Diffusion coefficient momentum

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

The Chemkin package deals with problems that can be stated in terms of equation of state, thermodynamic properties, and chemical kinetics, but it does not consider the effects of fluid transport. Once fluid transport is introduced it is usually necessary to model diffusive fluxes of mass, momentum, and energy, which requires knowledge of transport coefficients such as viscosity, thermal conductivity, species diffusion coefficients, and thermal diffusion coefficients. Therefore, in a software package analogous to Chemkin, we provide the capabilities for evaluating these coefficients. ... 

While electrical conductivity, diffusion coefficients, and shear viscosity are determined by weak perturbations of the fundamental diffu-sional motions, thermal conductivity is dominated by the vibrational motions of ions. Heat can be transmitted through material substances without any bulk flow or long-range diffusion occurring, simply by the exchange of momentum via collisions of particles. It is for this reason that in liquids in which the rate constants for viscous flow and electrical conductivity are highly temperature dependent, the thermal conductivity remains essentially the same at lower as at much higher temperatures and more fluid conditions. 

An analogy exists between mass transfer (which depends on the diffusion coefficient) and momentum transfer between the sliding hquid layers (which depends on the kinematic viscosity). Calculations show that the ratio of thicknesses of the diffnsion and boundary layer can be written as... 

If we define AT as the eddy diffusivity for momentum, the vertictil eddy diffusion coefficient under unstable conditions can be expressed as... 

At low momentum transfer A2 describes the translational Rouse diffusion coefficient of the whole diblock, considering N l-f) segments exerting the friction Q and A[fsegments exerting the friction In the high Q hmit, RPA predicts... 

It is apparent that the source and sink terms can be different between mass, heat, and momentum transport. There is another significant difference, however, related to the magnitude of the diffusion coefficient for mass, heat, and momentum. 

Turbulent diffusion occurs because turbulent eddies are transporting mass, momentum, and energy over the eddy scale at the rotational velocity. This transport rate is generally orders of magnitude greater than the transport rate due to molecular motion. Thus, when a flow is turbulent, diffusion is normally ignored because e Z). The exception is very near the flow boundaries, where the eddy size (and turbulent diffusion coefficient) decreases to zero. 

Prandtl s mixing length hypothesis (Prandtl, 1925) was developed for momentum transport, instead of mass transport. The end result was a turbulent viscosity, instead of a turbulent diffusivity. However, because both turbulent viscosity and turbulent diffusion coefficient are properties of the flow field, they are related. Turbulent viscosity describes the transport of momentum by turbulence, and turbulent diffusivity describes the transport of mass by the same turbulence. Thus, turbulent viscosity is often related to turbulent diffusivity as... 

The computational efficiency of a FF approach also enables simulations of dynamical behavior—molecular dynamics (MD). In MD, the classical equations of motion for a system of N atoms are solved to generate a search in phase space, or trajectory, under specified thermodynamic conditions (e.g., constant temperature or constant pressure). The trajectory provides configurational and momentum information for each atom from which thermodynamic properties such as the free energy, or time-dependent properties such as diffusion coefficients, can be calculated. 

Liquids with low viscosity or large 3 (high density or efficient momentum transfer across the boundary layer) have a rotational diffusion coefficient close to that of the Debye equation [220], eqn. (110). For viscous liquids, the rotational diffusion coefficient tends to saturate to a viscosity-independent value. Tanabe [235] has found perdeuterobenzene rotational diffusion to be well described by the Hynes et al. theory [221, 222]. 

In this text we are concerned exclusively with laminar flows that is, we do not discuss turbulent flow. However, we are concerned with the complexities of multicomponent molecular transport of mass, momentum, and energy by diffusive processes, especially in gas mixtures. Accordingly we introduce the kinetic-theory formalism required to determine mixture viscosity and thermal conductivity, as well as multicomponent ordinary and thermal diffusion coefficients. Perhaps it should be noted in passing that certain laminar, strained, flames are developed and studied specifically because of the insight they offer for understanding turbulent flame environments. 

This relationship is captured in differential-equation form as Eq. 7.60. Since the momentum and energy equations (Eqs. 7.59 and 7.62) explicitly involve r2, the radial coordinate has become a dependent variable, not an independent variable. A consequence of the Von Mises transformation is that the radial velocity v is removed as a dependent variable and the radial convective terms are eliminated, which is a bit of a simplification. However, the fact that the group of dependent variables pur2 appear within the diffusion terms is a bit of a complication. The factor pur2 plays the role of an apparent variable diffusion coefficient. ... 

It is clear that the viscosity, thermal conductivity, and diffusion coefficients transport coefficients are defined in analogous ways. They relate the gradient in velocity, temperature, or concentration to the flux of momentum, energy, or mass, respectively. Section 12.3 will present a kinetic gas theory that allows an approximate calculation of each of these coefficients, and more rigorous theories are given later in this chapter. 

In equation 13, C1 and Cs are the total concentrations in the liquid and solid phases, respectively. This statement of the problem assumes that the convective flux due to the moving boundary (growing surface) is small, the diffusion coefficients are mutual and independent of concentration, the area of the substrate is equal to the area of the solution, the liquid density is constant, and no transport occurs in the solid phase. Further, the conservation equations are uncoupled from the equations for the conservation of energy and momentum. Mass flows resulting from other forces (e.g., thermal diffusion and Marangoni or slider-motion-induced convective flow) are neglected. 

Yi and Ys - gyromagnetic ratio of spin 1 and spin S nuclear spin, rJS = intemuclear distance, tr= rotational correlation time, x< = reorientation correlation time, xj = angular momentum correlation time, Cs = concentration of spin S, Cq = e2qzzQ/h = quadrupole coupling constant, qzz = the electric field gradient, Q = nuclear electric quadrupole moment in 10 24 cm2, Ceff = effective spin-rotational coupling constant, a = closest distance of appropriate of spin 1 and spin S, D = (DA+DB)/2 = mutual translational self diffusion coefficient of the molecules containing I and S, Ij = moment of inertia of the molecule, Ao = a// - ol-... 

This model is applicable if the thickness of the boundary layer x 2(Dt)1/2 is small as compared to the distance between the reactor walls (where t is the residence time in the reactor and D the diffusion coefficient). In this case, component concentration equations are obtained from the mass and momentum conservation laws and the continuity equation... 

If one corrects for the effects of attractive forces, the corrected values are much smaller than the observed values the correction overestimates the importance of attractive forces on the value of the diffusion coefficient. Although attractive forces have an effect on the diffusion coefficient at low density, they are not nearly as Important as they are in determining the value of the angular momentum correlation time. Diffusion is primarily determined by the repulsive forces between molecules, even at the lowest densities. 

The analogy for transport processes is readily interpreted from Stokes theory if we consider the generalization that forces or fluxes of a property are proportional to a diffusion coefficient, the surface area of the body, and a gradient in property being transported. In the case of momentum, the transfer rate is related to the frictional and pressure forces on the body. The diffusion coefficient in this case is the kinematic viscosity of the gas (vg = p-g/pg, where pg is the gas density). The momentum gradient is Pjg Uoo/B. 

The three properties of viscosity, heat conductivity, and diffusion represent, respectively, the transfer of momentum, energy, and mass within a gas. The gas diffusion coefficient indicates the relative ability of one gas molecule to move with respect to its surroundings—the greater the value of the diffusion coefficient, the more rapid this movement. The diffusion coefficient Z)1>2 for a gas of species 1 diffusing into a gas of species 2 can be estimated from the expression... 

Equation 9.12 indicates that the diffusion coefficient of an aerosol particle is independent of particle density and hence is independent of particle mass. But is this really so Since particle mass is so much greater than molecular mass and the particles are continually undergoing bombardment by the molecules, one would expect changes in the direction of the particle to be gradual, compared to the rapid changes in direction with molecular diffusion. But if this is true, then particle momentum (mass) should be considered in the particle diffusion coefficient equation. 

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