Diffusional particle transport

In the respiratory tract, only ultrafine particles (particles smaller than 0.1 pm in diameter) are deposited solely due to diffusion, since those particles cover more than 30 pm s by diffusional transport (Fig. 2). For all ultrafine particles of the same size, deposition is the same regardless of their density. Because of the time-dependence of diffusional particle transport, it is anticipated that diffusional deposition of ultrafine particles occurs mainly in lung regions of maximum residence time of the tidal air—i.e., in small airways and in the lung periphery. 

Figure 3 Schematic filter characteristic of the human respiratory tract for aerosol particles. Three domains can be recognized the domain of deposition decreasing with particle size is solely due to diffusional particle transport, the domain of minimum deposition is due to simultaneous diffusional and gravitational particle transport, and the domain of deposition increasing with particle size due to gravitational and inertial particle transport.

Mechanisms of Mechanical Particle Transport. When particles do not follow, but diverge from, airflow streamlines and thereby come in contact with airspace surfaces, particle deposition occurs. This diverging from airflow streamlines and particle trajectories is mainly due to mechanisms of mechanical particle transport inertial, gravitational, and diffusional particle transport (Fig. 3). The... 

Numerical simulations of the coarsening of several particles are now possible, allowing the particles to change shape due to diffusional interparticle transport in a manner consistent with the local interphase boundary curvatures [17]. These studies display interparticle translational motions that are a significant phenomenon at high volume fractions of the coarsening phase. 

Studies with porous catalyst particles conducted during the late 1930s established that, for very rapid reactions, the activity of a catalyst per unit volume declined with increasing particle size. Mathematical analysis of this problem revealed the cause to be insufficient intraparticle mass transfer. The engineering implications of the interaction between diffusional mass transport and reaction rate were pointed out concurrently by Damkohler [4], Zeldovich [5], and Thiele [6]. Thiele, in particular, demonstrated that the fractional reduction in catalyst particle activity due to intraparticle mass transfer, r, is a function of a dimensionless parameter, 0, now known as the Thiele parameter. 

We note that Eqn. 198 consists of two parts the first involves diffusional substrate transport through the layer, and the other involves electrochemical kinetics at the particle surface. As previously noted when DsiR corresponding to Case 1, the reaction at the particle surface is diffusion-controlled, and Eqn. 198 reduces to... 

In the light of the transport mechanisms causing particle deposition in the respiratory system, the dependency of total deposition on particle diameter, as displayed in Figure lA, becomes clear. Minimal deposition occurs in the size range between 0.1 and 1 pm, because neither impaction or sedimentation nor diffusion are effective in particle displacement. With decreasing particle diameter, diffusional particle displacement increases so that particle deposition in the respiratory system increases. With increasing particle diameter, the distance covered by sedimentation or impaction increases such that total particle deposition is also enhanced. 

A further problem is possible if the reinforcements are very small. Coarsening of the particles or whiskers may occur driven by Ostwald ripening, in which large particles grow through diffusional transport at the expense of smaller ones. This can be minimized by choosing matrices in which the reinforcement elements have very low solid solubilities and diffusion coefficients. Platelets, however, have been shown to be more resistant to coarsening than particles or whiskers. 

Bisrat et al. concluded that for sparingly soluble, suspended drugs, diffusional transport plays a major role in the dissolution kinetics [19]. They studied the effect of particle size and viscosity on dissolution rate and apparent diffusional distance (.h-App) of oxazepam and griseofulvin. The term apparent diffusional distance was used as a simplified measure of the distance over which diffusion dominates and was calculated as follows ... 

In the general case, when arbitrary interaction profiles prevail, the particle deposition rate must be obtained by solving the complete transport equations. The first numerical solution of the complete convective diffusional transport equations, including London-van der Waals attraction, gravity, Brownian diffusion and the complete hydrodynamical interactions, was obtained for a spherical collector [89]. Soon after, numerical solutions were obtained for a panoplea of other collector geometries... 

In this study the ratio of the particle sizes was set to two based on the average value for the two samples. As a result, if the diffusion is entirely controlled by secondary pore structure (interparticle diffusion) the ratio of the effective diffusion time constants (Defl/R2) will be four. In contrast, if the mass transport process is entirely controlled by intraparticle (platelet) diffusion, the ratio will become equal to unity (diffusion independent of the composite particle size). For transient situations (in which both resistances are important) the values of the ratio will be in the one to four range. Diffusional time constants for different sorbates in the Si-MCM-41 sample were obtained from experimental ZLC response curves according to the analysis discussed in the experimental section. Experiments using different purge flow rates, as well as different purge gases... 

If the rate constants for the sorption-desorption processes are small equilibrium between phases need not be achieved instantaneously. This effect is often called resistance-to-mass transfer, and thus transport of solute from one phase to another can be assumed diffusional in nature. As the solute migrates through the column it is sorbed from the mobile phase into the stationary phase. Flow is through the void volume of the solid particles with the result that the solute molecules diffuse through the interstices to reach surface of stationary phase. Likewise, the solute has to diffuse from the interior of the stationary phase to get back into the mobile phase. 

Systematically speaking, so-called internal oxidation reactions of alloys (A,B) are extreme cases of morphological instabilities in oxidation. Internal oxidation occurs if oxygen dissolves in the alloy crystal and the (diffusional) transport of atomic oxygen from the gas/crystal surface into the interior of the alloy is faster than the countertransport of the base metal component (B) from the interior towards the surface. In this case, the oxidation product BO does not form a stable oxide layer on the alloy surface. Rather, BO is internally precipitated in the form of small oxide particles. The internal reaction front moves parabolically ( Vo into the alloy. Examples of internal reactions are discussed quantitatively in Chapter 9. 

Particles suspended in a nonuniform gas may be subject to absorption or loss of heat or material by diffusional transport. If the particle is suspended without motion in a stagnant gas, heat or mass transfer to or from the body can be estimated from heat conduction or diffusion theory. One finds that the net rate of transfer of heat to the particle surface in a gas is... 

The applicability of Maxwell s equation is limited in describing particle growth or depletion by mass transfer. Strictly speaking, mass transfer to a small droplet cannot be a steady process because the radius changes, causing a change in the transfer rate. However, when the difference between vapor concentration far from the droplet and at the droplet surface is small, the transport rate given by Maxwell s equation holds at any instant. That is, the diffusional transport process proceeds as a quasi-stationary process. 

When the particle is moving relative to the suspending fluid, transport of heat or matter is enhanced by convective diffusional processes. Under conditions where the particle exists in a rarified medium (Kn 0), the heat and mass tranfer relations are modified to account for surface accommodation or sticking of colliding molecules and the slippage of gas around the particle. 

In spite of their seeming variety, theoretical approaches of different authors to the consideration of solid-state heterogeneous kinetics can be divided into two distinct groups. The first group takes account of both the step of diffusional transport of reacting particles (atoms, ions or, in exceptional cases if at all, radicals) across the bulk of a growing layer to the reaction site (a phase interface) and the step of subsequent chemical transformations with the participation of these diffusing particles and the surface atoms (ions) of the other component (or molecules of the other chemical compound of a binary multiphase system). This is the physicochemical approach, the main concepts and consequences of which were presented in the most consistent form in the works by V.I. Arkharov.1,46,47... 

While the above criteria are useful for diagnosing the effects of transport limitations on reaction rates of heterogeneous catalytic reactions, they require knowledge of many physical characteristics of the reacting system. Experimental properties like effective diffusivity in catalyst pores, heat and mass transfer coefficients at the fluid-particle interface, and the thermal conductivity of the catalyst are needed to utilize Equations (6.5.1) through (6.5.5). However, it is difficult to obtain accurate values of those critical parameters. For example, the diffusional characteristics of a catalyst may vary throughout a pellet because of the compression procedures used to form the final catalyst pellets. The accuracy of the heat transfer coefficient obtained from known correlations is also questionable because of the low flow rates and small particle sizes typically used in laboratory packed bed reactors. 

Madon and Boudart propose a simple experimental criterion for the absence of artifacts in the measurement of rates of heterogeneous catalytic reactions [R. J. Madon and M. Boudart, Ind. Eng. Chem. Fundam., 21 (1982) 438]. The experiment involves making rate measurements on catalysts in which the concentration of active material has been purposely changed. In the absence of artifacts from transport limitations, the reaction rate is directly proportional to the concentration of active material. In other words, the intrinsic turnover frequency should be independent of the concentration of active material in a catalyst. One way of varying the concentration of active material in a catalyst pellet is to mix inert particles together with active catalyst particles and then pelletize the mixture. Of course, the diffusional characteristics of the inert particles must be the same as the catalyst particles, and the initial particles in the mixture must be much smaller than the final pellet size. If the diluted catalyst pellets contain 50 percent inert powder, then the observed reaction rate should be 50 percent of the rate observed over the undiluted pellets. An intriguing aspect of this experiment is that measurement of the number of active catalytic sites is not involved with this test. However, care should be exercised when the dilution method is used with catalysts having a bimodal pore size distribution. Internal diffusion in the micropores may be important for both the diluted and undiluted catalysts. 

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