Relative velocity average

ASpGr = Difference in specific gravity of the particle and the surrounding fluid tyvg = Average residence time based on liquid flow rate and vessel volume, min tmjp = Minimum residence time to allow particles to settle based on Stokes Law, min u = Relative velocity between particle and main body of fluid, ft/sec... 

Gal-Or and Resnick (Gl) have developed a simplified theoretical model for the calculation of mass-transfer rates for a sparingly soluble gas in an agtitated gas-liquid contactor. The model is based on the average gas residencetime, and its use requires, among other things, knowledge of bubble diameter. In a related study (G2) a photographic technique for the determination of bubble flow patterns and of the relative velocity between bubbles and liquid is described. 

The proposed technique will be used here to illustrate the case of interfacial heat and multicomponent mass transfer in a perfectly mixed gas-liquid disperser. Since in this case the holding time is also the average residence time, the gas and liquid phases spend the same time on the average. If xc = zd = f, then for small values of t, the local residence times tc and td of adjacent elements of the continuous and dispersed phases are nearly of the same order of magnitude, and hence these two elements remain in the disperser for nearly equal times. One may conclude from this that the local relative velocity between them is negligibly small, at least for small average residence times. Gal-Or and Walatka (G9) have recently shown that this is justified especially in dispersions of high <6 values and relatively small bubbles in actual practice where surfactants are present. Under this domain, Eqs. (66), (68), (69) show that as the bubble size decreases, the quantity of surfactants necessary to make a bubble behave like a solid particle becomes smaller. Under these circumstances (pd + y) - oo and Eq. (69) reduces to... 

When in suspension however cells tend to move at the minimum relative velocity with respect to the surrounding fluid. This means that a spherical cell will rotate at the velocity gradient of the suspension, or in proportion to it. For a spherical cell of diameter, d, the average shear rate, Yavg> on its surface is given by ... 

We calculate the averages of the absolute and relative velocities in the Maxwell-Boltzmann distribution ... 

Fig. 5. Four basic illumination programs and their outputs. The top row gives the program, intensity I (linear scale) versus time t. The bottom row gives the growth output, velocity (relative to average velocity) versus time. The second and third row give the level of adaptation A, and the subjective intensity, i = I/A, calculated according to the theory developed. Note that the scale used to plot i(t) is twenty times larger for the down than for the up programs. (From Delbriick and Rei-chardt, 1956)...

When velocity is not selected, an average over the relative velocity yields... 

For determination of reaction probability and reaction cross section, a large number of collision trajectories have to be considered and appropriate averages over the initial conditions performed. The reaction probability is calculated for a specified initial relative velocity vR (i.e. initial relative kinetic energy), rotational state /, and impact parameter b. The reaction probability is the ratio of number of reactive trajectories to the total number trajectories, i.e. 

The release location influences the vertical distribution of the time-averaged concentration and fluctuations. For a bed-level release, vertical profiles of the time-averaged concentration are self-similar and agreed well with gradient diffusion theory [26], In contrast, the vertical profiles for an elevated release have a peak value above the bed and are not self-similar because the distance from the source to the bed introduces a finite length scale [3, 25, 37], Additionally, it is clear that the size and relative velocity of the chemical release affects both the mean and fluctuating concentration [4], The orientation of the release also appears to influence the plume structure. The shape of the profiles of the standard deviation of the concentration fluctuations is different in the study of Crimaldi et al. [29] compared with those of Fackrell and Robins [25] and Bara et al. [26], Crimaldi et al. [29] attributed the difference to the release orientation, which was vertically upward from a flush-mounted orifice at the bed in their study. 

The reaction rate coefficients in the above equations may be related to reaction rates per pair of particles 2/, in nuclear physics (e.g., Fowler et al., 1975 Harris et al., 1983) by k = Xj/(1 + 5/ ), where 8 = 0 except for i= , for which 5/ = 1. That is, for Reactions 2-145 and 2-147 in which two identical particles collide to react, the definition of k is half of defined by nuclear physicists and for reactions in which different particles collide, the definition of k is the same as Xij. The reaction rate coefficients depend on temperature in a complicated way (Table 2-3) and may be calculated as the average value of the product of relative velocity times cross section. The concentrations of the intermediate species can be derived as follows. From Equation 2-155, 145 [ H] = ki4e[ H]pH]. That is. 

Spring-bead models relate frictional force to the relative velocity of the medium at the point of interaction. The entanglement friction coefficient above is defined in terms of the relative velocity of the passing chain. Since the coupling point lies, on the average, midway between the centers of the two molecules involved, the macroscopic shear rate must be doubled when applying the result to a spring-bead model. Substitution of 2 CE for Con in the Rouse expression for viscosity yields... 

These latter measurements led only to relative cross-section values. However, by comparison with absolute values of velocity-averaged cross sections, they can be put on an absolute scale. To do this, the absolute values obtained in FA measurements were used because here the velocity distribution is exactly known—a Maxwellian distribution /(t>, T) with the temperature of the buffer gas. Denoting the velocity-dependent relative total ionization cross section, obtained in the beam experiment, by oKl(v) and the absolute total ionization rate constant obtained in the FA experiment by R(T), then a normalization k may be determined by... 

The average relative velocity between particles 1 and 2 can be evaluated by... 

Figure 16.2. Evolution of a typical WIMP number density in the early universe. The number of WIMPs in a volume expanding with the universe (comoving density) first decreases exponentially due the Boltzmann factor e-m/T an(j then freezes out to a constant value when the WIMP annihilation reactions cannot maintain chemical equilibrium between WIMPs and standard model particles. In the figure, (av) is the thermally averaged annihilation cross section times relative velocity. WIMPs with larger annihilation cross section end up with smaller densities.

All the experiments described in this chapter were performed on molecules in gas cells, and, unlike in beam experiments, the results obtained are averaged over all possible orientations of the molecules with respect to the relative velocity vector, and over the Maxwellian distribution of the relative velocities of the colliding particles. Nevertheless, the experimental data stimulated interest in the development of theoretical models of elastic... 

In the above equation, e, n, and u are the internal energy, the number density and the average velocity of the fictitious particle, respectively, at the reflecting spherical surface. Since the net flux of the fictitious particles at the reflecting surface is. zero (i.e., u = 0), Eq. [ 14] can be written in terms of the relative velocity distributions of the incident and reflected colliding particles as,... 

---