Rotation coefficients sphere

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... 

To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

The diffusion coefficient is estimated using Stokes law, D = kTjQmrja, where rj is the viscosity and a the radius of the rotating sphere. This rough model allows a calculation of the correlation function. It turns out that a Boltzmann distribution remains a Boltzmann distribution so that the system of equations (12)... 

Here /(S2, t) is an arbitrary function of the two usual polar angles defining the sphere orientation (denoted by S2) and of time t. A is the angular Laplacian while D is the rotational diffusion coefficient ... 

Drag and lift coefficients for rotating spheres. All data plotted are for smooth spheres. 

The rotational diffusion coefficient of the fuzzy cylinder can be formulated in a similar way. For the rotational diffusion process, it is convenient to imagine a hypothetical sphere which has the diameter equal to Lc, just encloses the test fuzzy cylinder, and moves with the translation of the fuzzy cylinder. If the test cylinder and the portions of surrounding fuzzy cylinders entering the sphere are projected onto the spherical surface as depicted in Fig. 15b (cf. [108]), the rotational diffusion process of the test cylinder can be treated as the translational diffusion process of a circle on the hypothetical spherical surface with ribbon-like obstacles. 

Figure 6.5 Temperature dependence of the characteristics of sodium k-carrageenan particles dissolved in an aqueous salt solution (0.1 M NaCl). The cooling rate is 1.5 °C min-1, (a) ( ) Weight-average molar weight, Mw, and (A) second virial coefficient, A2. (b) ( ) Specific optical rotation at 436 nm, and ( ) penetration parameter, y, defined as tlie ratio of the radius of the equivalent hard sphere to the radius of gyration of the dissolved particles (see equation (5.33) in chapter 5). See the text for explanations of different regions I, II, III and IV.

The Debye theory [220] in which a sphere of volume V and radius a rotates in a liquid of coefficient of viscosity t has already been mentioned. There is angular momentum transfer across the sphere—liquid interface that is, the liquid sticks to the sphere so that the velocity of the sphere and liquid are identical at the sphere s surface. Solution of the rotational diffusion equation... 

When there is no tangential force (or no transmission of angular momentum) across the liquid—sphere interface, the sphere slips within the liquid. Within the hydrodynamic theory, the rotational relaxation time is negligible for an inertialess slippery sphere. A variable coefficient of slip (or stick), j3, may be introduced. As 3 tends from 0 to 00 the rotational relaxation time increases from 0 to r)V/kT [221, 222]. 

Fig. 17, The correction factor to the Smoluchowski rate coefficient for reaction between an isotropic reactant, B, and an axially symmetric reactant, A, which has a cap of reactivity of the spherical surface, subtending a semi-angle of 77/9, 77/2, 577/6 at the sphere s centre. The reaction radius is R and the radius of B is rB. Translational and rotational diffusion coefficients are given by eqn. (118), for reactions with G = 1, i.e,...

The mass transport rate coefficient, kd, for a RDE at the maximum practical rotation speed of 10000 per min"1 is approximately 2 x 10-2 cms-1 [28], which sets a limit of about 10 3 cms 1 for the electrode reaction kinetics. For the study of very fast electrode processes, such as some outer sphere redox reactions on noble metal electrodes under stationary conditions, higher mass transport rates in the solution adjacent to the electrode must be employed. 

While the clathrate model is attractive, it is not correct to assume that the water is organized in some long-lived structure the observation that the self-diffusion coefficient for co-sphere water is larger than that for the solute rules this out. However, the rotational correlation time is shorter for ethanol and t-butyl alcohol in water (in the clathrate cage ) than in the pure liquid (Goldammer and Hertz, 1970 Goldammer and Zeidler, 1969). Nmr experiments show that in water the solvent dipole moments point away from the apolar groups (Hertz and Radle, 1973). 

Forced convection heat transfer is probably the most common mode in the process industries. Forced flows may be internal or external. This subsection briefly introduces correlations for estimating heat-transfer coefficients for flows in tubes and ducts flows across plates, cylinders, and spheres flows through tube banks and packed beds heat transfer to nonevaporating falling films and rotating surfaces. Section 11 introduces several types of heat exchangers, design procedures, overall heat-transfer coefficients, and mean temperature differences. 

It has recently become more widely appreciated that the presence of rotational diffusional anisotropy in proteins and other macromolecules can have a significant affect on the interpretation of NMR relaxation data in terms of molecular motion. Andrec et al. used a Bayesian statistical method for the detection and quantification of rotational diffusion anisotropy from NMR relaxation data. Sturz and Dolle examined the reorientational motion of toluene in neat liquid by using relaxation measurements. The relaxation rates were analyzed by rotational diffusion models. Chen et al measured self-diffusion coefficients for fluid hydrogen and fluid deuterium at pressures up to 200 MPa and in the temperature range 171-372 K by the spin echo method. The diffusion coefficients D were described by the rough sphere (RHS) model invoking the rotation translational coupling parameter A = 1. 

Diffusion. - Distribution of the diffusivitity of fluid in a horizontally oriented cylinder was demonstrated by NMR imaging in two papers on a granular flow system and in the earth s magnetic field. Correlation time (ic) and diffusion coefficient (D = Xc) imaging (CTDCI) was applied to a granular flow system of 2 mm oil-filled sphere rotated in a half-filled horizontal cylinder, ie. to an Omstein-Uhlenbeck process with a velocity autocorrelation function. Time dependent apparent diffusion coefficients are measured, and Tc... 

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