Diffusion wall effect

In general, the flow rate F(t) consists of the following additive components the controlled flow rate Fd of the entering gas, the flow rate Fi which is due to parasitic leaks and/or diffusion, and the flow rate Fw resulting from possible adsorption-desorption processes on the system walls (in Section I, references are given to papers dealing with the elimination or control of the wall effects in the flash filament technique). In each of these flow rate components a particular ratio of the investigated adsorbate and of the inert gas exists and all these components contribute to the over-all mean values Fh(t) and F (t). 

Dispersion in packed tubes with wall effects was part of the CFD study by Magnico (2003), for N — 5.96 and N — 7.8, so the author was able to focus on mass transfer mechanisms near the tube wall. After establishing a steady-state flow, a Lagrangian approach was used in which particles were followed along the trajectories, with molecular diffusion suppressed, to single out the connection between flow and radial mass transport. The results showed the ratio of longitudinal to transverse dispersion coefficients to be smaller than in the literature, which may have been connected to the wall effects. The flow structure near the wall was probed by the tracer technique, and it was observed that there was a boundary layer near the wall of width about Jp/4 (at Ret — 7) in which there was no radial velocity component, so that mass transfer across the layer... 

The starting point is the convective-diffusion equation suitably modified to account for wall effects and potential field effects (25). 

Another kind of wall-effect was proposed by El perin (1967). He suggested that an adsorbed layer of polymer molecules could exist at the pipe wall during flow and this could lower the viscosity, create a slip, dampen turbulence pulsations, and prevent any initiation of vortices at the wall. Later work (Little 1969), however, with a transparent pipe and dyed polymer, showed that the adsorption could in be fact an experimental artifact (a quantity of polymer solution, trapped in pressure gage piping, slowly diffused back into the solvent flow). Although polymer molecules do more or less adhere to clean surfaces in thin films, there is no interaction with the bulk of the solution which could alter the flow properties (Gyr, 1974). Thus, it is evident that adsorption of the additives on surfaces is not the reason for the drag reducing effect. 

Alternatives to batch testing include the use of diffusion cells or flowthrough columns. Diffusion cells are easier to operate, but are less representative of field conditions where some advection may occur. However, operation of columns at very low flow rates is difficult and subject to artifacts. To minimize possible wall effects associated with shrink/swell behavior of low-permeability clay materials, several researchers have utilized column devices that provide a confining pressure, such as flexible wall permeameters (e.g., Acar and Haider, 1990 Smith and Jaffe, 1994 Shackelford and Redmond, 1995 Khandelwal et al., 1998 Khandelwal and Rabideau, 2000). 

It is important to note that the preparative separation is carried out at a low flow rate to allow for mass transfer, diffusion and column wall effects discussed previously. 

Figure 4. Effect of carbon on the wall heat flux with diffuse walls. Results of coupled radiation-aerodynamic model calculations by Toor and Boni (1974).

W. Steiner, Trans. Faraday Soc.y 31,623 (1935). The data are taken from flow-tube experiments on H atoms produced in a Wood s tube and may be in error owing to wall effects and diffusion. 

In the set of relations (3.182)-(3.188), P represents the coefficient for the velocity increase due to the species transport through the wall, Bi is the heat transfer Biot number (Bi = (arj)/ ), Bip is the mass transfer Biot number for the gaseous phase (Bi[) = (kri)/DA) and Bip is the Biot number for the porous wall (Bip = (k5xx,)/DAw)- Two new parameters and D w, respectively, represent the wall thickness and the wall effective diffusion coefficient of species. The model described by the set of relations (3.182)-(3.188) can easily be modified to respond to the situation of a membrane reactor when a chemical reaction occurs inside the cylindrical space and when one of the reaction products can permeate through the wall. The example particularized here concerns the heat and mass transfer of a... 

The wall effect will result in an appreciable spread in residence time of the reactant, unless its contribution is relatively unimportant at a sufficiently large ratio between bed and particle diameter (say, greater than 20) or unless there is a sufficiently strong effect of radial diffusion, as will be discussed below. 

The effect of inner diffusion can also be determined by analyzing the change in the rate of the process due to the change in the size of catalyst particles [ 11 ]. Comparison of the value of rate constant at 600 C (Tables 2 and 4), after taking into account the wall effect, gives the following values of effectiveness factors ... 

The essence of the side walls effect follows. The flow velocity turns to zero at the side walls as well as at the main (accumulation and depletion) walls of the FFF channel. Therefore, the flow profile is nonuniform, not only along the width of the channel but also along its breadth. The size of the regions near the side walls where the flow is substantially nonuniform is of the same order as w. The nonuniformity of the flow in both directions, combined with diffusion of solute particles, leads to Taylor dispersion and peak broadening that could be different from the one predicted by the 2D models. 

The Stefan-Maxwell (Maxwell, 1860 Stefan, 1872) equation gives implicit relations for the fluxes when the system is isothermal and the wall effects are negligible, this means negligible viscous transport (i.e. constant pressure) and Knudsen diffusion. For multicomponent mixture the equation has the form ... 

The main consideration in the design and construction of the equipment has been that the measurements obtained reflect the effect of the process variables on the chemical phenomena occurring and that perturbations arising from physical processes such as diffusion, heat transfer, and wall effects be minimized. 

The picture described is that of convective-diffusion of finite size spherical Brownian particles through a circular capillary. In consequence, this may be looked upon as a generalizaton of the Taylor problem for point size particles (Brenner Edwards 1993). A detailed analysis of this problem based on Brenner s moment analysis method has been carried out by Brenner Gaydos (1977), taking into account the tube wall effects on the motions of the particles. Neglecting wall interactions, the essential element of the chromatographic technique can be illustrated by a simple calculation for the average velocity of a particle. 

The finite particle size thus reduces the dispersivity in comparison with the Taylor value. The smaller dispersion coefficient results from the excluded volume which does not allow the particle center to sample the region of highest velocity gradient near the wall, thereby reducing the mean radial diffusion and hence dispersivity. Unlike with the mean particle velocity, the wall effects enter to first order in A, and reduce the values of all the numerical coefficients of the A terms in comparison with the values obtained by only accounting for the excluded volume effect. 

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