Collisional KE

The kinetic equation (KE) for homogeneous systems can be written by grouping together all the source terms appearing in Eq. (7.1) on the right-hand side  

The collision kernel depends upon the type of interactions between particles, and here we focus on the hard-sphere collision kernel Z (g, x) = rfp g x. An alternative kernel is, for example, the Maxwell kernel described in Chapter 6. 

Sectional and class methods for the solution of the collisional KE are generally called discrete-velocity methods (DVM). These methods are based on the simple idea of discretizing the velocity space into a grid constituted by a finite number of points. The existing methods are characterized by different grid structures (Aristov, 2001). For example, lattice Boltzmann methods discretize the velocity space into a regular cubic lattice with a constant lattice size (Li-Shi, 2000), whereas other methods employ different discretization schemes (Monaco Preziosi, 1990). By using a similar approach to that used with PBE, it is possible to define A,- as the number density of the particles with velocity and the discretized KE becomes 

In many problems we are not interested in the distribution of particle-velocity vectors, but under certain conditions the distribution of the particle-velocity magnitude gives an adequate description of the system. One such situation is when the hypothesis of isotropy holds. Under this simplifying hypothesis, the direction of the velocity vector loses any importance and as a consequence can be replaced by the scalar = (Vj + 

Equation (7.54) holds at the equilibrium and when the initial NDF is isotropic. In order to illustrate how the discretization for the kinetic equation is typically carried out, some additional simplifications must be introduced. It is convenient to write the collision integral as 

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The generic application of QBMM to the homogeneous collisional KE is straightforward. Recall that the generic definition of the moment of the velocity NDF is miju = m(i, j, k) = f and that its evolution obeys the following equation ... 

CgH (n = 6, 7, 8). A novel collision-induced isomerization of CgH7 (10a), which has a sttained allenic bond, to (lOyS) has been reported to occur upon SIFT injection of (10a) at elevated kinetic energies (KE) and collision with helium. In contrast, radical anions (9) and (11) undergo electron detachment upon collisional excitation with helium. Bimolecular reactions of the ions with NO, NO2, SO2, COS, CS2, and O2 have been examined. The remarkable formation of CN on reaction of (11) with NO has been attributed to cycloaddition of NO to the triple bond followed by eliminative rearrangement. 

Let the rate constant for collisional excitation to any particular vibrational level v be denoted ke (v). Then... 

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