In this work, the hydrodynamics of multicomponent ideal gas mixtures have been studied. Starting from kinetic equations, the Eulerian approach is used to derive equations of motion for a multicomponent system, where each component may have a different velocity and kinetic temperature. The equations are based on Grad's method of moment derived from the kinetic model, in a Relaxation Time Approximation (RTA). Based on this model, a computer code has been developed for numerical computation of compressible flows of a binary gas mixture in generalized curvilinear boundary conforming coordinates. Since these equations are similar to the Navier-Stokes equations for the single fluid systems, the same numerical methods are applied to these new equations. The Roe's numerical scheme is used to discretize the convective terms of governing fluid flow equations. The prepared algorithm and the computer code are capable of computing and presenting the flow fields of each component of the system separately, as well as the average flow field of the multicomponent gas system as a whole. A comparison between the present code results and those of a more common algorithm based on the mixture theory in a supersonic converging-diverging nozzle provides the validation of the present formulation. Afterwards, a more involved nozzle cooling problem with a binary ideal gas (Helium-Xenon) is chosen to compare the present results with those of the ordinary mixture theory. The present model provides the details of the flow fields of each component separately which is not available otherwise.