Rayleigh condition

The equation derived by Troelstra and Kruyt is only valid for coagulating dispersions of colloids smaller than a certain maximum diameter given by the Rayleigh condition, d 0.10 A0. Equation 4 applies in cases where particles are transported solely by Brownian motion. Furthermore, the kinetic model (Equations 2 and 3) has been derived under the assumption that the collision efficiency factor does not change with time. In the case of some partially destabilized dispersions one observes a decrease in the collision efficiency factor with time which presumably results from the increase of a certain energy barrier as the size of the agglomerates becomes larger. 

Figure 1 Isotopic change under open- and closed-system Rayleigh conditions for evaporation with a fractionation factor a =1.01 for an initial liquid composition of = 0. The of the remaining water (solid line A), the instantaneous vapor being removed (solid line B), and the accumulated vapor being removed (solid line C) all increase during singlephase, open-system, evaporation under equilibrium conditions. The of water (dashed line D) and...

Now the Rayleigh condition ensures that C > 0. Although C2 may be negative, it is always less negative than C is positive. Furthermore,... 

Hence the Rayleigh condition (12-139) ensures that the condition (12-140) is satisfied. From the Rayleigh form of the stability criteria we see that... 

However, the maximum surface charge (2r that the surface of a droplet can accommodate in vacuum is limited by the Rayleigh condition (Rayleigh 1882) ... 

Experimentally the Rayleigh ratio for benzene at 90° has been observed to equal about 1.58 X 10 m" under the conditions described in this example. By Eq. (10.6), r = (167t/3) so the value of R corresponding to this calculated turbidity is Rg = 5.41 X lO" m". The ratio between the observed value of Rq and that calculated in the example is called the Cabannes factor and equals about 2.9 in this case. 

Prepare a log-log plot of rx versus X and evaluate the slope as a test of the Rayleigh theory applied to air. The factor M/pN in Eq. (10.36) becomes 6.55 X 10 /No, where Nq is the number of gas molecules per cubic centimeter at STP and the numerical factor is the thickness of the atmosphere corrected to STP conditions. Use a selection of the above data to determine several estimates of Nq, and from the average, calculate Avogadro s number. The average value of n - 1 is 2.97 X 10" over the range of wavelengths which are most useful for the evaluation of N. ... 

The left-hand side of the inequality is the slope of the Rayleigh line, and the right-hand side is the slope of the isentrope centered on the initial state. We showed in Section 2.5 that the isentrope and Hugoniot are tangent at the initial state. Thus, the stability condition which requires that the shock wave be supersonic with respect to the material ahead of it is equivalent to the statement that the Rayleigh line must be steeper than the Hugoniot at the initial state. 

The stability condition that the shock wave is subsonic with respect to the shocked material behind it is equivalent to the statement that the Hugoniot must be steeper than the Rayleigh line at the final state. 

Natural Visibility Conditions visibility conditions attributable to Rayleigh scattering and aerosol associated with natural processes. 

Fig. 2.3. Experimental determination of shock-stress versus volume compression from propagating shock waves is accomplished by a series of experiments carried out at different loading pressures. In the figure, the solid lines connect individual pressure-volume points with the initial condition. These solid straight lines are Rayleigh lines. The dashed line indicates an extrapolation into an uninvestigated low pressure region. Such extrapolation is typical of much of the strong shock data.

The presence of D g 26 governing differential equation and the boundary conditions renders a closed-form solution impossible. That is, in analogy to both bending and buckling of a symmetric angle-ply (or anisotropic) plate, the variation in lateral displacement, 5vy, cannot be separated into a function of x alone times a function of y alone. Again, however, the Rayleigh-Ritz approach is quite useful. The expression... 

Residual minimization method (RMM-DIIS). Wood and Zunger [27] proposed lo minimize the norm of the residual vector instead of the Rayleigh quotient. This is an unconstrained minimization condition. Each minimization step starts with the evaluation of the preconditioned residual vector K for the approximate eigenstate... 

Combustion-generated noise is a problem in itself. However, if an acoustic wave can interact with the combustion zone, so that the heat release rate is a function of the acoustic pressure, q = f p ), then Equation 5.1.14 describes a forced oscillator, whose amplitude can potentially reach a high value. The condition for positive feedback was first stated by Rayleigh [23] ... 

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