Diffusion finite diameter

In actual practice, any tubular light source will have a finite diameter and will not behave as a true line source. Radiation from an extended light source will emanate from points displaced from the lamp s axis, causing the lamp to appear rather like a diffuse light source. In addition, imperfections in the... 

The boundary condition for particle diffusion differs from the condition for molecular dilTusion becau.se of the finite diameter of the particle. For certain classes of problems, such as flows around cylinders and spheres, the particle concentration is assumed to vanish at one particle radiu.s from the surface ... 

DIFFUSION AT LOW REYNOLDS NUMBERS SIMILITUDE LAW FOR PARTICLES OF FINITE DIAMETER... 

For particles of finite diameter, the interception effect becomes important. A useful similitude law that takes both diffusion and interception into account can be derived as follows (Friedlander, 1967) It is assumed that the concentration boundary layer is thin and falls... 

This is the similitude law for the diffusion of particles of finite diameter but with R < 1 in low-speed Rows. For fixed Rc, the group should be a single-valued function... 

The success of the analysis in correlating experimental data for clean filters offers convincing support for the theory of convective diffusion of particles of finite diameter to surfaces. As particles accumulate in the filter, both the efficiency of removal and the pressure drop increase, and the analysis no longer holds. Some data on this effect are available in the literature. Care must be taken in the practical application of these results because of pinhole leaks in the filters or leaks iU ound the frames. 

Figure 3.6 Efficiency minlmuni for single fiber removal efficiency for particles of finite diameter. For very small particles, diffusion controls according to f3.38) and ijr The different curves nesuli from the effects of velocity. In the interception range according to (3.39), rjtt is...

Tlie use of the term "membrane for these filters is somewhat misleading. Membranes are normally used to separate the components of a gas mixture which have different permeabilities through the membrane material. The permeabilities. In turn, can be related to the solubilities and diffusion coefficients in the membrane which differ for different gases. However, for a membrane filter, the gas passes through the pores of the film by a macroscopic flow process, driven by the pressure gradient. No gas separation takes place. The principal mechani.sms of panicle deposition for both fibrous and membrane filters are the dilTusion and impaction of particles of finite diameter. Settling and electro.siaiic effects may contribute to removal. 

Extension of the hydrodynamic theory to explain the variation of detonation velocity with cartridge diameter takes place in two stages. First, the structure of the reaction zone is studied to allow for the fact that the chemical reaction takes place in a finite time secondly, the effect of lateral losses on these reactions is studied. A simplified case neglecting the effects of heat conduction or diffusion and of viscosity is shown in Fig. 2.5. The Rankine-Hugoniot curves for the unreacted explosive and for the detonation products are shown, together with the Raleigh line. In the reaction zone the explosive is suddenly compressed from its initial state at... 

Here, p is the local (transverse) Peclet number, which is the ratio of transverse diffusion time to the convection time. Per is the radial Peclet number (ratio of transverse diffusion time to a convection time based on pipe radius). We assume that p <4 1 while Per is of order unity. (Remark The parameter Pe /p — ux)L/Dm is also known as the axial Peclet number. Also note that for any finite Per or tube diameter, the axial Peclet number tends to infinity as p tends to zero.) When such scale separation exists, we can average the governing equation over the transverse length scale using the L-S technique and obtain averaged model in terms of axial length and time scales. 

When the polymer flhn is oxidized, its electronic conductivity can exceed the ionic conductivity due to mobile counterions. Then, the film behaves as a porous metal with pores of limited diameter and depth. This can be represented by an equivalent circuit via modified Randles circuits such as those shown in Figure 8.4. One Warburg element, representative of linear finite restricted diffusion of dopants across the film, is also included. The model circuit includes a charge transfer resistance, associated with the electrode/fllm interface, and a constant phase element representing the charge accumulation that forms the interfacial double... 

The GC results are compared in fig. 10.18 with Monte Carlo calculations of Boda et al. [32]. These were carried out assuming that the electrolyte ions are hard spheres with a diameter of 300 pm in a dielectric continuum. The estimates of < ) using the Monte Carlo technique fall below the GC estimates. They demonstrate the importance of including finite ion size in a model of the diffuse layer. 

This means that as Pe increases, the mesh size must decrease. Since the mesh size decreases, it takes more elements or grid points to solve the problem, and the problem may become too big. One way to avoid this is to introduce some numerical diffusion, which essentially lowers the Peclet number. If this extra diffusion is introduced in the flow direction only, the solution may still be acceptable. Various techniques include upstream weighting (finite difference [10]) and Petrov-Galerkin (finite element [11]). Basically, if a numerical solution shows imphysical oscillations, either the mesh must be refined, or some extra diffusion must be added. Since it is the relative convection and diffusion that matter, the Peclet number should always be calculated even if the problem is solved in dimensional units. The value of Pe will alert the chemist, chemical engineer, or bioengineer whether this difficulty would arise or not. Typically, is an average velocity, x is a diameter or height, and the exact choice must be identified for each case. 

A wormlike chain is specified by the persistence length A and the contour length Lp. However, it does not have a thickness. We need to give it a diameter b for the chain to have a finite diffusion coefficient. The model is called a wormlike cylinder (Fig. 3.62). The expressions for the center-of-mass diffusion coefficient and the intrinsic viscosity were derived by Yamakawa et al. in the rigid-rod asymptote and the flexible-chain asymptote in a series of h/A and A/A-... 

Predicted saturation levels S ) within the MPL (a), and GDL (b) of a bi-layer gas diffusion media at positions across the depth of the network (Z) demonstrating the effect of having finite-sized droplets with diameter in the catalyst layer as a boundary condition in a bi-layer GDL (SourceiWu etal.")... 

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