Diffusion axial

The net transport of component A in the +2 direction in the centrifuge is equal to the sum of the convective transport and the axial diffusive transport. At the steady state the net transport of component A toward the product withdrawal point must be equal to the rate at which component A is being withdrawn from the top of the centrifuge. Thus, the transport of component is given by equation 72 ... 

The first term may be considered as the contribution of the internal circulation or convective flow to the stage length, the second term as the contribution of the axial diffusion to the stage length. The stage separation factor is given by... 

Axial diffusion Mass transfer by diffusion along streamlines that occurs at... 

It seems probable that a fruitful approach to a simplified, general description of gas-liquid-particle operation can be based upon the film (or boundary-resistance) theory of transport processes in combination with theories of backmixing or axial diffusion. Most previously described models of gas-liquid-particle operation are of this type, and practically all experimental data reported in the literature are correlated in terms of such conventional chemical engineering concepts. In view of the so far rather limited success of more advanced concepts (such as those based on turbulence theory) for even the description of single-phase and two-phase chemical engineering systems, it appears unlikely that they should, in the near future, become of great practical importance in the description of the considerably more complex three-phase systems that are the subject of the present review. 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

The relatively short distance in the radial direction leads to much higher diffusion rates. In most of what follows, axial diffusion will be ignored. 

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... 

The use of dimensionless variables will be illustrated using Equation (8.12) but with an added term for axial diffusion ... 

When expressed in the scaled variables, the ajd and ajd terms have the same magnitude, but the d ajd term is multiplied by a factor of that will not be larger than 0.01. Thus, this term, which corresponds to axial diffusion, may be neglected, consistent with the conclusion in Section 8.2. 

The thoughtful reader may wonder about a real reactor with a high level of radial diffusion. Won t there necessarily be a high level of axial diffusion as well and won t the limit of oo really correspond to a CSTR rather than... 

This section derives a simple version of the convective diffusion equation, applicable to tubular reactors with a one-dimensional velocity profile V (r). The starting point is Equation (1.4) applied to the differential volume element shown in Figure 8.9. The volume element is located at point (r, z) and is in the shape of a ring. Note that 0-dependence is ignored so that the results will not be applicable to systems with significant natural convection. Also, convection due to is neglected. Component A is transported by radial and axial diffusion and by axial convection. The diffusive flux is governed by Pick s law. 

A simple correction to piston fiow is to add an axial diffusion term. The resulting equation remains an ODE and is known as the axial dispersion model ... 

It is also assumed that axial diffusion of mass, heat and momentum are negligible. The velocity profiles are assumed to be locally fully developed in length. For this simplified case the polymer concentration and secondary flows are determined by three dimensionless groups ... 

Table 1.6 Characteristic quantities to be considered for micro-reactor dimensioning and layout. Steps 1, 2, and 3 correspond to the dimensioning of the channel diameter, channel length and channel walls, respectively. Symbols appearing in these expressions not previously defined are the effective axial diffusion coefficient D, the density thermal conductivity specific heat Cp and total cross-sectional area S, of the wall material, the total process gas mass flow m, and the reactant concentration Cg [114].

Surface Polarization in TFF The simplified model of polarization shown in Fig. 20-47 is used as a basis for analyzing more complex systems. Consider a single component with no reaction in a thin, two-dimensional boundary layer near the membrane surface. Axial diffusion is negligible along the membrane surface compared to convection. 

Fig. 2.4p shows three types of post-column reactor. In the open tubular reactor, after the solutes have been separated on the column, reagent is pumped into the column effluent via a suitable mixing tee. The reactor, which may be a coil of stainless steel or ptfe tube, provides the desired holdup time for the reaction. Finally, the combined streams are passed through the detector. This type of reactor is commonly used in cases where the derivatisation reaction is fairly fast. For slower reactions, segmented stream tubular reactors can be used. With this type, gas bubbles are introduced into the stream at fixed time intervals. The object of this is to reduce axial diffusion of solute zones, and thus to reduce extra-column dispersion. For intermediate reactions, packed bed reactors have been used, in which the reactor may be a column packed with small glass beads. 

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