Stokes number relationship

Figure 8 compares the Reynolds number versus Stokes number relationship between the jet and no-jet formation case from both experimental and theoretical data [3]. The small discrepancy between the two is attributed to the orifice geometry or curvature effect. 

In the Stokes number relationship x is the particle relaxation time defined as... 

Combining the previous two relationships allows us to estimate the extent of granulation as a function of the governing group of the Stokes number, or... 

Let us establish a relationship between the Stokes numbers of the initial and last collisions and estimate the collision number. The decrease in the Stokes number, which corresponds to the repetitive collision St, can be estimated by substituting the particle velocity v, (at its repetitive touch with the bubble surface) into the equation for St. v, can be determined by inserting the time interval T between two collisions (cf Eq. (11.61)) into Eq. (11.60). The second term in Eq. (11.60) can be neglected and we get co,t = n. Taking this into account, from Eqs (11.59) and (11.68) we obtain... 

Even within the group of dynamic methods, one can find in the recent literature entirely different hydration numbers, for instance, those presented in Table 12.2 for biologically important ions [157]. Surprisingly, the fundamental Stokes-Einstein relationship between the hydrodynamic radius and the diffusion coefficient of the ion is being used in several different manners in the calculation of the effective hydrated radius of an ion (compare [158] and [159]). 

In accordance with the value of the Stokes number, particle deposition takes place on the front side (see Fig. IX.5). An analysis of the experimental data shows that in flow around cylindrical and spherical surfaces, the number of solid particles held on the surface will always be less than the number of particles in the impinging stream this is a result of rebound, the probability of which increases with increasing particle velocity. Hence, the Stokes number can be used to characterize particle adhesion only on the front side of the object and only with relatively low flow velocities. Moreover, the relationship between the capture coefficient and the Stokes number has thus far been expressed only qualitatively. 

The term is sometimes called the Best number. " An analytic expression for the terminal velocity corresponding to Stokes law is also available at low Re [Eq. (3-18)]. Outside these ranges of Re, or when more accurate predictions are required, vs. Re relationships are inconvenient for determining terminal velocities since both groups involve Uj. Hence an iterative procedure is needed. It is more convenient to express Re as a function of the latter being independent of Uj. Empirical correlations of this form, based on the same data... 

Figure 8-26 The structure of the 994-residue Ca2+-ATPase of the endoplasmic reticulum of rabbit muscle at 0.8-nm resolution. (A) Predicted topology diagram organized to correspond to the electron density map prepared by electron crystallography of frozen-hydrated tubular crystals. The number of amino acid residues in each connecting loop is marked. (B) The electron density map with the predicted structure embedded. The relationships of the helices in (B) to those in (A) are not unambiguous. The helices marked B, D, E, and F in (B) may form the Ca2+ channel. The large cytoplasmic loops, which are black in (A), were not fitted. From Zhang et al.553 Courtesy of David L. Stokes.

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers theory, and in contrast to the inertial motion in Eyring s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or diffusion) in a rod of variable cross-section or in a stream of variable velocity. The magnitude of the diffusion coefficient is established by comparison with the macroscopic kinetics of the change of nuclei, which does not consider fluctuations (cf. Einstein s application of Stokes law to diffusion). The steady rate of nucleus formation is calculated (the number of nuclei per cubic centimeter per second for a given supersaturation). For condensation of a vapor, the results do not differ from those of Volmer. 

Abnormal Stokes Shift.—The shift of an emission band to frequencies lower than those expected from the usual mirror image relationship between absorption and fluorescence bands (i.e., lack of a common 0-0 band). The phenomenon must always indicate that the emitting state is not the one produced by absorption. A number of processes may be responsible for example, the first formed excited state may undergo a chemical transformation, such as isomerization, so that emission involves a chemical species different from that which originally absorbed the light. 

Because the Reynolds number is much smaller than 1 and a Newtonian flow behavior is being observed in the first place, the Navier-Stokes equations convert to Stokes equations, and we obtain a system of linear equations for the flow calculations. It therefore follows that there must be a linear relationship both between the flow rate and the pressure and between the flow rate and the power. This is demonstrated in Figs. 8.10 and 8.11 in which the dimensionless conveying and power characteristic are illustrated, respectively. The red lines reflect the Newtonian cases. As expected, a linear relationship is revealed. 

Determination of the PSD requires knowledge of the relationship between Do and particle size analogous to the Stokes-Einstein Eq. (39). The number distribution of particle size also requires an expression for P(q). For roughly spherical particles that do not absorb light at the laser wavelength, the Raylcigh-Debye-Gans (RDG) (43,44) approximation for spheres... 

The simplest case to consider is the settling of a single homogeneous sphere, under gravity, in a fluid of infinite extent. Many experiments have been carried out to determine the relationship between settling velocity and particle size under these conditions and a unique relationship between drag factor (C ) and Reynolds number Re) has been found that reduces to a simple equation, known as the Stokes equation, at low Reynolds number. 

At high velocities, due to the onset of turbulence, the drag on the sphere increases above that predicted by Stokes equation and particles settle more slowly than the law predicts. However, settling velocities can be related to particle diameters by applying Newton s equation which is available as an empirical relationship between C, and Re. The upper size limit for Stokes equation is limited, due to the onset of turbulence, to Reynolds number smaller than 0.25. 

Another intriguing quality of Raman spectroscopy is its capability to measure local temperature quantitatively and precisely. This is possible in two distinct ways, arising due to two different characteristics of the Raman spectra in crystalline solids. The first characteristic is the presence of the phonon occupation number in the Raman scattering cross section in accordance with (17.3). While the relation to temperature of the strict intensity of a particular phonon peak is obfuscated by the numerous other components of the Raman scattering cross section, taking the ratio of integrated intensities of the Stokes (1 ) and anti-Stokes (Ias) peaks provides the following relationship by which to measure temperature ... 

It is generally recognized that the ionic conductivity of a low molecular weight liquid is inversely proportional to the bulk viscosity using Stokes law for the drift of a spherical ion through a viscous medium. The conductivity is the sum of the products of the numbers of ions and their mobilities. The relationship between the DC conductivity (o) and the melt viscosity (q) is expressed in the following form [79]. 

The relationship (7.4) can also be derived, if the equation of motion (Navier-Stokes differential equations) are drawn up and dimensionlessly formulated under given boundary conditions (here the continuity and energy equations). W. Nusselt followed this path (1909/1915). The thus derived pi-numbers were later named by... 

Insofar as the mass-transfer coefficient for clean bubbles is concerned (see, for example, the review by Clift et al. (1978)), in the case of spherical bubbles moving under creeping (or Stokes) flow conditions, the following correlation has been proposed Sh = 1 + (1 + 0.564Pe ). For spherical particles with Rep > 70 the Sherwood number can be expressed by the following relationship (Lochiel Calderbank, 1964) ... 

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