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In the present work, numerical simulation of steady, compressible and supersonic airflow in a magneto-hydrodynamic (MHD) generator has been studied. This flow considered to be ideal with low magnetic Reynolds number. A two-dimensional channel with four-pair electrodes and with various geometries and boundary conditions were utilized as a MHD Faraday generator model. The computational model consists of the Navier-Stokes equations coupled with electromagnetic source terms, Maxwell's equations and Ohm's law. Implicit based on density solver is used to solve the Navier-Stokes and the electric potential method is used to solve the Poisson's equation. First, the boundary conditions of constant temperature and constant heat flux were compared. Due to the less Joule heating and generation of higher electrical power, constant heat flux boundary condition was selected to continue working.