C.Jiang et al /Inte ass Tranfer91(2015)98-109 65 and The man onof theretr 3.0 (1)For the ousenclol-g ondition the fow of the air in th cellular icture 2.0 ong the left hot wall and the right cold r increase ntal c (2)n the gravity c e increas of the magneti 0.5 a clockwise vo at firs appears the stratified for Ra=10P an nsities.Finally (Theod phase temperature is affected degreea Conflict of interest None declared. s are very weak References 飞 21 5 M Ami of the 3引 co vection of r in thep rous enc losu orTeh5itecmtonhcohctcransifcrteoattheoraS ime,the average Nusselt number respects the trend d of de I81 of the the mag eemGAeteteoa2getcne number. 112引 5.Conclusion Thermoma gnetic convection of air was investigated numeri- in the presenforce 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 gravita￾tional 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 equilib￾rium 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 num￾ber 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 counterac￾tion 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 por￾ous enclosure increases with the increase of the magnetic force number. 5. Conclusion Thermomagnetic convection of air was investigated numeri￾cally in a two-dimensional porous square enclosure under a mag￾netic quadrupole field in the presence or absence of a gravity field. The scalar magnetic potential method was used to calculate the 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: (1) For the non-gravity condition, the flow of the air in the por￾ous 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. References [1] K. Vafai, C.L. Tien, Boundary and inertia effects on flow and heat transfer in porous media, Int. J. 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Hashim, Natural convection in a porous trapezoidal enclosure with an inclined magnetic field, Comput. Fluids 47 (2011) 155–164. [13] T. Grosan, C. Revnic, I. Pop, D.B. Ingham, Magnetic field and internal heat generation effects on the free convection in a rectangular cavity filled with a porous medium, Int. J. Heat Mass Transfer 52 (2009) 1525–1533. [14] D.A. Nield, Impracticality of MHD convection in a porous medium, Transp. Porous Media 73 (2008) 379–380. [15] M. Sathiyamoorthy, Analysis of convective heat transfer in a square cavity filled with porous medium under a magnetic field, Spec. Topics Rev. Porous Media – Int. J. 2 (2007) 171–180. Fig. 11. Effect of Darcy number on the average Nusselt number for Ra = 105 and e = 0.5 under the gravitational condition. 108 C. Jiang et al. / International Journal of Heat and Mass Transfer 91 (2015) 98–109