Radial dispersion effects

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. 

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 ... 

The basic message in all of the above is essentially that one may avoid axial dispersion effects if it is possible to design with the necessary axial aspect ratio, n, but there is no way in general to avoid radial dispersion effects. 

Therefore, there are axial and radial dispersion effects due to mass and heat transfer phenomena in isothermal and nonisothermal systems. 

To calculate the mass of the catalyst, we use the PFR model, assuming a pseudohomogeneous rate and without axial or radial dispersion effects. Thus ... 

Figure 3.3 Nonidealities in a PFR boundary layer development. The nonuniformities in the velocity fields cause mixing problems, giving rise to axial and/or radial dispersion effects.

The precise and, where needed, short setting of the residence time allows one to process oxidations at the kinetic limits. The residence time distributions are identical within various parallel micro channels in an array, at least in an ideal case. A further aspect relates to the flow profile within one micro channel. So far, work has only been aimed at the interplay between axial and radial dispersion and its consequences on the flow profile, i.e. changing from parabolic to more plug type. This effect waits to be further exploited. 

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. 

However, the two mechanisms interact and molecular diffusion can reduce the effects of convective dispersion. This can be explained by the fact that with streamline flow in a tube molecular diffusion will tend to smooth out the concentration profile arising from the velocity distribution over the cross-section. Similarly radial dispersion can give rise to lower values of longitudinal dispersion than predicted by equation 4.39. As a result the curves of Peclet versus Reynolds number tend to pass through a maximum as shown in Figure 4.6. 

It has been demonstrated that radial dispersion contributes more significantly to the dilution of the sample in the flow than does axial dispersion. This type of fluid movement, termed secondary flow by Tijssen [43], results in a washout effect accounting for the low mutual contamination of samples successively injected into a carrier stream. TTiis advantageous feature is a result of the use of low flow rates and small tubing bores, and results in decreased peak-width and hence to increased sampling rate. 

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. 

It should be pointed out that for a low pressure gas the radial- and axial diffusion coefficients are about the same at low Reynolds numbers (Rediffusion effects may be important at velocities where the dispersion effects are controlled by molecular diffusion. For Re = 1 to 20, however, the axial diffusivity becomes about five times larger than the radial diffusivity [31]. Therefore, the radial diffusion flux could be neglected relative to the longitudinal flux. If these phenomena were also present in a high-pressure gas, it would be true that radial diffusion could be neglected. In dense- gas extraction, packed beds are operated at Re > 10, so it will be supposed that the Peclet number for axial dispersion only is important (Peax Per). 

The importance of dispersion and its influence on flow pattern and conversion in homogeneous reactors has already been studied in Chapter 2. The role of dispersion, both axial and radial, in packed bed reactors will now be considered. A general account of the nature of dispersion in packed beds, together with details of experimental results and their correlation, has already been given in Volume 2, Chapter 4. Those features which have a significant effect on the behaviour of packed bed reactors will now be summarised. The equation for the material balance in a reactor will then be obtained for the case where plug flow conditions are modified by the effects of axial dispersion. Following this, the effect of simultaneous axial and radial dispersion on the non-isothermal operation of a packed bed reactor will be discussed. 

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... 

The use of the Coanda effect is based on the desire to have a second passive momentum to speed up mixing in addition to diffusion [55, 163], The second momentum is based on so-called transverse dispersion produced by passive structures, which is in analogy with the Taylor convective radial dispersion ( Taylor dispersion ) (see Figure 1.180 and [163] for further details). It was further desired to have a flat ( in-plane ) structure and not a 3-D structure, since only the first type can be easily integrated into a pTAS system, typically also being flat A further design criterion was to have a micro mixer with improved dispersion and velocity profiles. 

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... 

For non-adiabatic reactors, along with radial dispersion, heat transfer coefficient at the wall between the reaction mixture and the cooling medium needs to be specified. Correlations for these are available (cf. % 10) however, it is possible to modify the effective radial thermal conductivity (k ), by making it a function of radial position, so that heat transfer at the wall is accounted for by a smaller k value near the tube-wall than at the tube center (11). 

Gas-phase reacdotis are carried out primarily in tubular reactors where the flow is generally turbulent. By assuming that there is no dispersion and ttiere are no radial gradients in either temperature, velocity, or concentration, we can model the flow in the reactor as plug-flow. Laminar reactors are discussed in Chapter 13 and dispersion effects in Chapter 14. The differential form of the design equation... 

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 . ... 

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