A solution of the Boltzmann equation in the presence of inelastic collisions

The effect of inelastic collisions between two molecules on the solution of the Boltzmann equation is taken into account by presenting the change of state of molecules after collisions as a random (with uniform probability distribution) movement along a surface of an NN-dimensional sphere, the squared radius of which is equal to the total energy of the molecules before and after the collision in the reference system of the centre of mass. The projection of a point on the surface of this sphere in each of N   directions gives the root square of the kinetic energy in one of three directions in the physical space, or the internal energy of one of degrees of freedom, of one of two molecules. The kinetic energies of two molecules are described by the first six dimensions of the system, and the remaining (N-6)(N-6) dimensions describe the internal energies. This approach is applied to three test problems: shock wave structure in nitrogen, one-dimensional heat transfer through a mixture of n-dodecane and nitrogen and one-dimensional evaporation of n-dodecane into nitrogen. In the first problem, the predictions of the model are shown to be close to experimental data and also to the predictions of the earlier developed model, based on a different approach to taking into account the effects of inelastic collisions. The predicted heat flux for the second problem and mass flux for the third problem are shown to be very weak functions of the number of internal degrees of freedom when this number exceeds about 15. These results open the way for considering systems with arbitrarily large numbers of internal degrees of freedom by reducing the analysis of these systems to the analysis of systems with relatively small numbers of internal degrees of freedom.