Time-averaged distribution

FIG. 23-44 Schematic representation of time-averaged distribution and spread for a continuous plume. and o2 are the statistical measures of crosswind and vertical dimensions 4.3oy is the width corresponding to a concentration 0.1 of the central value when the distribution is of gaussian form (a corresponding cloud height is 2.15o2). (Redrawn from Pasquill and Smith, Atmospheric Diffusion, 3d ed., Ellis Norwood Limited, Chichester, U.K, 1983). 

Molecules that have no permanent dipole still have their electrons in movement. Although the time-averaged distribution of electrons is symmetrical, at any instant the electrons are not uniformly distributed, so the molecule has a small instantaneous dipole, p. This instantaneous dipole can polarize electrons in a neighboring molecule, giving a small dipole in the molecules. This is the dispersion attraction responsible for molecules sticking together. These dispersions forces are the weakest of all inter-... 

The genius of Debye and Hiiekel lay in their formulation of a very simple but powerful model for the time-averaged distribution of ions in very dilute solutions of electrolytes. From this distribution they were able to obtain the electrostatic potential contributed by the surrounding ions to the total electrostatic potential at the reference ion and hence the chemical-potential change arising from ion-ion interactions [Eq.(3.3)]. Attention will now be focused on their approach. 

Hence, the complicated problem of the time-averaged distribution of ions inside an electrolytic solution reduces, in the Debye-Hiickel model, to the mathematically simpler problem of finding out how the excess charge density q varies with distance r from the central ion. 

Fig. 5 illustrates a physical model of the chromatography process. Initially, there is a dynamic equilibrium of molecules between the phases. Then, one phase is moved relative to the other with an average velocity, v. In the stationary phase, molecules do not move while in the mobile phase, molecules move with a velocity equal to v. Provided that the interphase mass transfer rate is fast relative to the flow rate of the mobile phase, the time-average distribution of a molecule between the phases is statistically equal to the equilibrium distribution as determined by the distribution constant. 

With the time-average distribution equal to the equilibrium distribution, the fraction of time spent by a particular molecule in the mobile phase, Ti, can be evaluated mathematically as... 

Consider the interaction of two electrons, e and c2, that are located in the AOs and (frv. We do not exclude the possibility that the two electrons are in the same AO, pi = v, provided that they have opposite spin. The time-averaged distribution of electron 1 is given by 2(ci)dvi and that of electron 2 by 2(c2)dv2 (Bom interpretation). Therefore, the... 

Figure 15.8 shows the time-averaged distributions of heat releeise with and without control. With strong oscillations and no control, the flame is anchored to the flame holder. Thus, the pressure anti-node is associated with high heat release, a condition that will promote thermoacoustic coupling. With control, the flame is slightly lifted, and heat release at the flame holder or pressure anti-node is considerably lower. 

From the low-temperature fluorescence work of Vanderkooi, we would expect that if the red edge of the absorption spectrum of the complex is excited, then only a few of the collective (Debye) modes will be excited. Work of Vanderkooi was steady state, and thus only the time-averaged distribution of collective modes is seen in a time-resolved experiment with a subpicosecond time scale resolution the actual occupation of the collective mode can be observed directly, since the fact that the mode is coupled to the chromophore makes the chromophore absorbance a function of the collective mode occupation. If the coherence time of the collective mode is sufficiently long, one might actually expect to find a periodic modulation of the chromophore absorbance as collective mode energy oscillates between the few modes excited. If we assume that the Debye spectrum has a maximum at about 100 cm , then these modes should show up with a frequency of about 3 x 10 Hz, or a period of about... 

Distribution and Correlation Functions.—We consider a single spherical particle with position r and velocity v at time t in a concentrated dispersion of mean number density p. The distribution function measures the probability of finding a particle (the same or another particle) with position r" and velocity v" at time t". The osmotic equation of state is related to a time-averaged distribution function that depends on r alone, whereas the dynamic behaviour depends on time-dependent functions. A basic premise of statistical mechanics is that a time-average is equivalent to an ensemble average at fixed time the ensemble average is denoted by angular brackets (...). 

Figure 17 Predicted time-averaged distribution of solids volume fraction in (A) bubbling fluidized bed, (B) turbulent fluidized bed, and (C) CFB riser (Hong et al, 2015).

The theory goes as follows in the stable liquid state there are two distinct structural species of the liquid, a high density liquid (HDL) and a low density liquid (LDL). Due to fluctuations each component will vary between the two species, with only the time averaged distribution of species staying constant for a specific temperature and pressure. Upon an increase in pressure the proportion of the more dense species increases, which leads to an increase in the density of the liquid above that of the corresponding crystal. This causes the melting curve to have a negative relationship with pressure, as can be seen from the Clausius-Clapeyron relationship ... 

Figure 22 Time-averaged distribution of the particle velocities in the whole apparatus (all particles) and inside the tube (h<320 mm) for the case study 1.

A quantitative comparison of the particle dynamics in the Wurster tube for all three case studies is presented in Table 8. The time-averaged distributions of the vertical component of translational velocity and angular velocity of particles inside the tube are shown in Figs. 25 and 26, respectively. 

Figure 25 Time-averaged distribution of the vertical velocity of the particles inside the tube.

---