Maxwellian diffusion

Despite the fact that relaxation of rotational energy in nitrogen has already been experimentally studied for nearly 30 years, a reliable value of the cross-section is still not well established. Experiments on absorption of ultrasonic sound give different values in the interval 7.7-12.2 A2 [242], As we have seen already, data obtained in supersonic jets are smaller by a factor two but should be rather carefully compared with bulk data as the velocity distribution in a jet differs from the Maxwellian one. In the contrast, the NMR estimation of a3 = 30 A2 in [81] brought the authors to the conclusion that o E = 40 A in the frame of classical /-diffusion. As the latter is purely nonadiabatic it is natural that the authors of [237] obtained a somewhat lower value by taking into account adiabaticity of collisions by non-zero parameter b in the fitting law. 

Two other attempts, without the use of a distribution function, are worth mentioning, as these are operationally related to experiments and serve to give a rough estimate of the thermalization time. Christophorou et al. (1975) note that in the presence of a relatively weak external field E, the rate of energy input to an electron by that field is (0 = eEvd, where vd is the drift velocity in the stationary state. Under equilibrium, it must be equal to the difference between the energy loss and gain rates by an electron s interaction with the medium. The mean electron energy is now approximated as (E) = (3eD )/(2p), where fl = vd /E is the drift mobility and D is the perpendicular diffusion coefficient (this approximation is actually valid for a Maxwellian distribution). Thus, from measurements of fl and D the thermalization time is estimated to be... 

MeV H, and 0.8 MeV He3 formed by the D-D reactions in advanced fuel reactors. The energy spectrum of the central plasma can be expected to be Maxwellian at the Tokamak operating temperature ( 10 keV). There is little or no information on the interaction between blistering and externally induced stress. Thermal and mechanically induced stress will influence diffusion and this will have a direct effect on blistering. 

The parameter Tr is the temperature corresponding to the half-range Maxwellian distribution of the particles reflected diffusively. The temperature 7 r is related to the temperature of the reflecting surface through the thermal accommodation coefficient E, defined as... 

In this case no detailed collision kinetics are involved. The collision parameters rnm and the parameters in the Maxwellian post-collision velocity distribution f m are derived from experimentally determined gas viscosity or diffusivity, and the collisional invariants, respectively. Usually this term is negligible in present experiments, but exceptions exist [16]. In particular for ITER, and the high collisionality there, these terms are expected to become more relevant. However, due to the BGK-approximations made, their implementation into the models does not require further discussion here. 

From these time-scales, it may be assumed in most circumstances that the free electrons have a Maxwellian distribution and that the dominant populations of impurities in the plasma are those of the ground and metastable states of the various ions. The dominant populations evolve on time-scales of the order of plasma diffusion time-scales and so should be modeled dynamically, that is in the particle number continuity equations, along with the momentum and energy equations of plasma transport theory. The excited populations of impurities on the other hand may be assumed relaxed with respect to the instantaneous dominant populations, that is they are in a quasi-equilibrium. The quasi-equilibrium is determined by local conditions of electron temperature and electron density. So, the atomic modeling may be partially de-coupled from the impurity transport problem into local calculations which provide quasi-equilibrium excited ion populations and effective emission coefficients (PEC coefficients) and then effective source coefficients (GCR coefficients) for dominant populations which must be entered into the transport equations. The solution of the transport equations establishes the spatial and temporal behaviour of the dominant populations which may then be re-associated with the local emissivity calculations, for matching to and analysis of observations. 

The thermal diffusion factor a is proportional to the mass difference, (mi — mo)/(mi + m2). The thermal diffusion process depends on the transport of momentum in collisions between unlike molecules. The momentum transport vanishes for Maxwellian molecules, particles which repel one another with a force which falls off as the inverse fifth power of the distance between them. If the repulsive force between the molecules falls off more rapidly than the fifth power of the distance, then the light molecule will concentrate in the high temperature region of the space, while the heavy molecule concentrates in the cold temperature region. When the force law falls off less rapidly than the fifth power of the distance, then the thermal diffusion separation occurs in the opposite sense. The theory of the thermal diffusion factor a is as yet incomplete even for classical molecules. A summary of the theory has been given by Jones and Furry 15) and by Hirschfelder, Curtiss, and Bird 14), Since the thermal diffusion factor a for isotope mixtures is small, of the order of 10", it remained for Clusius and Dickel (8) to develop an elegant countercurrent system which could multiply the elementary effect. 

In addition to colliding with other molecules, the particles in a DSMC simulation may interact with solid surfaces. An extensive review of the state of gas-surface interaction modeling is provided by Hurlbut. In almost all DSMC computations, a combination of two ideal limits is employed that may be characterized by surface accommodation coefficients, a. These limits are (1) specular reflection (a = 0) in which reversal of the velocity component normal to the surface is the only change made and (2) diffuse reflection with full thermal accommodation (a = 1), in which new velocity components are sampled from a Maxwellian distribution function at the surface temperature. In reality, the dynamics of particle interaction with real surfaces lies somewhere between these limits and it is common to try and capture such behavior phenomenologically through the use of... 

This balance equation can also be derived from kinetic theory [101], In the Maxwellian average Boltzman equation for the species s type of molecules, the collision operator does not vanish because the momentum mgCs is not an invariant quantity. Rigorous determination of the collision operator in this balance equation is hardly possible, thus an appropriate model closure for the diffusive force is required. Maxwell [65] proposed a model for the diffusive force based on the principles of kinetic theory of dilute gases. The dilute gas kinetic theory result of Maxwell [65] is generally assumed to be an acceptable form for dense gases and liquids as well, although for these mixtures the binary diffusion coefficient is a concentration dependent, experimentally determined empirical parameter. 

Keywords self-diffusion coefficient, thermal hydrodynamic fluctuations, dynamic viscosity, kinematic viscosity, Maxwellian relaxation time... 

It may be instructive to develop a more molecular picture of diffusion in a liquid by considering the concepts of molecular and diffusional velocity (21). In a Maxwellian gas, a particle of mass m and average one-dimensional velocity, has an average kinetic energy of This energy can also be shown to be 4T/2, (22, 23) thus the average molecular velocity is For an O2 molecule (m — 5 X 10 g) at 300 K, one... 

To determine the coefficient of Brownian diffusion, it is necessary to find the root-mean-square (instead of the mean) displacement of the particle (see (8.62)). These values are easy to derive with the help of the expressions (8.81) and (8.82). Suppose the distribution of particles initial velocities is Maxwellian, as in the collisionless discharged rarefied gas, with the zero average value (this can be the case if the velocity of the liquid along the x-axis is zero). Then... 

Fig. 9. Repeller curve obtained for a dc repeller field showing how the relative ion intensity for argon ions varies with the exit energy of the ions. Discriminatory problems in the ion accelerating system are avoided by collecting the ions on a Faraday cup immediately outside the ion exit slit. The theoretical curve, which is normalized to the plateau region, considers lateral diffusive loss of the ions during their passage through the ionization chamber as a consequence of their initial Maxwellian velocity distribution.

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