Effective radial dispersion coefficient

When the effective radial dispersion coefficient (De)r can be neglected in comparison to the effective axial dispersion coefficient (De)1 Equation 8-120 reduces... 

The data were plotted, as shown in Fig. 11, using the effective diameter of Eq. (50) as the characteristic length. For fully turbulent flow, the liquid and gas data join, although the two types of systems differ at lower Reynolds numbers. Rough estimates of radial dispersion coefficients from a random-walk theory to be discussed later also agree with the experimental data. There is not as much scatter in the data as there was with the axial data. This is probably partly due to the fact that a steady flow of tracer is quite easy to obtain experimentally, and so there were no gross injection difficulties as were present with the inputs used for axial dispersion coefficient measurement. In addition, end-effect errors are much smaller for radial measurements (B14). Thus, more experimentation needs to be done mainly in the range of low flow rates. 

Taylor (T2) and Westhaver (W5, W6, W7) have discussed the relationship between dispersion models. For laminar flow in round empty tubes, they showed that dispersion due to molecular diffusion and radial velocity variations may be represented by flow with a flat velocity profile equal to the actual mean velocity, u, and with an effective axial dispersion coefficient Djf = However, in the analysis, Taylor... 

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... 

Like axial dispersion, radial dispersion can also occur. Radial-dispersion effects normally arise from radial thermal gradients that can dramatically alter the reaction rate across the diameter of the reactor. Radial dispersion can be described in an analogous manner to axial dispersion. That is, there is a radial dispersion coefficient. A complete material balance for a transient tubular reactor could look like ... 

Many authors pointed out that although the axial dispersion coefficient is much higher than the radial dispersion coefficient in a CFB, the back-mixing effect is still negligible because of the large axial convective flow . ... 

The axial mixing in a tubular reactor can sometimes be described by a dispersion model. This model is based on the assumption that the RTD may be considered to result from piston flow on which is superimposed an axial dispersion. The latter is taken into account by means of a constant effective axial dispersion coefficient, Dax, which has the same dimensions as the molecular diffusion coefficient, Dm. Usually Dax is much larger than the molecular diffusion coefficient because it incorporates all effects that cause deviations from plug flow, such as variations in radial velocities, eddies, and vortices. 

Flow in microchannels with diameters between 10 and 500 pm is mostly laminar and characterized by a parabolic velocity profile. Therefore, the molecular diffusion in axial and radial directions plays an important role influencing the RTD. Diffusion in the radial direction tends to diminish the spread of the parabolic velocity profile, whereas diffusion in the axial direction increases the spread [72,73]. Taylor [72] and Aris [73] established the following relation to predict the effective axial dispersion coefficient for laminar flow ... 

In particular, equation (7-146) expresses that there is no mass transfer at the wall, since the concentration derivative is zero, and that heat transfer occurs with a constant wall temperature, Tw, and a local heat-transfer coefficient, This heat-transfer coefficient is now appearing in a boundary condition and is not equivalent to the overall heat-transfer coefficient used in nonisothermal axial dispersion models. The radial dispersion coefficient, Z) is, as the name implies, the radial counterpart to the axial dispersion coefficient, and while we expect a different correlation for it there are no new conceptual boundaries set here. The effective bed thermal conductivity, A however, is another matter altogether and we will worry about it more later. 

In this model the effects of all mechanisms which contribute to axial mixing are lumped together into a single effective axial dispersion coefficient. More detailed models which include, for example, radial dispersion are generally not necessary and in many cases it is in fact possible to neglect axial dispersion altogether and assume ideal plug flow. 

Axial dispersion model In the dispersion model, deviation from plug flow is expressed in terms of a dispersion or effective axial diffusion coefficient. The mathematical derivation is similar to that for plug flow except that a term is now included for diffusive flow in addition to that for convective flow. This term appears as (d[A]ldz), where is the effective axial dispersion coefficient. The continuity equation in the absence of radial variations takes the form... 

Effective dispersion is affected by both axial and radial dispersion, as well as by Taylor dispersion due to the radial profile of axial gas velocities. Schiigerl (1967) related the effective dispersion coefficient, Dge, axial dispersion coefficient, Dg, radial dispersion coefficient, Z)gr, and the radial velocity profile by... 

Radial (transverse) dispersion of mass During passage of the fluid through the fixed bed, the repeated lateral displacement combined with mixing of fluid elements of different streamlines lead to radial mixing perpendicular to the main flow in axial direction. This can be characterized by an apparent radial dispersion coefficient (effective radial diffusivity) D a. which is in most practical cases much higher than the molecular diffusion coefficient. 

As a rule of thumb, axial dispersion of heat and mass (factors 2 and 3) only influence the reactor behavior for strong variations in temperature and concentration over a length of a few particles. Thus, axial dispersion is negligible if the bed depth exceeds about ten particle diameters. Such a situation is unlikely to be encountered in industrial fixed bed reactors and mostly also in laboratory-scale systems. Radial mass transport effects (factor 1) are also usually negligible as the reactor behavior is rather insensitive to the value of the radial dispersion coefficient. Conversely, radial heat transport (factor 4) is really important for wall-cooled or heated reactors, as such reactors are sensitive to the radial heat transfer parameters. 

Especially in the case of strongly exothermic reactions, radial temperature gradients appear in the reactor tube. The existence of these gradients implies that the chemical reaction proceeds at different velocities in various radial positions and, consequently, radial concentration gradients emerge. Because of these concentration gradients, dispersion of the material is initiated in the direction of the radial coordinate. Dispersion of heat and material can be described with radial dispersion coefficients, and the mathematical formulation of dispersion effects resembles that of Pick s law (Chapter 4) for molecular diffusion. 

If the heat effect that is caused by the chemical reactions is considerable and if the heat conductivity of the catalyst material is low, radial temperature gradients emerge in a reactor tube. This implies, accordingly, that the rate of the chemical reaction varies in the radial direction, and, as a result, concentration gradients emerge in a reactor tube. This phenomenon is illustrated in Figure 5.28. Radial heat conduction can be described with the radial dispersion coefficient as will be shown below. 

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. 

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 magnitude of the dispersion effect due to transverse or radial mixing can be assessed by relying on theoretical predictions " and experimental observations " which confirm that the value of the Peclet number Pe(= udp/D, where dp is the particle diameter) for transverse dispersion in packed tubes is approximately 10. At bed Reynolds numbers of around 100 the diffusion coefficient to be ascribed to radial dispersion effects is about four times greater than the value for molecular diffusion. At higher Reynolds numbers the radial dispersion effect is correspondingly larger. 

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