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