Hydrodynamically equivalent

Note The size of a hydrodynamically equivalent sphere may be different for different types of motion of the macromolecule, e.g., for diffusion and for viscous flow. 

Several theoretical tentatives have been proposed to explain the empirical equations between [r ] and M. The effects of hydrodynamic interactions between the elements of a Gaussian chain were taken into account by Kirkwood and Riseman [46] in their theory of intrinsic viscosity describing the permeability of the polymer coil. Later, it was found that the Kirdwood - Riseman treatment contained errors which led to overestimate of hydrodynamic radii Rv Flory [47] has pointed out that most polymer chains with an appreciable molecular weight approximate the behavior of impermeable coils, and this leads to a great simplification in the interpretation of intrinsic viscosity. Substituting for the polymer coil a hydrodynamically equivalent sphere with a molar volume Ve, it was possible to obtain... 

The hydrodynamic equivalent molecular dimensions can be related to the unperturbed dimensions by the introduction of a hydrodynamic expansion factor ah. Then Eq. (9.15) reads... 

Figure 13 shows the data for the three phenolic groups of ribonuclease which ionize reversibly (Tanford etal., 1955a), based on spectrophotometric titration curves such as Fig. 11. A straight-line plot is obtained, in agreement with Eq. (14). The values of w are 0.112, 0.093, and 0.061, respectively, at ionic strengths 0.01, 0.03, and 0.15. (The salt used to produce the ionic strength was KCl, and there is evidence that neither K" nor CF is bound to an appreciable extent. The use of Zn as abscissa is therefore presumably acceptable.) Comparison with the calculated values of Table III shows that the experimental values are lower than predicted by about 20%. Such a deviation must be considered almost within the error of calculation. [If the radius of the hydrodynamically equivalent sphere (19 A) had been used as the basis of calculation, the calculated values of w would have become 0.119, 0.096, and 0.066, respectively.]...

The ionic radius r, is that of the hydrodynamically equivalent sphere consisting of both the molecular ion and its attached hydration shell. Given that the volume of a sphere is (4/3)jtr, Eqs. 15 and 18 predict that the logarithm of the iontophoretic permeability coefficient, for a given solute should be linearly related to the logarithm of its molal volume (MV) with a slope of -1/3 ... 

Xu, G. W. Kato, K. 1999 Hydrodynamic equivalent diameter for clusters in heterogeneous gas-solid flow. Chemical Engineering Science 54, 1837-1847. 

Since the coaxial cell has hydrodynamics equivalent to those of the... 

Table V. Estimated Size of Cellulases of Various Fungi Calculated from Given Are for Hydrodynamically Equivalent Spheres or Ellipsoids with...

Z-average mean The hydrodynamic (equivalent spherical) diameter of a particle in a liquid, determined by dynamic light scattering... 

The solution to the problem of hydrodynamie interaction between two rigid spherieal partieles, approaehing each other aeross a viseous fluid, was obtained by Taylor (122). Two spherieal emulsion drops of tangentially immobile sur-faees (due to the presenee of dense surfactant adsorption monolayers) are hydrodynamically equivalent to the two rigid partieles eonsidered by Taylor. The hydrodynamic interaction is due to the dissipation of kinetic energy when the liquid is expelled from the gap between the two spheres. The resulting friction force decreases the velocity of the two spherical drops proportionally to the decrease in the surface-to-surface distance h in accordance with the Taylor (122) equation ... 

The radius (or hydrodynamic equivalent of the radius) of the retained species determines the distance of the center of such a species from the accumulation waU. There is no distribution of concentration of the retained component across the channel. Consequently, the ratio a is a decisive parameter determining the retention in steric EET ... 

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. 

The most practical parameter for the size of random coil molecules would be the radius of a hydrodynamically equivalent sphere (Stokes radius). This quantity can be directly determined through the measurements of viscosity, diffusivity or sedimentation velocity, and abundant data have been accumulated. The results of SE measurement can be conveniently expressed in terms of this quantity. This way of presentation at the same time serves as the "universal calibration curve" for SEC columns. 

Let us assume the random coil in the solution as a hard sphere of the radius / h as in the thermodynamic sphere (Figure 2.8). This hypothetical sphere is not the representative of the segment distribution, but shows the region inside the coil where the solvent flow cannot pervade. It is called the hydrodynamically equivalent sphere (Figure 2.12). Its volume is vh =4ttR /3. The radius /fn is not the same as the radius of gyration, but is... 

In the case of polymers diffusing in a solvent, we can replace the random coil by the hydrodynamically equivalent sphere of radius /fn- We find... 

The diffusion coefficients are used to define the hydrodynamic equivalent diameters for translation and rotation ... 

Based on the solution of the velocity field, the force and torque acting on the aggregates can be calculated and the hydrodynamic equivalent diameters for translation (xh,t) and rotation (xi -) can be derived. They scale with the aggregate mass via a power law that reflects the stmctural properties of the aggregate. That means, while for hep aggregates the aggregate mass (N) is proportional to the third power of Xh, there is a fractal-tike relationship for DLCA aggregates with a hydrodynamic dimension 4i close to the fiactal dimension (Fig. 4.17, left cf. discussion to Kirkwood-Riseman theory on pp. 164). This fractal relationship... 

Intensity weighted harmonic mean size from DLS experiments (m) Hydrodynamic (equivalent) diameter of translation/rotation (m) Stokes diameter or sedimentation equivalent diameter (-)... 

In a gas—sohd CFB with heterogeneous reactions and mass transfer, in Hne with the structural characteristics of the SFM model (Hong et al, 2012), as shown in Fig. 12, the mass transfer and reaction in any local space can be divided into components of the dense cluster (denoted by subscript c), the dilute broth (denoted by subscript f), and in-between (denoted by subscript i), respectively. And these terms can be represented by Ri (1 = gc, gf, gi, sc, sf, si). Both the dense and dilute phases are assumed homogenous and continuous inside, and the dense phase is fiarther assumed suspended uniformly in the dilute phase in forms of clusters of particles. Then the mass transfer terms can be described with Ranz-Marshall-hke relations for uniform suspension of particles (Haider and Basu, 1988). In particular, the mesoscale interaction over the cluster will be treated as is for a big particle with hydrodynamic equivalent diameter of d. Due to dynamic nature of clusters, there are mass exchanges between the dilute and dense phases with rate ofTk (k = g, s), pointing outward from the dilute to the dense phase. 

The dimensions and the molecular weight of copolymer micelles can be determined by quite a number of techniques, especially scattering and hydrodynamic characterization techniques as summarized in Table 7.3. In general practice the hydrodynamic radius Ry is determined by DLS techniques. By treating the micelles as hydrodynamically equivalent spheres and using the Stokes-Einstein relation, Ry can be evaluated from the translational diffusion coefficient extrapolated to infinite dilution D -. 

This theory has been partially confirmed by sedimentation experiment (Langevin and Rondelez, 1978). The value of the slope so far found was —0.50 0.10. We now have some evidence to believe that in the semidilute range of polymer solution the solvent is forced through in orderly fashion around the blob of radius C but still cannot penetrate the interior of the blob. Note that this theory is reminiscent of the pearl necklace model and the hydrodynamic equivalent sphere. 

If the micelles are treated as hydrodynamically equivalent spheres then the translational diffusion coefficient for the limiting case of infinitely dilute solution is given by the Stokes-Einstein relation... 

Suppose that a polymeric ion is represented by a hydrodynamically equivalent sphere. That is, suppose that a polymeric ion is represented as a sphere of radius R inside which all fixed charged groups are distributed uniformly. The density of the fixed charges in the sphere, v, is given by... 

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