Hydrodynamic potential distribution

A classical example of this strategy is the capillary gap cell (see Sec. IV.B), which has been run by BASF for more than a quarter of a century [7]. Several dozen electrodes are piled onto each other to give a stocky pillar. The electrodes consist of graphite rings. The diameter of the central hole is a tenth of the outer diameter of the electrodes. The hydrodynamics and the current and potential distribution are far from optimal. 

In contrast to free bubble surfaces, the role of boundary layers for the strong or complete retardation of a bubble surface is of substantial interest. Levich (1962), who introduced the notion of the hydrodynamic potential field of bubbles in the absence of surfactant pointed out that the velocity drop across the boundary layer is small. For completely or strongly retarded surfaces, the velocity distribution beyond the hydrodynamic boundary layer is a potential one. 

The Reynolds number for a particle Rep of supercritical size, deposited on the surface of a sufficiently large bubble (for which a potential distribution of the liquid velocity field is valid), is much larger than imity. In this case, the hydrodynamic resistance is expressed by a resistance coefficient. In aerosol mechanics a technique is used (Fuks, 1961) in which the non-linearity from the resistance term is displaced by the inertia term. As a result, a factor appears in the Stokes number which, taking into account Eq. (11.20), can be reduced to (l + Rep /b). This allows us to find the upper and the lower limits of the effect by introducing K instead of K " into Eq. (10.47) and the factor X in the third term. 

For the first time Mileva (1990) has considered the effect of a hydrodynamic boundary layer on the elementary act of inertia-free microflotation based on a mobile bubble surface free of an adsorption layer and at high Reynolds numbers. The velocity distribution is a potential one, Eq. (8.117), with an additional contribution of the velocity differential along the cross-section of the boundary layer, Eq. (8.127). The difference between the velocity distribution along the bubble surface and the potential distribution is given by Eq. (8.128). In contrast, Mileva (1990) used the formulas of a related theory by Moore (1963) which are the solution of the same Eq. (8.122) under the same boundary condition (8.118). 

Integration of Eq. (10.28) along the cross-section of the hydrodynamic layer allows us to check whether within its limits the radial velocity component is proportional to the tangential derivative of the velocity distribution along the bubble surface, which differs slightly from the potential distribution. The effect of a boundary layer on the normal velocity component and on inertia-free deposition of particles should be therefore very small. The formula for the collision efficiency given by Mileva as an inertia-free approximation is thus VRc times less than the collision efficiency according to Sutherland, which is definitely erroneous. 

DLS is a technique used in conjunction with -potential measurements to analyze polyplex size (average hydrodynamic diameter) distribution. Because the formation of polyplexes can result in a polydisperse mixture of nanoparticles of different size and shape (induding uncomplexed components), multiple peaks can result. Thus, typical reported data include average hydrodynamic diameter and polydispersity. Most software associated with DLS instmmentation permits assessment of a multimodal size distribution, but such data are often unreported or underreported. The default assumption of a single Gaussian distribution may not be appropriate for the majority of polymer-nucleic add complex formulations that are analyzed by DLS. Also, the analysis of the particle sizes assumes a uniform sphere, which again is not necessarily... 

The approximate experimental determination of xl), is based on measurement of the velocity of a charged particle in a solvent subjected to an applied voltage. Such a particle experiences an electrical force that initiates motion. Since a hydrodynamic frictional force acts on the particle as it moves, a steady state is reached, with the particle moving with a constant velocity U. To calculate this electrophoretic velocity U theoretically, it is, in general, necessary to solve Poisson s equation (Equation 3.19) and the governing equations for ion transport subject to the condition that the electric field is constant far away from the particle. The appropriate viscous drag on the particle can be calculated from the velocity field and the electrical force on the particle from the electrical potential distribution. The fact that the sum of the two is zero provides the electrophoretic velocity U. Actual solutions are complex, and the electrical properties of the particle (e.g., polarizability, conductivity, surface conductivity, etc.) come into play. Details are given by Levich (1962) (see also Problem 7.8). 

Basically, there are three approaches to stack modelling. The first is fully 3D modelling, taking into account the hydrodynamics of flows in channels, heat transport and potential distribution over the stack volume. This approach leads to extremely time-consuming CFD codes to our knowledge there is only one model of that type (Liu et al., 2006). 

Thus far, these models cannot really be used, because no theory is able to yield the reaction rate in terms of physically measurable quantities. Because of this, the reaction term currently accounts for all interactions and effects that are not explicitly known. These more recent theories should therefore be viewed as an attempt to give understand the phenomena rather than predict or simulate it. However, it is evident from these studies that more physical information is needed before these models can realistically simulate the complete range of complicated behavior exhibited by real deposition systems. For instance, not only the average value of the zeta-potential of the interacting surfaces will have to be measured but also the distribution of the zeta-potential around the mean value. Particles approaching the collector surface or already on it, also interact specifically or hydrodynamically with the particles flowing in their vicinity [100, 101], In this case a many-body problem arises, whose numerical... 

---