An asymptotic new method of solving the generalized Boltzmann equation previously developed by the authors is presented, and its difference from the traditional Chapman-Enskog method is discussed. The application of this new method is considered by an example of an O2/O dissociating mixture flow with separation around a double cone. A comparison with available literature and experimental data is presented. The computed and measured surface heat fluxes are found to be in reasonable agreement. The computed surface heat flux is analyzed through comparisons to numerical results of other researchers and codes.