Hydrodynamic correction factor

TABLE 4.6. Hydrodynamic Correction Factors, G for Diffusion in Two-Particle or Partide-Snrface Systems... 

In the previous chapters, the diffusion equation has been used extensively to model fast chemical reaction in solution. By addition of various correction factors (such as intermolecular forces, long-range transfer, solvent structure, hydrodynamic repulsion, etc.), the agreement between experiment and theory can be improved as the model becomes more realistic. Nevertheless, the reactants have been presumed to execute Brownian motion. This is only the long-time limit of their actual behaviour. 

The disadvantages of using empirical correction factors, which lump many parameters together, becomes clear when one considers that a and 0 have been found to change depending on not only the concentration and type of contaminants, but also on the hydrodynamics of the system. Clearly, a better understanding of the relationship between physical properties and kLa and the quantification of these physical properties in (waste-)water is necessary, so that correlations based on dimensional analysis can be made. However, from the practical point of view, the empirical correction factors have proven their worth, when measured and used appropriately. 

As shown earlier, the direct conversion from retention time to particle diameter values [Eq. (1)] requires that the correction factor y is predicted or experimentally estimated. It is known that, in GrFFF, y can be influenced by either hydrodynamic or other effects as those due to the mobile phase [8] and the channel walls nature [9]. All of these effects can influence particle size determination by GrFFF. 

The role of hydrodynamic interaction in Brownian diffusion was discussed in Section 8.2. Consider now its effect on turbulent coagulation. Formally, it can be taken into account in the same manner as in Brownian motion, by introducing a correction multiplier into the factor of turbulent diffusion (10.57). Another, more correct way (see Section 11.3) is to use the Langevin equation that helped us determine the factor of Brownian diffusion in Section 8.2. As was demonstrated in [60], the factor of turbulent diffusion is inversely proportional to the second power of the hydrodynamic resistance factor ... 

This approach is very simple and however has some limitations. LTM is valid if the particle size is small compared to the device dimensions, and in this version, it is valid for spherical particles. For the Stokes law to be valid, the particle needs to be several diameters away from the solid boundaries and the other particles (to model the hydrodynamic interaction of the particle with the wall, a correction factor needs to be introduced [6]) Moreover, if the disturbance of the flow field is significant due to the presence of the particle (e.g., the number of particles may be high within the domain or the size of the particle may be comparable with the microchannel size), the validity of the LTM is questionable. Since LTM does not include the presence of the particle, the simulation of the flow field can be performed with any standard software which handles the solution of PDFs, and the trajectory of the particles can be obtained at the postprocessing step. 

As a second benchmark problem, the motion of spherical and elliptical particles are analyzed in a channel with a hurdle at the middle. The results for spherical (Fig. 6) and elliptical particles (Fig. 7) are shown in the figures below. The rotation of the particles can also be realized. The particles realesed from 10 pm follows a streamline which is different than that of 10 pm after the hurdle due to hydrodynamic interaction of the particles at the comers and within the hurdle section. As the released location increases, this issue diminishes. Although, it is not simulated, the interaction of the particles with the comers has size dependence. So, the location of the particles after the hurdle depends on the size of the particle. BEM has clearly the ability to model this hydrodynamic interaction with the wall without any need for correction factor. The dependence of the equilibrium position after the hurdle on particle size is the key ingredient for the microfluidic devices for hydrodynamic separation of bioparticles. With the ability of BEM, this issue can be explored in details to come up with efficient microfluidics bioparticle separators. 

Indeed this volume effect enters here only as a correction factor, in the case in which the macromolecular skein is relatively dense, so that the occluded solvent acts for a part as hydrodynamically immobilized. Takii the ideal case, that the occluded solvent is not immobilized, theoretically the- specific viscosity increase... 

On the other hand, many sharp fractions are available with several homologous series of random coil molecules. Common parameters to indicate the size of random coils are the root-mean-square of end-to-end distance, mean radius of gyration and the radius of the hydrodynamically equivalent sphere. Various discussions have been presented in the previous works with regard to the appropriate choice of the parameter or the correction factor for it (ref. 14, 25, 27, 29, 30, 31, 34). These discussions, however, have all ignored the wall effect described above and hence their significance is limited. 

In an unrestricted channel (i.e., open sea or large channel), over squat is negligible, and the previous numerical models work well unless the UKC is relatively small h/T < 1.1). Under a 3-m UKC or in a very restricted channel, the numerical model has to check that the hydrodynamics are not being modified by the squat. Some empirical formulas try to make such a correction with a restriction factor that multiplies the squat calculated for unrestricted water. For instance, the K coefficient in Barrass and for Huuska are examples of these types of correction factors. 

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