Particles kinetic theory

We proceed now to the problems (Problem 2) and (Problem 3). At least two levels of description are involved in direct molecular simulations. The first one is the level of the np-particle kinetic theory and the second is the level of fluid mechanics on which the external forces and the final results that we seek are formulated. We shall use the multiscale formulation developed above and combine the two levels. The two levels that we consider in this section are... 

C2-np-particle kinetic theory with the state variables... 

C / -classical fluid mechanics and np-particle kinetic theory... 

Filtration, in the most general sense, may be defined as the removal of particles from the aerosol. This occurs either by their attachment to nonaerosol media (walls, vegetation, "fabric filters", etc.) or to larger particles which are subsequently removed. Since particle transport in the gas is intimately involved, a characterization of the gas flow field and the detailed mechanisms of particle kinetic theory near a surface must be invoked. Classically, filtration was treated as the simple adhesion of a single particle to a surface. However, it is now known that after the first particles adhere, subsequent ones tend to be captured by the initial ones to form chains. Impaction of a large particle upon such a chain or other break-off processes can cause resuspension. Thus, filtration is dependent upon properties of the aerosol and gas as a whole [1.9,10]. 

We will almost always treat the case of a dilute gas, and almost always consider the approximation that the gas particles obey classical, Flarniltonian mechanics. The effects of quantirm properties and/or of higher densities will be briefly commented upon. A number of books have been devoted to the kinetic theory of gases. Flere we note that some... 

The kinetic theory of gases has been used so far, the assumption being that gas molecules are non-interacting particles in a state of random motion. This... 

This rule conforms with the principle of equipartition of energy, first enunciated by Maxwell, that the heat capacity of an elemental solid, which reflected the vibrational energy of a tliree-dimensional solid, should be equal to 3f JK moH The anomaly that the free electron dreory of metals described a metal as having a tliree-dimensional sUmcture of ion-cores with a three-dimensional gas of free electrons required that the electron gas should add anodier (3/2)7 to the heat capacity if the electrons behaved like a normal gas as described in Maxwell s kinetic theory, whereas die quanmtii theory of free electrons shows that diese quantum particles do not contribute to the heat capacity to the classical extent, and only add a very small component to the heat capacity. 

The traditional unipolar diffusion charging model is based on the kinetic theory of gases i.e., ions are assumed to behave as an ideal gas, the properties of which can described by the kinetic gas theory. According to this theory, the particle-charging rate is a function of the square of the particle size dp, particle charge numbers and mean thermal velocity of tons c,. The relationship between particle charge and time according White s... 

White s equation is widely used mainly because it is easy to use and because it gives values which are in reasonable agreement with the experimental ones. However, because this model is based on the kinetic theory of gases, it should be used for small particles only. This model (as many others) assumes that particle charge can be described with a continuous function. Especially in the case of small particles, only the lowest charge numbers (0, 1, 2) are possible, and therefore the model—which does not take into account the discrete charge numbers—is somewhat questionable. 

There is a reasonable explanation for this type of deviation. The kinetic theory, which explains the pressure-volume behavior, is based upon the assumption that the particles exert no force on each other. But real molecules do exert force on each other The condensation of every gas on cooling shows that there are always attractive forces. These forces are not very important when the molecules are far apart (that is, at low pressures) but they become noticeable at higher pressures. With this explanation, we see that the kinetic theory is based on an idealized gas—one for which the molecules exert no force on each other whatsoever. Every gas approaches such ideal behavior if the pressure is low enough. Then ihe molecules are, on the average, so far apart that then-attractive forces are negligible. A gas that behaves as though the molecules exert no force on each other is called an ideal gas or a perfect gas. 

Kelvin temperature scale, 58 Ketones, 334 Kerosene. 231, 341 Kilo, 40 Kilocalorie, 40 Kinetic energy, 53, 114 billiard ball analogy, 6, 114 distribution, 130, 131 formula for, 59 of a moving particle, 59 relation to temperature, 56, 131 Kinetic theory, 49, 52, 53 and Avogadro s Hypothesis, 58 review, 61... 

In its most elementary aspects, kinetic theory is developed on the basis of a hard sphere model of the particles (atoms or molecules) making up the gas.1 The assumption is made that the particles are uniformly distributed in space and that all have the same speed, but that there are equal numbers of particles moving parallel to each coordinate axis. This last assumption allows one to take averages over... 

In its more advanced aspects, kinetic theory is based upon a description of the gas in terms of the probability of a particle having certain values of coordinates and velocity, at a given time. Particle interactions are developed by the ordinary laws of mechanics, and the results of these are averaged over the probability distribution. The probability distribution function that is used for a given macroscopic physical situation is determined by means of an equation, the Boltzmann transport equation, which describes the space, velocity, and time changes of the distribution function in terms of collisions between particles. This equation is usually solved to give the distribution function in terms of certain macroscopic functions thus, the macroscopic conditions imposed upon the gas are taken into account in the probability function description of the microscopic situation. 

The quantities n, V, and (3 /m) T are thus the first five (velocity) moments of the distribution function. In the above equation, k is the Boltzmann constant the definition of temperature relates the kinetic energy associated with the random motion of the particles to kT for each degree of freedom. If an equation of state is derived using this equilibrium distribution function, by determining the pressure in the gas (see Section 1.11), then this kinetic theory definition of the temperature is seen to be the absolute temperature that appears in the ideal gas law. 

As described above, the magnitude of Knudsen number, Kn, or inverse Knudsen number, D, is of great significance for gas lubrication. From the definition of Kn in Eq (2), the local Knudsen number depends on the local mean free path of gas molecules,, and the local characteristic length, L, which is usually taken as the local gap width, h, in analysis of gas lubrication problems. From basic kinetic theory we know that the mean free path represents the average travel distance of a particle between two successive collisions, and if the gas is assumed to be consisted of hard sphere particles, the mean free path can be expressed as... 

The friction coefficient of a large B particle with radius ct in a fluid with viscosity r is well known and is given by the Stokes law, Q, = 67tT CT for stick boundary conditions or ( = 4jit ct for slip boundary conditions. For smaller particles, kinetic and mode coupling theories, as well as considerations based on microscopic boundary layers, show that the friction coefficient can be written approximately in terms of microscopic and hydrodynamic contributions as ( 1 = (,(H 1 + (,/( 1. The physical basis of this form can be understood as follows for a B particle with radius ct a hydrodynamic description of the solvent should... 

The two fundamental theories for calculating the attachment coefficient, 3, are the diffusion theory for large particles and the kinetic theory for small particles. The diffusion theory predicts an attachment coefficient proportional to the diameter of the aerosol particle whereas the kinetic theory predicts an attachment coefficient proportional to the aerosol surface area. The theory... 

The kinetic theory of radon progeny attachment to aerosol particles assumes that unattached atoms and aerosol particles undergo random collisions with the gas molecules and with each other. The attachment coefficient, 3(d), is proportional to the mean relative velocities between progeny atoms and particles and to the collision cross section (Raabe, 1968a) ... 

From equation (3) is is clear that the kinetic theory predicts an attachment rate of radon daughters to aerosol particles proportional to the square of the diameter of the aerosol particle. 

When the radius of an aerosol particle, r, is of the order of the mean free path, i, of gas molecules, neither the diffusion nor the kinetic theory can be considered to be strictly valid. Arendt and Kallman (1926), Lassen and Rau (1960) and Fuchs (1964) have derived attachment theories for the transition region, r, which, for very small particles, reduce to the gas kinetic theory, and, for large particles, reduce to the classical diffusion theory. The underlying assumptions of the hybrid theories are summarized by Van Pelt (1971) as follows 1. the diffusion theory applies to the transport of unattached radon progeny across an imaginary sphere of radius r + i centred on the aerosol particle and 2. kinetic theory predicts the attachment of radon progeny to the particle based on a uniform concentration of radon atoms corresponding to the concentration at a radius of r + L... 

For small particle sizes the kinetic theory is applicable, whereas for large particle sizes the diffusion theory applies. A useful approximation is therefore to use the kinetic theory in the small particle range and the diffusion theory in the large size region. 

Figures 3 and 4 show the variation of the average attachment coefficient with CMD. It can be seen that for particles of CMD less than 0.06 ym and Og = 2 the kinetic theory predicts an attachment coefficient similar to the hybrid theory, whereas for CMD greater than about 1 ym the diffusion theory and the hybrid theory give approximately the same results. For a more polydisperse aerosol (Og = 3) the kinetic theory deviates from the hybrid theory even at a CMD = 0.01 ym. The diffusion theory is accurate for a CMD greater than about 0.6 ym. These results are easily explained since as the aerosol becomes more polydisperse, there are more large diameter particles (CMD >0.3 ym) which attach according to the diffusion theory. In contrast, the kinetic theory becomes more inaccurate as the aerosol becomes more polydisperse.

In this section, we will only discuss the basic principles of kinetic theory, where for detailed derivations we refer to the classic textbook by Chapman and Cowling (1970), and a more recent book by Liboff (1998). Of central importance in the kinetic theory is the single particle distribution function /s(r, v), which can be defined as the number density of the solid particles in the 6D coordinate and velocity space. That is, /s(r, v, t) dv dr is the average number of particles to be found in a 6D volume dv dr around r, v. This means that the local density and velocity of the solid phase in the continuous description are given by... 

Furthermore, the closures for the fluid—particle drag and the particle-phase stresses that we discussed were all derived from data or analysis of nearly homogeneous systems. In what follows, we refer to the TFM equations with closures deduced from nearly homogeneous systems as the microscopic TFM equations. The kinetic theory based model equations fall in this category. 

Fig. 29. Snapshots of particle volume fraction fields obtained while solving a kinetic theory-based TFM. 75 pm fluid catalytic particles in ambient air. Simulations were done over a 16 x 32 cm periodic domain. The average particle volume fraction in the domain is 0.05. Dark (light) color indicates regions of high (low) particle volume fractions. (See Refs. Agrawal et al., 2001 Andrews et al., 2005) for other parameter values.) Source Andrews and Sundaresan (2005).

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