Of molecular diffusivity

Neglecting flow nonuniformities, the contributions of molecular diffusion and turbulent mixing arising from stream sphtting and recombination around the sorbent particles can be considered additive [Langer et al., Int. ]. Heat and Mass Transfer, 21, 751 (1978)] thus, the axial dispersion coefficient is given by ... 

Dispersion Model An impulse input to a stream flowing through a vessel may spread axially because of a combination of molecular diffusion and eddy currents that together are called dispersion. Mathematically, the process can be represented by Fick s equation with a dispersion coefficient replacing the diffusion coefficient. The dispersion coefficient is associated with a linear dimension L and a linear velocity in the Peclet number, Pe = uL/D. In plug flow, = 0 and Pe oq and in a CSTR, oa and Pe = 0. 

In chemicals like salol the molecules are elongated (non-spherical) and a lot of energy is needed to rotate the randomly arranged liquid molecules into the specific orientations that they take up in the crystalline solid. Then q is large, is small, and the interface is very sluggish. There is plenty of time for latent heat to flow away from the interface, and its temperature is hardly affected. The solidification of salol is therefore interface controlled the process is governed almost entirely by the kinetics of molecular diffusion at the interface. 

Figure 3.2.1 illustrates the mixing in packed beds (Wilhelm 1962). As Reynolds number approaches the industrial range Rep > 100, the Peclet numbers approach a constant value. This means that dispersion is influenced by turbulence and the effect of molecular diffusion is negligible. 

The distribution of tracer molecule residence times in the reactor is the result of molecular diffusion and turbulent mixing if tlie Reynolds number exceeds a critical value. Additionally, a non-uniform velocity profile causes different portions of the tracer to move at different rates, and this results in a spreading of the measured response at the reactor outlet. The dispersion coefficient D (m /sec) represents this result in the tracer cloud. Therefore, a large D indicates a rapid spreading of the tracer curve, a small D indicates slow spreading, and D = 0 means no spreading (hence, plug flow). 

Hydrodynamic Dispersion Macroscopic dispersion is produced in a capillar) even in tlie absence of molecular diffusion because of the velocity profile produced by the adherence of the fluid to tlie wall. Tlris causes fluid particles at different radial positions to move relative to one anotlier, witli tlie result tliat a series of mixing-cup samples at tlie end of tlie capillary e.xhibits dispersion. 

An open bowl, 0.3 m in diameter, contains water at 350 K evaporating into the atmosphere, if the air currents are sufficiently strong to remove the water vapour as it is formed and if the resistance to its mass transfer in air is equivalent to that of a 1 mm layer for conditions of molecular diffusion, what will be the rate of cooling due to evaporation The water can be considered as well mixed and the water equivalent of the system is equal to 10 kg. The diffusivity of water vapour in air may be taken as 0.20 ctn2/s and the kilogram molecular volume at NTP as 22.4 in3. 

P. Clavin and F.A. Williams. Effects of molecular diffusion and of thermal expansion on the structure and dynamics of premixed flames in turbulent flows of large scale and low intensity. Journal of Fluid Mechanics, 116 251-282,1982. 

Flayduk, W. and Minhas, B.S. (1982) Correlations for prediction of molecular diffusivities in liquids. Can.]. Chem. 

Solutions of surfactant-stabilized nanogels share both the advantage of gels (drastic reduction of molecular diffusion and of internal dynamics of solubilizates entrapped in the micellar aggregates) and of nonviscous liquids (nanogel-containing reversed micelles diffuse and are dispersed in a macroscopicaUy nonviscous medium). Effects on the lifetime of excited species and on the catalytic activity and stability of immobilized enzymes can be expected. 

Because of the low rates of molecular diffusion, assessment of workplace air quality using passive samplers usually entails sampling for a working shift, and exposure periods of one to four weeks tend to be needed to measure concentrations in ambient air. 

To study the effects of molecular diffusion and formation/destabilization of reaction fronts it is advised to rely on small flow-through chambers such as capillaries or cells of sheet-type cross-section [68]. These micro reactors simply provide the small-scale environment needed for such laboratory investigations. 

Direct observation of molecular diffusion is the most powerful approach to evaluate the bilayer fluidity and molecular diffusivity. Recent advances in optics and CCD devices enable us to detect and track the diffusive motion of a single molecule with an optical microscope. Usually, a fluorescent dye, gold nanoparticle, or fluorescent microsphere is used to label the target molecule in order to visualize it in the microscope [31-33]. By tracking the diffusive motion of the labeled-molecule in an artificial lipid bilayer, random Brownian motion was clearly observed (Figure 13.3) [31]. As already mentioned, the artificial lipid bilayer can be treated as a two-dimensional fluid. Thus, an analysis for a two-dimensional random walk can be applied. Each trajectory observed on the microscope is then numerically analyzed by a simple relationship between the displacement, r, and time interval, T,... 

In accordance with Pick s Law, diffusive flow always occurs in the direction of decreasing concentration and at a rate, which is proportional to the magnitude of the concentration gradient. Under true conditions of molecular diffusion, the constant of proportionality is equal to the molecular diffusivity of the component i in the system, D, (m /s). For other cases, such as diffusion in porous matrices and for turbulent diffusion applications, an effective diffusivity value is used, which must be determined experimentally. 

Each film is in stagnant or laminar flow, such that mass transfer across the films is by a process of molecular diffusion and can therefore be described by Pick s Law. 

J. Karger, C. M. Papadakis, F. Stallmach 2004, (Structure-mobility relations of molecular diffusion in interface systems), in Molecules in Interaction with Surfaces and Interfaces, eds. R. Haber-landt, D. Michel, A. Poppel, R. Stannar-ius, Springer, Heidelberg. 

J. Karger, H. Pfeifer 1994, (NMR studies of molecular diffusion), in NMR Techniques in Catalysis, eds. a. Pines, A. Bell, Marcel Dekker, New York. 

Fig. 5.1.2 Non-ideal capillary flow reactor (a) propagators [13] and (b) corresponding RTDs calculated from the propagator data, (a) The propagators indicate the distribution of average velocities over each observation time (A) ranging from 50 ms to 1 s. As the observation time increases the spins exhibit a narrowing distribution of average velocities due to the motional narrowing effect of molecular diffusion across the streamlines. The dashed vertical line represents the maximum velocity that would be present in the absence of molecular...

The study of molecular diffusion in solution by NMR methods offers insights into a range of physical molecular properties. Different mobility rates or diffusion coefficients may also be the basis for the separation of the spectra of mixtures of small molecules in solution, this procedure being referred to as diffusion-ordered spectroscopy (DOSY) [271] (Figure 5.11). In this 2D experiment, the acquired FID is transformed with respect to 2 (the acquisition time). 

Chemicals can pass through soil liners by molecular diffusion as well as by advective transport. One can study the molecular diffusion of chemicals in the soil by compacting soil at the bottom of an impermeable beaker and ponding waste liquid or leachate on top of the soil. At the start of the experiment, the concentration c is equal to c0 in the waste liquid. The soil is clean. Even though no water flows into the soil by advection, chemicals move into the soil by the process of molecular diffusion. Eventually, the concentration of the waste liquid and the soil will be one and the same (see Figure 26.12). 

The drainage system for removing leachate or other aggressive liquids from landfills, surface impoundments, and waste piles is critically important. Even if a liner has no leaks, the phenomenon of molecular diffusion will allow some of the organics from the liquids ponded on top of the liner system to leach through the FML and the clay. The timely collection and removal of that leachate is at the heart of this section. 

Effect of molecular diffusion and vapor-phase chemical reactions Liquid metal vapors consist of molecules and gaseous atoms. Working with alkali metals, Ewing et al. (1967) found that the molecules are principally dimers and tetramers. The... 

NA Peppas, PJ Hansen, PA Buri. A theory of molecular diffusion in the intestinal mucus. Int J Pharm 20 107-118, 1984. 

The only instances in which external mass transfer processes can influence observed conversion rates are those in which the intrinsic rate of the chemical reaction is so rapid that an appreciable concentration gradient is established between the external surface of the catalyst and the bulk fluid. The rate at which mass transfer to the external catalyst surface takes place is greater than the rate of molecular diffusion for a given concentration or partial pressure driving force, since turbulent mixing or eddy diffusion processes will supplement ordinary molecular diffusion. Consequently, for porous catalysts one... 

In Chapter 11, we indicated that deviations from plug flow behavior could be quantified in terms of a dispersion parameter that lumped together the effects of molecular diffusion and eddy dif-fusivity. A similar dispersion parameter is usefl to characterize transport in the radial direction, and these two parameters can be used to describe radial and axial transport of matter in packed bed reactors. In packed beds, the dispersion results not only from ordinary molecular diffusion and the turbulence that exists in the absence of packing, but also from lateral deflections and mixing arising from the presence of the catalyst pellets. These effects are the dominant contributors to radial transport at the Reynolds numbers normally employed in commercial reactors. 

Hydrodynamic dispersion is in many cases taken to be a Fickian process, one whose transport law takes the form of Fick s law of molecular diffusion. If flow is along x only, so that vx = v and vy = 0, the dispersive fluxes (mol cm-2 s-1) along x and y for a component i are given by,... 

The effective diffusivity De is a characteristic of the particle that must be measured for greatest accuracy. However, in the absence of experimental data, De may be estimated in terms of molecular diffusivity, Dab (for diffusion of A in the binary system A + B), Knudsen diffusivity, DK, particle voidage, p, and a measure of the pore structure called the particle tortuosity, Tp. 

In general, liquid-phase reactions (Sc > 1) and fast chemistry are beyond the range of DNS. The treatment of inhomogeneous flows (e.g., a chemical reactor) adds further restrictions. Thus, although DNS is a valuable tool for studying fundamentals,4 it is not a useful tool for chemical-reactor modeling. Nonetheless, much can be learned about scalar transport in turbulent flows from DNS. For example, valuable information about the effect of molecular diffusion on the joint scalar PDF can be easily extracted from a DNS simulation and used to validate the micromixing closures needed in other scalar transport models. 

Since the molecular diffusivities are used in (5.254), the interval length L(t) and the initial conditions will control the rate of molecular diffusion and, subsequently, the rate of chemical reaction. In order to simulate scalar-gradient amplification due to Kolmogorov-scale mixing (i.e., for 1 < Sc), the interval length is assumed to decrease at a constant rate ... 

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