Stokes number density dependence

The impact of scale on growth and consolidation depends upon the choice of collisional velocity (Table 21-22). Granule consolidation and density likely increase with the maximum peripheral speed, or scale-up by For growth, on the basis of a critical viscous Stokes number... 

The mean free path of molecules in air at atmospheric pressure is /free — 1 /(Niiyg), where Nl 2.69 10 cm is the number density of gas molecules and cTg 10 " cm is the cross section for elastic collisions of molecules. These numbers result in /free — 3.7 10 cm, or 37 nm. The mean pore radius of the GDL is in the order of 10 pm, which means that the flow in the GDL pores occurs in a continuum regime. Thus, pressure-driven oxygen transport in a dry porous GDL can be modeled as a viscous Hagen-Poiseuille flow in an equivalent duct. However, determination of the equivalent duct radius and the dependence of this radius on the GDL porosity is a nontrivial task (Tamayol et al., 2012). Much workhas recently been done to develop statistical models of porous GDLs and to calculate viscous gas flows in these systems using Navier-Stokes equations (Thiedmann et al., 2012). 

The impaction of spherical particles depicted in Figure 9.1 depends on the fluid properties and the particle diameter and density as described by the Stokes number. Those particles with an aerodynamic diameter larger than a well-defined cutoff size will collide with an impaction plate due to their larger inertia. Smaller particles will follow the flow field and pass by the plate. The impaction of nanoflbers, however, depends on their length in addition to diameter and density. Cheng et al. [14] have outlined a theoretical approach to this and confirmed their theory via experimental studies. For nanofibers, the aerodynamic equivalent diameter is defined as... 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

A commonly used size dependent property is the equivalent spherical diameter. The equivalent spherical diameter is the diameter of a sphere with the same volume as the particle. For a cube this sphere would have a diameter 1.24 times the edge length of the cube. Another common equivalent spherical diameter is the Stokes diameter. The Stokes diameter is the diameter of a sphere that has the same terminal settling velocity as an irregular particle. (Note Settling has to be under laminar flow [i.e., Re3molds number less than 0.2] in both cases and the density of both the particle and the sphere are assumed to be the same). 

The value of Sp depends on several parameters, including the hydrodynamic properties of the channels, the centrifugal force (Sp increases to reach a maximum with the centrifugal force), the mobile-phase flow rate (Sp decreases linearly with the mobile-phase flow rate), the physical properties of the solvent system (such as viscosity, density, interfacial tension), the sample volume, the sample concentration, and the tensioactive properties of solutes to separate [2,3]. It is necessary to precisely monitor Sp because various chromatographic parameters depend on it, in particular the efficiency, the retention factor, and the resolution. Foucault proposed an explanation for the variation of Sp with the various parameters previously described. He modeled the mobile phase in a channel as a droplet and applied the Stokes law which relies on the density difference between the two phases, the viscosity of the stationary phase, and the centrifugal force. Then, he applied the Bond number, derived from the capillary wavelength which was formerly introduced for the hydrodynamic mode [4] and which relies on the density difference between the two phases the interfacial tension and the centrifugal force [3]. 

Thus the condition for observation of Raman spectrum is a nonzero cf which is the selection rule. The intensity of Raman scattering, however depends on a number of factors, which includes the frequency of the scattered light (energy separation and hence populations), the density of vibrational states, the damping constant etc. In the case of glasses, an expression due to Shuker and Gammon (1971) for the Raman intensity of the Stokes lines is given by. 

Sedimentation can be settling or creaming, depending on the sign of the density difference between particles and surrounding liquid. Theory for sedimentation of a single sphere is well developed. The simple Stokes equation can be used if a number of conditions is fulfilled. The most... 

The main distinction of the theory of a dynamic adsorption layer formed under weak and strong retardation arises when formulating the convective diffusion equation. At weak retardation the Hadamard-Rybczynski hydrodynamic velocity field is used while at strong retardation the Stokes velocity field. Different formulas for the dependence of the diffusion layer thickness on Peclet numbers are obtained. The problem of convective diffusion in the neighbourhood of a spherical particle with an immobile surface at small Reynolds numbers and condition (8.74) is solved, so that the well-known expression for the density distribution of the diffusion flow along the surface can be used. As a result, Eq. (8.10) takes the form (Dukhin, 1982),... 

A similar case may be made for the use of density in Stokes law, the buoyancy of particles in the separation zone must be taken into account. The fine particles displace the continuous phase and hence it is the density of the liquid that is used in the model. In any case, the suspension density in the zone is not known but is likely to be much less than that of the feed. The second use of fluid density is in the resistance coefficient, Eu. The density to be used there depends on how we define the Euler number the dynamic pressure in the denominator (equation 6.9) is simply a yardstick against which we measure the pressure loss through a cyclone. We have used the clean liquid density in the dynamic pressure alternatively, the feed suspension density may be used. It is immaterial which of the two densities is used (they are both equally unrealistic) provided the case is clearly defined conversion from one to the other is a simple matter. 

The rate of diffusion controlled reaction is typically given by the Smoluchowski/Stokes-Einstein (S/SE) expression (see Brownian Dynamics), in which the effect of the solvent on the rate constant k appears as an inverse dependence on the bulk viscosity r), i.e., k oc (1// ). A number of experimental studies of radical recombination reactions in SCFs have found that these reactions exhibit no unusual behavior in SCFs. That is, if the variation in the bulk viscosity of the SCF solvent with temperature and pressure is taken into accounL the observed reaction rates are well described by S/SE theory. However, these studies were conducted at densities greater than the critical density, and, in fact, the data is inconclusive very near to the critical density. Additionally, Randolph and Carlier have examined a case in which the observed diffusion controlled, free radical spin exchange rates are up to three times faster than predicted by S/SE theory, with the deviations becoming most pronounced near the critical point. This deviation was attributed to some sort of solvent-solute clustering effect. It is presently unclear why this system is observed to behave differently from those which were observed to follow S/SE behavior. Possible candidates are differences in thermodynamic conditions or molecular interactions, or even misinterpretation of the data arising from other possible processes not considered. 

Finite-element simulations are useful to understand the mechanism of NDR and its dependence on the composition in the internal and external solutions, pore geometry, and nanopore surface charge density. Similar to modeling flow effects on nanopore ICR described earlier, the Nernst-Planck equation governing the diffusional, migrational, and convective fluxes of ions (Equation 2.18), the Navier-Stokes equation for low-Reynolds number flow engendered by the external pressure and electroosmosis (Equation 2.20), and Poisson s equation relating the ion distributions to the local electric field (Equation 2.19) were simultaneously solved to obtain local values of the fluid... 

For typical lubrication situations with a density of the lubricant of p = 10 kg m , a sliding velocity Vo = Im s, a viscosity of the lubricant of t) = 1 Pa s, and a gap width ho = 10 [xm, we obtain a Reynolds number Re = pvoL/r] 0.01 (Eq. (6.42)). This means, we can safely assume laminar flow. For constant Vq also, the explicit time dependence vanishes, d /dX k, 0, and we have creeping flow. If we exclude a side leakage in y-direction and due to symmetry we can assume that Vy = 0. In addition, the flow velocity of lubricant in z-direction is negligible. As a result, the Navier-Stokes equation for creeping flow (Eq. (6.10)) reduces to... 

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