In this paper, by means of the finite element method, the MHD free convection flow of non-Newtonian nanoliquid inside a halved annulus enclosure considering a constant heat flux is being applied to the straight walls has been analyzed. The enclosure is loaded with water and nanoparticles of alumina (Al₂O₃) the combination of these tows produces a liquid that shows shear-thinning behavior. The governing parameters of this study are the Rayleigh number (10³ ≤ Ra ≤ 10⁶), inclination angle (0 ≤ α ≤ 90), power-law index (0.6 ≤ n ≤ 1) and Hartmann number (0 ≤ Ha ≤ 100). The outcomes indicate that the inclination angle of the cavity and the Hartmann number can be considered as effective control parameters at different Rayleigh numbers. While the magnetic field is applied the velocity field retarded and hence convection heat transfer and the Nusselt number diminished, and the increment of power-law index prompts the heat transfer to drop. It is shown that the power-law index and Hartmann reduces the local Nusselt number. Also, the average Bejan numbers corresponding of active parameters are discussed in this study.