Spheres roughness effects

A number of refinements and applications are in the literature. Corrections may be made for discreteness of charge [36] or the excluded volume of the hydrated ions [19, 37]. The effects of surface roughness on the electrical double layer have been treated by several groups [38-41] by means of perturbative expansions and numerical analysis. Several geometries have been treated, including two eccentric spheres such as found in encapsulated proteins or drugs [42], and biconcave disks with elastic membranes to model red blood cells [43]. The double-layer repulsion between two spheres has been a topic of much attention due to its importance in colloidal stability. A new numeri-... 

The Hamaker constant A can, in principle, be determined from the C6 coefficient characterizing the strength of the van der Waals interaction between two molecules in vacuum. In practice, however, the value for A is also influenced by the dielectric properties of the interstitial medium, as well as the roughness of the surface of the spheres. Reliable estimates from theory are therefore difficult to make, and unfortunately it also proves difficult to directly determine A from experiment. So, establishing a value for A remains the main difficulty in the numerical studies of the effect of cohesive forces, where the value for glass particles is assumed to be somewhere in the range of 10 21 joule. 

Roughly speaking the ( -factor describes how deeply a particle is drained by the solvent a deep draining causes a reduction of the hydro dynamically effective sphere radius and c becomes small, if on the other hand only a shallow draining is possible increases and can become much larger than R. ... 

Here Kjj is obtained from Fig. 9.5. Equation (9-27) and the equations of Chapter 5 can be used to determine the decrease in Sh for a rigid sphere with fixed settling on the axis of a cylindrical tube. For example, for a settling sphere with 2 = 0.4 and = 200, Uj/Uj = 0.76 and UJUj = 0.85. Since the Sherwood number is roughly proportional to the square root of Re, the Sherwood number for the settling particle is reduced only 8%, while its terminal velocity is reduced 24%. As in creeping flow, the effect of container walls on mass and heat transfer is much smaller than on terminal velocity. 

Davies (D3) found that roughened spheres behave rather differently at Re = 9 X 10". Both Cl> and Cl rose steadily with increasing vJU, presumably due to the effect of roughness in displacing the critical transition to lower Re (see Section II). It is therefore possible that rough spheres show negative lift at somewhat lower Re, but this has not been confirmed. 

Nu with intensity up to 0.01 (E2, R2). As /r increases further, but still /r < /rc, Nu increases roughly linearly, but more slowly. Similar effects have been observed for cylinders (MIO). For spheres at higher Re, the average Nusselt number increases linearly with /r for /r < /r (R2). Few reliable data are available for /r > /r. Figure 10.13 presents a tentative correlation for the effect of turbulence on the average Nusselt number for spheres. The ordinate is Nu/Nuo, the ratio of the Nusselt number at /r to the value in the absence of turbulence, while the abscissa is the ratio of /r, the intensity, to /r, the critical intensity. The value of Nuq was calculated from the correlations in Table 5.4 and from Eqs. (10-44) and (10-45). The correlation is divided into... 

There are numerous reports of investigations of the effects of roughness on flow in open channels. For instance, Reinius (R4) has reported on the effects of surfaces covered with various types of roughnesses (spheres, sand, etc.) on the flow of water in open channels, while Hama (HI) has reported... 

To obtain a rough estimate of the multiple scattering effects a model proposed by M. H. Cohen is useful (18). This model is based on the application of the Wigner-Seitz scheme to an electron in a helium crystal. Each helium atom is represented as a hard sphere characterized by a radius equal to the scattering length. The electron wave function will then be... 

Structure maker ions increase the concentration of micelles and reduce the concentration of monomers. In a very rough model one can assume the fixed hydration sphere around the ions cannot solve the ethylenoxide products and increase its concentrations in the rest bulk water phase (salt-out effect). In this model one can estimate the size of the hydration sphere of the ions. The hydration numbers gained by this method are surprisingly large3 They start at 200 water molecules per ion pair at 0.1 mole solutions of ions and decrease about 20 at 1 mole solutions31,72,130). 

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