Mean free path aerosol

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... 

Once particles have been formed, they grow by the combined effect of vapor deposition and continued coagulation. The transport of vapor species to the surfaces of the aerosol particles depends on the size of the aerosol particle relative to the mean free path of the gas molecules. A. that is, it depends on the Knudsen number characteristic of vapor diffusion to the particle, Kn ... 

In some cases, particularly in the growth of aerosol particles, the assumption of equilibrium at the interface must be modified. Frisch and Collins (F8) consider the diffusion equation, neglecting the convective term, and the form of the boundary condition when the diffusional jump length (mean free path) becomes comparable to the radius of the particle. One limiting case is the boundary condition proposed by Smoluchowski (S7), C(R, t) = 0, which presumes that all molecules colliding with the interface are absorbed there (equivalent to zero vapor pressure). A more realistic boundary condition for the case when the diffusion jump length, (z) R, has been shown by Collins and Kimball (Cll) and Collins (CIO) to be... 

Two-dimensional trajectories of a typical gas molecule and a typical aerosol particle can be compared in Fig. 9.2. The molecule shows sharp changes in direction, each change occurring when it strikes another molecule. As discussed in Chap. 3, the average distance between hits is defined as the mean free path of the molecule. For the particle, a hit by a single molecule does not appreciably affect its motion. Therefore, its path is not characterized by sharp changes in direction, but by smooth curves representing the combined effect of hits by many molecules. 

Since aerosol particles are continually undergoing molecular bombardment, their paths are smooth curves rather than segments of straight lines. It still is possible to define an apparent mean free path for the aerosol particles (Fuchs, 1964). This is the distance traveled by an average particle before it changes its direction of motion by 90°. The apparent mean free path represents the distance traveled by an average particle in a given direction before particle velocity in that direction equals zero. But this is just the stop distance. 

At normal pressure and temperature, lB reaches a minimum at an aerosol particle diameter of 2 x 10 e cm, but increases only by about a factor of 5 for particles 2 orders of magnitude larger or smaller than this size. Thus, the pseudo mean free path is essentially constant over the size range of interest, having a value of about 10 6 cm. 

Jennings (1988) reviewed the literature on viscosity with regard to compiling exact measurements for the mean free path of air molecules. He considered both dry air and moist air. At 20°C Jennings gives a value of 1.8193 x 10-4 cP for the viscosity of dry air, 1.815 x 10-4 cP for air at 50 percent relative humidity, and 1.8127 X 10-4 cP for air at 100 percent relative humidify. These figures indicate that for most aerosol work, a value for viscosity at 20°C of 1.82 x 10 4 cP is reasonably accurate regardless of the humidity. 

This process governs the rate of deposition of the molecules of nonvolatile compounds on the surface of gas ducts, and contributes to broadening of the chromatographic zones. Being of the order of 0.1 pm at STP, the mean free path of molecules, which is inversely proportional to pressure, reaches 1 cm only at about 0.01 mmHg. In dense enough gas, in the absence of convective flow, the macroscopic picture of migration of molecules (as well as of aerosol particulates) is described by the equations of diffusion. The mean squared diffusional displacement z2D of molecules, the time of diffusion t and the mutual diffusion coefficient >i 2 are related by ... 

Figure 7.11 Coagulation of aerosol panicles much smaller than the mean free path. Size distributions measured with the electrical mobility analyzer (Husar. 1971).

Suppose the aerosol contained in a large chamber is composed of particles larger than the mean free path of the gas. The surface-to-volume ratio of the chamber is sufficiently small to neglect deposition on the walls, and the composition of the system is uniform. Coagulation takes place, and at the same time the particles grow as a result of diffusion-controlled condensation but sedimentation can be neglected. Homogeneous nucleation does not occur and the system is isothermal. A system of this type has been used to model aerosol formation in photochemical air pollution. 

Before I began writing. I considered the pos.sibility of a general text covering small particle behavior in both gases and liquids. Much of the theory of physical behavior is the same or very similar for both aerosols and hydrosols. almost as much as in the fluid mechanics of air and water. The differences include double layer theory in the case of aqueous solutions and mean free path effects in gases. There are other important, specifically chemical differences. 

The physical meaning of the above criteria is as follows. The external force acting on an aerosol is generally gravitation. This means that the lifetime of a particle in the system is determined by its sedimentation velocity. If the particle radius is greater than the mean free path of gas molecules, the vs falling velocity is given by the well-known Stokes equation ... 

Another feature of aerosols is related to the fact that the particle size of dispersed matter, r, is comparable to the mean free path of molecules in a gas, Am. The choice of a particular method used to adequately describe the particle motion depends on the ratio between r and AM, which is given by the Knudsen number, Kn=AM/2r [1-4]. When Kn<10 2 one can apply the laws of mechanics (aerodynamics) of continuous media, and in particular the Stokes law, eq. (V.4) ... 

FIGURE 9.10 Removal rale of particles as a function of time for the conditions of Figure 9.9. 9.5.3 Mean Free Path of an Aerosol Particle... 

The concept of mean free path is an obvious one for gas molecules. In Ihe Brownian motion of an aerosol particle there is not an obvious length that can be identified as a mean free path. This is depicted in Figure 9.11 showing plane projections of the paths followed by an air molecule and an aerosol particle of radius roughly equal to 1 pm. The trajectories... 

To obtain the mean free path Xp, we recall that in Section 9.1, using kinetic theory, we connected the mean free path of a gas to measured macroscopic transport properties of the gas such as its binary diffusivity. A similar procedure can be used to obtain a particle mean free path A,p from the Brownian diffusion coefficient and an appropriate kinetic theory expression for the diffusion flux. Following an argument identical to that in Section 9.1, diffusion of aerosol particles can be viewed as a mean free path phenomenon so that... 

Certain quantities associated with the Brownian motion and the dynamics of single aerosol particles are shown as a function of particle size in Table 9.5. All tabulated quantities in Table 9.5 depend strongly on particle size with the exception of the apparent mean free path A,p, which is of the same order of magnitude right down to molecular sizes, with atmospheric values Xp 10-60nm. 

Transition and Free Molecular Regime When the mean free path Xp of the diffusing aerosol particle is comparable to the radius of the absorbing particle, the boundary... 

It should be noted that deposition processes in manufacture of nanoparticles often take place in a regime very far from equilibrium conditions [1]. The model for description of the impurity molecule trapping by growing nanoparticles should be valid for high non-equilibrium conditions. It should also describe the deposition process for arbitrary relation between the mean free path of gas molecules and the particle radius and take into account the trapping of non-condensable molecules. It is known that the gas-to-particle conversion can be realized by ordinary condensation (physical deposition) and by chemical deposition. Further we will consider the trapping of molecules by a small aerosol particle in physical deposition. 

Transition and Free Molecular Regime When the mean free path of the diffusing aerosol particle is comparable to the radius of the absorbing particle, the boundary condition at the absorbing particle surface must be corrected to account for the nature of the diffusion process in the vicinity of the surface. The correction is appreciable when the apparent free path of the aerosol particles kp is of the same order of magnitude as the radius Rp as we will see this occurs for Rp <0. /im. 

The physical meaning of this can be explained as follows. As we have seen the diffusion equations can be applied to Brownian motion only for time intervals that are large compared to the relaxation time, r, of the particles or for distances that are large compared to the aerosol mean free path kp. Diffusion equations cannot describe the motion of particles inside a layer of thickness kp adjacent to an absorbing wall. If the size of the absorbing sphere is comparable to kp, this layer has a substantial effect on the kinetics of coagulation. 

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