Collision parameter

As a practical method, designers have employed other methods such as / -pentane conversion as a key component, kinetic severity factor (31), or molecular collision parameter (32) to represent severity. Alternatively, molecular weight of the complete product distribution has been used to define conversion (A) for Hquid feeds. 

Inverse Collisions.—The particle velocities resulting from a collision between particles of velocities vx and v2, having collision parameters 6 and e, have been denoted as v[ and v they may be found from Eqs. (1-21). Consider now the particle velocities resulting from a collision between particles of velocities v[ and v2, with collision parameters b and e let these final velocities be denoted by v[ and v . 

To determine the collisions that bring particles into the volume around (r,vx), the inverse collisions must be considered it was shown previously that collisions of particles of velocities vx and v2 with collision parameters b and e give rise to a particle of velocity vx. As before, in the volume element g At bib de, a particle of velocity V2 will collide with the particle of velocity vj in the desired manner (to give a vx particle) there are /(r,v, % Af bdbde particles colliding with f(r,v,1,t)drdv 1 particles in drdy 1 so that the total collisions in At, which produce particles of velocity vx at r, are... 

Since the Arrhenius diagram is linear and the collision parameter A is constant over the whole temperature range, the activation energy can be cal-... 

The contact force between two particles is now determined by only five parameters normal and tangential spring stiffness kn and kt, the coefficient of normal and tangential restitution e and et, and the friction coefficient /if. In principle, kn and k, are related to the Young modulus and Poisson ratio of the solid material however, in practice their value must be chosen much smaller, otherwise the time step of the integration needs to become unpractically small. The values for kn and k, are thus mainly determined by computational efficiency and not by the material properties. More on this point is given in the Section III.B.7 on efficiency issues. So, finally we are left with three collision parameters e, et, and which are typical for the type of particle to be modeled. 

Figure 1. Electric field, E(t), and corresponding frequency spectrum, I(v), associated with distant collision of fast electron and molecular target (a) collision parameters—t), electron velocity and b, impact parameters (b) idealized case for very fast electron (c, d) realistic picture.

It should be noted that in electron scattering the amplitudes (or collision parameters) are defined and computed in a laboratory space-fixed coordi-... 

At once, the previous discussion explains why the polar molecules do not exhibit a polarization effect at all the strong anisotropy of the CO-Na potential leads to a complete mixing of 2 and II states. Somewhat less easily explained is the dependence of the polarization effect on CM and its disappearance at larger scattering angles. One possibility is to ascribe small quenching cross sections to small collision parameters and thus to deeper penetration, where the molecular anisotropy is dominant and thus mixes the initial state preparation as discussed previously. 

The collision parameters can be specified further if the double differential cross section is measured. This is usually written as d2a /dEdQ, where E and Q refer to the energy and solid angle of either the scattered positron or the ejected electron. Measurements of this quantity have been made for positron impact and will be described below and compared with data for electrons. 

Table 4.1 Collision parameters. The collision frequency is calculated at 1 atm and 300 K.

The rate of coagulation is considered to be dominated by a binary process involving collisions between two particles. The rate is given by bn,nj, where nl is the number of particles of z th size and b a collision parameter. For collision between i - and / -sized particles during Brownian motion, the physicist M. Smoluchowski derived the relation ... 

We have examined the nature of LIFS in some detail. The response of an atomic or molecular system is described in terms of appropriate rate (or balance) equations whose individual terms represent the rate at which individual quantum states are populated and depopulated by radiative and collisional processes. Given the response of a system to laser excitation, one may use the rate equations to recover information about total number density, temperature and collision parameters. 

Another method, devised by Cohen et al. to determine oxygen-rate gas collision parameters is to define an effective spin-orbit operator that includes r dependence, Zeff/r3, where the value of Zeff is adjusted to match experimental data (76). Langhoff has compared this technique with all-electron calculations using the full microscopic spin-orbit Hamiltonian for the rare-gas-oxide potential curves and found very good agreement (77). This operator has also been employed in REP calculations on Si (73), UF6 (78), U02+ and Th02 (79), and UF5 (80). The REPs employed in these calculations are based on Cowen-Griffin atomic orbitals, which include the relativistic mass-velocity and Darwin effects but do not include spin-orbit effects. Wadt (73), has made comparisons with calculations on Si by Stevens and Krauss (81), who employed the ab initio REP-based spin-orbit operator of Ermler et al. (35). 

These experiments are important because they are performed on a reaction for which a priori calculations of V(rAB, rBC, rCA) are likely to have their best chance of success as only three electrons are involved. Even here the accurate computation of V, frequently termed the potential-energy hypersurface, is extremely difficult. Porter and Karplus [19] have determined a semiempirical hypersurface, and Karplus, Porter, and Sharma [20] have calculated classical trajectories across it. This type of computer experiment has been mentioned before and will be described in greater detail later. The objective of Karplus et al. was to calculate aR(E) and E0. Collisions were therefore simulated at selected values of E, with other collision parameters selected by Monte Carlo procedures, and the subsequent trajectories were calculated using the classical equations of motion. Above E0, oR was found to rise to a maximum value, of the same order of magnitude as the gas-kinetic cross section, and then gradually to decrease to greater energies. 

In order to obtain properly averaged results, either collision parameters for each trajectory must be selected by Monte Carlo methods or, when starting values are systematically chosen, the final results must be integrated over complete statistical distributions. The purpose of a Monte Carlo selection technique is to ensure that the distributions of each parameter within a sample of trajectories approach the true statistical distributions as the size of the sample grows. Some examples of how this can be done for different types of distribution function will be described below. Before starting the integration, it is generally necessary to transform the selected values of the collisional... 

In this case no detailed collision kinetics are involved. The collision parameters rnm and the parameters in the Maxwellian post-collision velocity distribution f m are derived from experimentally determined gas viscosity or diffusivity, and the collisional invariants, respectively. Usually this term is negligible in present experiments, but exceptions exist [16]. In particular for ITER, and the high collisionality there, these terms are expected to become more relevant. However, due to the BGK-approximations made, their implementation into the models does not require further discussion here. 

Molecular Collision Parameter. For high severity cracking, especially in the case of heavy feed stocks, hydrocarbon partial pressure and short residence time are necessary to get high yields and economic running times. Hydrocarbon partial pressure and residence time are interrelated for a fixed coil design. The molecular collision parameter combines these parameters to a single number (2) (see Equation 1),... 

Figure 3a. Methane yields as a function of molecular collision parameter. Feedstock naphtha.

DPMs can also be used to understand the influence of particle properties on fluidization behavior. It has been demonstrated that ideal particles with restitution coefficient of unity and zero coefficient of friction, lead to entirely different fluidization behavior than that observed with non-ideal particles. Simulation results of gas-solid flow in a riser reactor reported by Hoomans (2000) for ideal and nonideal particles are shown in Fig. 12.8. The well-known core-annulus flow structure can be observed only in the simulation with non-ideal particles. These comments are also applicable to simulations of bubbling beds. With ideal collision parameters, bubbling was not observed, contrary to the experimental evidence. Simulations with soft-sphere models with ideal particles also indicate that no bubbling is observed for fluidization of ideal particles (Hoomans, 2000). Apart from the particle characteristics, particle size distribution may also affect simulation results. For example, results of bubble formation simulations of Hoomans (2000) indicate that accounting... 

In this expression n stands for the reduced mass, l0 for the orbital angular momentum for which the phase shift rj assumes its maximum value rj0 the absolute value of the second derivative of the phase shift with respect to / at 1 = l0 is denoted by rjo, (10) is linear in the anisotropy parameters q2.6 and 9212 °f (5) the quantities S(n, l - /, , b) are integrals which are calculated and tabulated by Franssen (1973) for a range of values of the reduced energy and the reduced collision parameter b with b0 = 21/6(/0 + 1/2)/(Rmk). The parameter n takes the values 12 and 6 for the Lennard-Jones potential actually used for / - / one has to insert the values 0 and 2, the difference of the orbital angular momentum of channels which are coupled by the interaction of (5). 

That the anisotropy A is a source of only restricted information can be seen from (10) and (11). For Acrg only the solution of the radial Schrodinger equation enters with / /0. The corresponding collision parameter is b0 lTRm, Olson et al. (1968). Because the influence of the IP is largest at the moment of closest approach, we may conclude that Acrg is sensitive to the AIP mostly at R MRB. 

In (11) the main influence comes from l l with corresponding collision parameters, Reuss (1964) and (1965) and Stolte (1972),... 

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