Radial Peclet numbers

FIGURE 9.1 Existing data for the radial Peclet number in large-diameter packed beds, (Pe) = usdpID,) versus pd u./u. 

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. 

The radial Peclet number may now be estimated using Figure 12.15 at NRe = 41,... 

The mixing properties in a fluidized bed are a strong function of the fraction voids. Minimum values of radial Peclet numbers udp/Djt) are observed at e = 0.7, corresponding to a transition in the type of particle circulation in the bed. 

Liquid-solid fixed beds. According to Gunn (1968), for random beds of spheres (e = 0.4), the radial Peclet numbers are from about 10-40 for 0.08 < Rep < 1000. 

From Figure 3.38, it is clear that the radial Peclet number is greater than the axial Peclet number in gas-solid systems for the same Reynolds number. 

According to Gunn (1968), the radial Peclet number in particulate fluidization (liquid-solid systems) ranges between 1 and 10 for values of Rep in the range 4—1000. Furthermore, the maximum mixing coefficient is found for sf = 0.7. Finally, the lateral (radial) mixing coefficients in gas-solid fluidized beds decrease constantly (for Rep > 10) from about 10 - 0.05 by increasing the expansion ratio from 0.01 - 0.2. 

Dispersion coefficients for packed beds are usually plotted in the form of a similar dimensionless group, the Peclet number udp/DL, which uses the diameter of the particles of the bed dp as the characteristic length rather than the bed diameter. Graphs of experimentally measured values of both axial and radial Peclet numbers plotted against Reynolds number, also based on particle diameter, are shown in Volume 2, Chapter 4. 

Effective Wall Biot Number, Bi, = 0.84 Effective Radial Peclet Numbers = 0.083 Pm = 0.075... 

Here, p is the local (transverse) Peclet number, which is the ratio of transverse diffusion time to the convection time. Per is the radial Peclet number (ratio of transverse diffusion time to a convection time based on pipe radius). We assume that p <4 1 while Per is of order unity. (Remark The parameter Pe /p — ux)L/Dm is also known as the axial Peclet number. Also note that for any finite Per or tube diameter, the axial Peclet number tends to infinity as p tends to zero.) When such scale separation exists, we can average the governing equation over the transverse length scale using the L-S technique and obtain averaged model in terms of axial length and time scales. 

We note that when Dm < ux)2tn, or equivalently, the radial Peclet number Per 6.93, axial diffusion can be neglected. The local Peclet number p, which is... 

Here, t and z have been scaled with respect to the convection time and length of monolith, respectively. The transverse or local Peclet number p and the radial Peclet number Per are defined as in the case of the Taylor problem, e is the volume fraction of the fluid phase and... 

In all the studies described above, only the axial dispersion was considered. Anderson et al.1 measured the radial dispersion for the dispersed water phase in an air-water system. The measurements were carried out in a 30.48-cm-diameter Lucite tube packed with 91.44 cm of 1.27-cm Raschig rings. A continuous source of tracer was used. The radial Peclet number decreased with the increase in both the gas and liquid Reynolds number. Some typical results are shown in Fig. 8-5. The measurements were carried out up to the flooding point. The entire results were correlated graphically, as shown in Fig. 8-6. In Figs. 8-5 and 8-6 the Peclet number was based on the fluid velocity through the column, the nominal packing... 

Figure 8-5 Effect of gas flow rate on radial Peclet number for a water phase (Jrom data of Anderson et al. ).

Pep effective radial Peclet number (based on R), GCpR/k. ... 

Axial and radial Peclet numbers as a function of Reynolds number for packed-beds. [Adapted from R, H. Wilhelm, Pure App. Chem., 5 (1962) 403, with permission of the International Union of Pure and Applied Chemistry.]... 

The conditions will only be maintained uniformly across the section of the bed if there is good lateral dispersion. This is often represented by a radial eddy diffusion coefficient In a packed bed division and recombination of streams around the particles promote this radial dispersion, and this has been analyzed as a random walk problem to predict a radial Peclet number Pe = of about 8. For Reynolds numbers above 10, a value... 

Radial Peclet numbers have been measured, and some of the results are shown in Fig. 13-5. Above a modified Reynolds number dpGjp of about 40, Pe,. is independent of flow rate and has a magnitude of about 10. The two terms involving radial gradients in Eq. (13-10) are generally small for isothermal operation. The only way that concentration gradients can... 

Figure 13-5 summarizes data for in terms of Pe,.. In terms of the radial Peclet number determined from mass-transfer data, if dp is the particle diameter and u is the superficial velocity, the turbulent-diffusion contribution is... 

FIGURE 14.11 Axial and radial Peclet numbers for single-phase flow of a gas or liquid through a packed bed of spherical particles. The limits of molecular diffusion are shown hy sohd hues that represent Pe = Re Sc Xg/g, where Xg = bed tortuousity =1.4 and Eg = bed voidage = 0.4 (Sherwood et al., 1975). 

Radial dispersion of mass and heat in fixed bed gas-solid catalytic reactors is usually expressed by radial Peclet number for mass and heat transport. In many cases radial dispersion is negligible if the reactor is adiabatic because there is then no driving force for long range gradients to exist in the radial direction. For non-adiabatic reactors, the heat transfer coeflScient at the wall between the reaction mixture and the cooling medium needs also to be specified. 

A numerical study of the free-radical polymerization of styrene (Scheme 6.15) compared the behavior of an interdigital micromixer with a T-junction and a straight tube [37, 48], The diffusion coefficient of the reactive species was varied to simulate the viscosity increase during a polymerization. The performance of the polymerization turned out to be largely dependent on the radial Peclet number. This dimensionless number is defined as the ratio of the characteristic time of diffusion in the direction perpendicular to the main flow to the characteristic time of convection in the flow direction (i.e., the mean residence time) and, therefore, is directly proportional to the characteristic length of the reactor. 

Fig. 6.32 Comparison of the polydispersity index (DPI) obtained in a multilamination micromixer (open symbols) and in a tube reactor (filled symbols) as a function of the radial Peclet number. (Courtesy of the Royal Society of Chemistry [48].)...

Note that in the inequality of (5-33) that the quantity uR/Dm) can be considered a radial Peclet number defined analogously to the longitudinal quantity defined previously. It has been suggested that the right inequality is a little tight, and that iiR/Dm) > 50 is a more comfortable approximation [V. Ananthakrishnan, W.N. Gill and A.I. Barduhn, Amer. Inst. Chem. Eng. J., 11, 1063 (1965)). 

Values of the radial dispersion coefficient, or the corresponding radial Peclet number, udp/Dr), in packed beds have been determined for both liquids and gases by a number of researchers they are definitely not the same as those in the axial direction. These results are shown in Figure 5.10a and b for liquids and gases, respectively. The corresponding empirical equations fitting these data are... 

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