Diffusion radial molecular

Effective diffusivity in Knudsen regime Effective diffusivity in molecular regime Knudsen diffusion coefficient Diffusion coefficient for forced flow Effective diffusivity based on concentration expressed as Y Dispersion coefficient in longitudinal direction based on concentration expressed as Y Radial dispersion coefficient based on concentration expressed as Y Tube diameter Particle diameter... 

G.I. Taylor (1953, 1954) first analyzed the dispersion of one fluid injected into a circular capillary tube in which a second fluid was flowing. He showed that the dispersion could be characterized by an unsteady diffusion process with an effective diffusion coefficient, termed a dispersion coefficient, which is not a physical constant but depends on the flow and its properties. The value of the dispersion coefficient is proportional to the ratio of the axial convection to the radial molecular diffusion that is, it is a measure of the rate at which material will spread out axially in the system. Because of Taylor s contribution to the understanding of the process of miscible dispersion, we shall, as is often done, refer to it as Taylor dispersion. 

The dispersion calculation is considerably simplified when the radial molecular diffusion D(d cldr ) is supposed large in comparison with the axial molecular diffusion D d cldx ). With the characteristic lengths as before, this implies Lla) > 1. But with L = Ut and t a lD, the criterion for the neglect of the axial gradients is... 

At constant fluid dynamic residence time, the RTD becomes narrower with decreasing chaimel dimension and thus decreasing diffusion time. The ratio of fluid dynamic residence time to radial diffusion time describes the Fourier number, Fo, which is a measure of the intensity of the radial molecular diffusion and thus radial mixing in den chaimels [12] ... 

In the mathematical treatment, the extreme cases of no radial diffusion and very rapid radial diffusion were considered. It was mentioned that these conditions correspond not only to zero and infinite radial diffusivities but also to finite longitudinal diffusivity. Since molecular diffusivities are independent of direction or flow, the radial concentration distribution can be expected to be intermediate between the distributions predicted by the two models. The radial concentration gradients at the axis and wall of the pipe must be zero, so that the true radial distribution curve may look somewhat like that shown in Fig. 10, passing close to the point of intersection of the two limiting theoretical curves. Near the outlet, the true curve is expected to be similar to that predicted by the no-radial-diffusion model. The actual curve becomes more like that given by the rapid-radial-diffusion model as locations further upstream are considered. 

Pipe reactors can be operated in laminar or turbulent flow. In laminar flow radial diffusivity is molecular only, which is very slow, particularly if the viscosity is high. In turbulent flow the radial fluctuating velocity component produces the radial turbulent diffusivity which is much faster than molecular diffusivity. Many devices have been developed to promote fast radial mixing in laminar flow, such as static mixers, which are discussed below and in Chapter 7. Besides static mixers, a number of methods exist to promote faster radial mixing in turbulent flow, since even in turbulent flow it takes 50 to 100 pipe diameters to achieve... 

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. 

Mass transfer from a single spherical drop to still air is controlled by molecular diffusion and. at low concentrations when bulk flow is negligible, the problem is analogous to that of heat transfer by conduction from a sphere, which is considered in Chapter 9, Section 9.3.4. Thus, for steady-state radial diffusion into a large expanse of stationary fluid in which the partial pressure falls off to zero over an infinite distance, the equation for mass transfer will take the same form as that for heat transfer (equation 9.26) ... 

To account for molecular diffusion, Equation (8.2), which governs reactant concentrations along the streamlines, must be modihed to allow diffusion between the streamlines i.e., in the radial direction. We ignore axial diffusion but add a radial diffusion term to obtain... 

Axial and radial dispersion or non-ideal flow in tubular reactors is usually characterised by analogy to molecular diffusion, in which the molecular diffusivity is replaced by eddy dispersion coefficients, characterising both radial and longitudinal dispersion effects. In this text, however, the discussion will be limited to that of tubular reactors with axial dispersion only. Otherwise the model equations become too complicated and beyond the capability of a simple digital simulation language. 

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

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. 

At high Reynolds numbers where molecular diffusion effects are negligible, experimental evidence confirms the general validity of equation 12.7.5. Figure 12.15 indicates how the Peclet number for radial mixing varies with the fluid Reynolds number. Above a Reynolds number of 40, the radial Peclet number is approximately 10. 

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... 

Points (3) and (4) above imply no molecular diffusion in the axial and radial... 

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