Diffusion finite length

In the second case (limit of fast kinetics at the gas-solid interface), the film becomes entirely bulk transport limited, corresponding to the limit of Hebb— Wagner polarization. Since electronic conduction is fast, this situation yields a Warburg impedance for finite length diffusion ... 

The release location influences the vertical distribution of the time-averaged concentration and fluctuations. For a bed-level release, vertical profiles of the time-averaged concentration are self-similar and agreed well with gradient diffusion theory [26], In contrast, the vertical profiles for an elevated release have a peak value above the bed and are not self-similar because the distance from the source to the bed introduces a finite length scale [3, 25, 37], Additionally, it is clear that the size and relative velocity of the chemical release affects both the mean and fluctuating concentration [4], The orientation of the release also appears to influence the plume structure. The shape of the profiles of the standard deviation of the concentration fluctuations is different in the study of Crimaldi et al. [29] compared with those of Fackrell and Robins [25] and Bara et al. [26], Crimaldi et al. [29] attributed the difference to the release orientation, which was vertically upward from a flush-mounted orifice at the bed in their study. 

Strictly speaking, equations 19-52 and 19-53 are valid only if the pollutant cloud is infinitely long. A more realistic situation is treated in Box 19.2 here a pollutant patch of finite length L along the x-axis is eroded on both edges due to diffusion processes (turbulence, dispersion, etc.). Again, the boundary is of the diffusive type since the transport characteristics on both sides of the boundary are assumed to be identical. 

Band electrodes were the first to be fabricated with true nanometer dimensions [17,93]. Although small in only one dimension, the width, this controls the magnitude of the flux, while the finite length leads to easily measured currents. These electrodes can be fabricated with a smallest dimension as small as 2 nm by sealing deposited metal films between insulators. While voltammograms obtained with electrodes with widths greater than 50 nm follow the theory expected for diffusion-controlled mass transport, narrower electrodes show less... 

Since the specie Mg " diffuses toward the electrode surface, the resulting concentration perturbation (0) is obtained from the finite-length diffusion impedance, represented... 

Equation (11.70) can be considered to be a finite-length diffusion impedance. As tanh(oo) = 1, the impedance response asymptotically approaches the response for an infinite domain at high frequencies, i.e.. 

Pollard and Present [32], on the basis of molecular kinetics and dynamics, rigorously evaluated the effective diffusion coefficient of nonadsorbable gases in capillaries of finite length to find that it increases with higher lc and its limiting value is ... 

After the tubes are brought together diffusion proceeds, but since the tubes are sealed and are of finite length there can be no mass transfer across the ends of the tube the boundary conditions therefore are... 

Total electrode impedance consists of the contributions of the electrolyte, the electrode solution interface, and the electrochemical reactions taking place on the electrode. First, we consider the case of an ideally polarizable electrode, followed by semi-infinite diffusion in linear, spherical, and cylindrical geometry and, finally a finite-length diffusion. 

In many cases the diffusion is not semi-infinite. This case, for example, is observed for polymer electrodes, for a thin mercury layer deposited on surfaces, and for rotating disk electrodes. In such cases, in Eq. (52) parameters B and B are not equal to 0. Two cases may be distinguished for finite-length diffusion, depending on the condition at the boundary located at a distance I from the electrode ... 

The problem of finite-length diffusion in spherical and cylindrical symmetry was solved by Jacobsen and West. ... 

Ideally, first the measurement modeling should be carried out. The number and the nature of the circuit elements should be identified and then the process modeling should be carried out. Such a procedure is relatively elementary for a circuit containing simple elements R, C, and L. It may also be carried out for circuits containing distributed elements that can be described by a closed-form equation CPE, semi-infinite, finite length, or spherical diffusion, etc. However, many different conditions arise from the numerical calculations (e.g., for correct solution for porous electrodes, for... 

Problems involving unsteady diffusion in more then one coordinete direction, such as a cylinder or finite length or a long slab of comparable width and depth dimensions, cun usually be solvnd by separation of variables. For example, for a cylinder of radius R and length L the governing equation and boundary conditions for an ideal gas mixture and constant surface conditions would be... 

This element describes finite length diffusion with a boundary condition of dc/dx = 0. The variables Yj and B are defined as... 

Fig. II.5.5 Nyquist impedance plot due to finite length diffusion with a transmissive boundary condition...

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