Radial dispersion coefficient

Radial dispersion coefficient for heat in a packed-bed 9.3 Axial dispersion coefficient for temperature in PDE Sec. 9.1 model... 

Bubble size in the circulating beds increases with Ug, but decreases with Ul or solid circulation rate (Gs) bubble rising velocity increases with Ug or Ul but decreases with Gs the ffequeney of bubbles increases with Ug, Ul or Gs. The axial or radial dispersion coefficient of liquid phase (Dz or Dr) has been determined by using steady or unsteady state dispersion model. The values of Dz and D, increase with increasing Ug or Gs, but decrease (slightly) with increasing Ul- The values of Dz and Dr can be predicted by Eqs.(9) and (10) with a correlation coefficient of 0.93 and 0.95, respectively[10]. 

The differential equation for dispersion in a cylindrical bed of voidage e may be obtained by taking a material balance over an annular element of height SI, inner radius r, and outer radius r + Sr (as shown in Figure 4.5). On the basis of a dispersion model it is seen that if C is concentration of a reference material as a function of axial position /, radial position r, time t, and DL and DR are the axial and radial dispersion coefficients, then ... 

If the axial and radial dispersion coefficients are each taken to be independent of position, we get Eq. (1-3) for which an analytical solution will sometimes be possible. AVe shall call the coefficients of this model the uniform dispersion coefficients thus the parameters of this model are DjtJ and u R). 

Rough estimates for axial dispersion coefficients can be made using random walk techniques, and these will be discussed in Section II,E. Also, a theory can be developed for predicting axial dispersion coefficients from radial dispersion coefficients which is the source of the dotted line of Figure 8. This will be discussed in Section II,D. Bischoff (B13), Fro-ment (F9), and Hofmann (Hll) have presented summaries of packed-bed data. 

Another method has been proposed by Blackwell (B16) and by Hiby and Schiimmer (H8) that avoids the necessity of measuring the complete concentration profile. A pipe with a diameter smaller than the system, thus forming an annular region, is used at the sampling point. A mixed mean sample from the annular region is now sufficient to enable one to determine the radial dispersion coefficient. From Eq. (55) this concentration will be, for an annular region of dimensionless radius a,... 

Packed Beds. Data on liquid systems using a steady point source of tracer and measurement of a concentration profile have been obtained by Bernard and Wilhelm (B6), Jacques and Vermeulen (Jl), Latinen (L4), and Prausnitz (P9). Blackwell (B16) used the method of sampling from an annular region with the use of Eq. (62). Hartman et al. (H6) used a bed of ion-exchange resin through which a solution of one kind of ion flowed and another was steadily injected at a point source. After steady state conditions were attained, the flows were stopped and the total amount of injected ion determined. The radial dispersion coefficients can be determined from this information without having to measure detailed concentration profiles. 

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. 

The radial dispersion coefficient for this case is, of course, the average eddy diffusivity as discussed in works on turbulence (H9). If the various analogies between momentum, heat, and mass transport are used. 

The solution of Eq. (173) poses a rather formidable task in general. Thus the dispersed plug-flow model has not been as extensively studied as the axial-dispersed plug-flow model. Actually, if there are no initial radial gradients in C, the radial terms will be identically zero, and Eq. (173) will reduce to the simpler Eq. (167). Thus for a simple isothermal reactor, the dispersed plug flow model is not useful. Its greatest use is for either nonisothermal reactions with radial temperature gradients or tube wall catalysed reactions. Of course, if the reactants were not introduced uniformly across a plane the model could be used, but this would not be a common practice. Paneth and Herzfeld (P2) have used this model for a first order wall catalysed reaction. The boundary conditions used were the same as those discussed for tracer measurements for radial dispersion coefficients in Section II,C,3,b, except that at the wall. 

Radial dispersion coefficient, dispersed plug flow model... 

Db R) Radial dispersion coefficient, general dispersion model in cylindrical coordinates Molecular diffusivity Exit age distribution function, defined in Section I... 

Axial and radial dispersion coefficients are equal at low Reynolds numbers because the dispersion is due to the molecular diffusion and the axial and radial structures of the bed are similar (Gunn, 1968). However, at high Reynolds numbers, the convective dispersion dominates and the values are different because the axial dispersion is primarily caused by differences in the fluid velocity in the flow channels, whereas the radial dispersion is primarily caused by deviations in the flow path caused by the particles. 

In general, the longitudinal and radial dispersion coefficients D r and D[ will differ. The moles leaving the element in unit time can be written similarly as a series of components ... 

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... 

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... 

This equation is dimensional, and cm/s for G, cm for dB, and cm2/s for DaG should be used. The radial mixing can be represented by radial dispersion coefficients for the gas and the liquid. For instance, the liquid radial dispersion coefficient is estimated at less than one-tenth of the axial one. 

Radial dispersion coefficient of benzene and poison respectively. 

Experimental results show that the radial dispersion coefficient increases with solids circulation rate and with bed density, and decreases, though slightly, with gas velocity. Experiments are yet inconclusive as to the dependence of Dt on column diameter. 

Values of radial dispersion coefficients from literature sources are listed in Table III. 

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 . ... 

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