-sumed to be constant. Combining Eqs. (20), (21) and (18),the scalar magnetic potential satisfies the Laplace’s equa-tion∇ϕm = 0 (22)3.3 Boundary ConditionsNon-slip condition is imposed for the two velocity compo-nents on the solid walls. The temperature boundary condi-tions are: (1) Wall of the enclosure (U = V = 0); (2) Leftvertical wall (X = 0) : θ = 0.5; (3) Right vertical wall (X =1) : θ = −0.5; (4) Top and bottom horizontal adiabatic walls(Y = 0,1) : ∂θ/∂Y = 0.3.4 Nusselt Number CalculationIn order to compare total heat transfer rate, Nusselt num-ber is used. The average Nusselt number at the hot wall isdefined asNum = −∫ 10∂θ∂XX=0dY (23)3.5 Numerical Procedure and Code VerificationThe governing Eqs. (14)–(17) are discretized by the finite-volume method (FVM) based on a non-uniform grid system.The third-order Quick scheme and the second-order centraldifference scheme are implemented for the convection anddiffusion terms. The set of discretized equations for eachvariable is solved by a line-by-line procedure, combining thetridiagonal matrix algorithm (TDMA) with the successiveover-relaxation (SOR) iteration method. The coupling be-tween velocity and pressure is solved by the SIMPLE algo-rithm. The convergence criterion is that the maximal resid-ual of all the governing equations is less than 10−6.The reliability and accuracy of the mathematical modeland code must be checked before calculation. Three gridTable 1 Comparison of present results with Lijun Yang et al. [11]T (K) NumLijun Yang et al. [11] Present Relative error (%)1 1.003 1.003 010 1.214 1.244 2.4750 2.120 2.166 2.17J Supercond Nov Magn (2014) 27:519–525 523Fig. 2 Effect of γ Ra on thestreamlines (left) and isotherms(right) at Da = 10−3 andε = 0.5524 J Supercond Nov Magn (2014) 27:519–525sizes 50 × 50, 60 × 60 and 70 × 70 are selected for analysis.The average Nusselt number for all three grid sizes is mon-itored at ε = 0.5, Pr = 0.71, Ra = 1 × 105, Da = 1 × 10−3,γ = 10. The results showed insignificant differences forthe 60 × 60 grids to the above. Therefore, for all com-putations in this article, a 60 × 60 uniform grid was em-ployed.In order to validate the numerical methods and codes ofthe present work, a recent, similar work by Lijun Yang et al.[11] was selected as the benchmark solution for compari-son. Lijun Yang considered the thermomagnetic convectionin an air-filled 2-D square enclosure confined to a mag-netic quadrupole field under zero-gravity. Table 1 presentscomparisons between the present results for Da = 107, ε =0.9999, g = 0 and those of Lijun Yang for the average Nus-selt number, temperature field and velocity field [11]. It isseen that the present results are in very good agreementwith those obtained by previous authors, which validates thepresent numerical code.4 Results and DiscussionRa becomes zero and γ becomes infinity when g = 0, so thatfinite product γ Ra has to be used to describe non-gravitycases.Figure 2 shows the streamlines and isotherms at vari-ous magnetic force numbers when Da = 10−3 and ε = 0.5.In each graphic shown, the porous enclosure was heatedisothermally from left-hand side vertical wall and cooledisothermally from opposing wall. Obviously, the convectionin the porous enclosure is strengthened with the increase ofγ Ra. The distribution of streamlines suggests that the mag-netic buoyancy force drives the air moving from the left hotwall to the right cold wall along the horizontal middle lineof the porous enclosure; then the air is moving downwardsand upwards to the top and bottom walls and returns to themiddle of the hot wall, so that the flow in the enclosure isof two cellular structures with horizontal symmetry aboutthe middle of the enclosure. The distribution of isothermssuggests that the isotherms in the porous enclosure are hor-izontally symmetric about the middle of the porous enclo-sure; isotherms are dense at the top and bottom of the hotwall and the middle of the cold wall. When γ Ra is relativelysmall, such as γ Ra = 1 × 105, the convection of air in theporous enclosure is very weak so that the heat transfer isdominated by the conduction mechanism, when isothermsapproximately exhibit a linear trend from the hot wall to thecold wall. With the increase of γ Ra, the convection in theporous enclosure is strengthened and the isotherms show se-vere deformation.Figure 3 illustrates the variations of the local Nusseltnumbers along the left hot wall and the right cold wall at var-ious γ Ra when Da = 10−3, ε = 0.5. It can be seen that theFig. 3 Effect of γ Ra on the local Nusselt numbers of the left (top) andright (bottom) side walls at Da = 10−3 and ε = 0.5local Nusselt numbers are increased as the γ Ra increases;the local Nusselt numbers are symmetric about the horizon-tal centerline, which is consistent with the isotherms. For thehot wall, the minimum local Nusselt number appears in themiddle position and the maximum appears in the top andbottom of the hot wall. For the cold wall, the local Nus-selt number is maximal in the middle position and graduallypresents monotonic decrease from the middle of the coldwall to its ends.5 ConclusionThe thermomagnetic convection of air in a two-dimensionalporous square enclosure under a magnetic quadrupole fieldis numerically investigated under non-gravitational condi-tions. The results show that the flow of the air in the enclo-sure is of two cellular structures with horizontal symmetryabout the middle of the enclosure. The local Nusselt num-J Supercond Nov Magn (2014) 27:519–525 525bers along the left hot wall and the right cold wall are in-creased as the magnetic force number increases and are sym-metric about the horizontal centerline. The magnetic fieldintensity, the Darcy number and the Rayleigh number havea significant effect on the flow field and heat transfer in aporous square enclosure.Acknowledgements This work was financially supported by theScientific Research Fund of Hunan Provincial Education Department(11C0060), Scientific Research Fund of Changsha Science and Tech-nology Bureau (K1203003)Open Access This article is distributed under the terms of the Cre-ative Commons Attribution License which permits any use, distribu-tion, and reproduction in any medium, provided the original author(s)and the source are credited.References1. Al-Amiri, A.M.: Numer. Heat Transf., Part a Appl. 41, 817 (2002)2. Gobin, D., Goyeau, B., Neculae, A.: Int. J. Heat Mass Transf. 48,1898 (2005)3. Tagawa, T., Ozoe, H.: Numer. Heat Transf., Part B Fundam. 41, 1(2002)4. Braithwaite, D., Beaugnon, E., Tournier, R.: Nature 354, 134(1991)5. Carruthers, J.R., Wolfe, R.: J. Appl. Phys. 39, 5718 (1968)6. Shigemitsu, R., Tagawa, T., Ozoe, H.: Numer. Heat Tranf., Part AAppl. 43, 449 (2003)7. Tagawa, T., Ujihara, A., Ozoe, H.: Int. J. Heat Mass Transf. 46,4097 (2003)8. Bednarz, T., Tagawa, T., Kaneda, M., et al.: Prog. Comput. FluidDyn. 5, 261 (2005)9. Wang, Q.W., Zeng, M., Huang, Z.P., et al.: Int. J. Heat MassTransf. 50, 3684 (2007)10. Zeng, M., Wang, Q.W., Ozoe, H.: Prog. Comput. Fluid Dyn. 9, 77(2009)11. Yang, L.J., Ren, J.X., Song, Y.Z., et al.: J. Magn. Magn. Mater.261, 377 (2003)

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Numerical Simulation of Thermomagnetic Convection of Air in a Porous Square Enclosure Under a Magnetic Quadrupole Field

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Numerical Simulation of Thermomagnetic Convection of Air in a Porous Square Enclosure Under a Magnetic Quadrupole Field

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