Particle Peclet number

In order to evaluate the efficacy of the expanded bed technique the plate height (HETP), plate number (N), resolution (Rs), Bodenstein number (Bo), particle Peclet number (Pep) and axial dispersion coefficient (DJ have been calculated and compared with the corresponding values of a traditional HPLC column. N can be expressed by... 

The main parameter in this model characterizing the quality of the flow is the axial dispersion coefficient. The term axial is used to distinguish mixing in the direction of flow from mixing in the radial direction. Then, based on this parameter, the particle Peclet number is introduced ... 

Liquid-solid fixed beds In the related literature, there are correlations for the evaluation of the particle Peclet number (Pefi for materials that are frequently used in adsorption and... 

Generally, for spherical and other irregular-shaped particles (intalox saddles, rasching rings, berl saddles), the particle Peclet number is found to be between 0.3 and 0.8 for Reynolds number between 0.01 and 150 (Ebach and White, 1958). For a wide range of values of the Reynolds number, the Chung equation can be used (Chung and Wen, 1968) ... 

This correlation has been derived using glass beads, aluminum beads, and steel beads. Furthermore, according to Chung and Wen (1968), the particle Peclet number is between 0.06 and 0.3, showing no particular trend, for 0.01 < Rep < 10, whereas it steadily increases for/ ep>10. 

It is interesting to check the Peclet number of the fixed bed. The Reynolds number is 0.154, and for this low value, the most appropriate correlation is that of Chung (eq. (3.314)). The resulting particle Peclet number is 0.39 and thus, the bed Peclet number is 151.98, which is fairly high, and we can assume that the plug-flow condition is assured. 

From Table 1-15 of Appendix I, we find that the diffusion coefficient of toluene in air is 8.7 X 10-6 m2/s. Then, using the properties of ah at 25 °C (Table 1-6, Appendix I), we find that Sc = 1.74 and Rep = 6.92, and using the Edwards-Richardson correlation (eq. (3.317)) the particle Peclet number is found to be 1.98 and thus, the bed Peclet number is 609.2, which is fairly high, and plug-flow condition can be assumed. 

For the evaluation of the particle Peclet number and the liquid holdup, the correlations proposed by Inglezakis et al. are used, i.e. eqs. (3.313) and (3.332), respectively. The Biot number, liquid holdup, and bed Peclet number for downflow operation versus relative volumetric flow rate are presented in Figure 4.35. 

To have near-ideal plug flow in a small-scale bed, the particle Peclet number should be high enough to cause a high bed Peclet number, whereas in a large-scale unit this particle Peclet number is of minimal importance, since it is multiplied by Z/dp, and thus the bed Peclet number is expected to be high enough. 

Some interesting conclusions can be drawn from the form of Equation (9). The K s can be determined only by comparison with data they cannot, to our knowledge, be predicted from first principles or existing correlations. The assumed value of 10 for the particle Peclet number is completely absorbed into making that an uncritical assumption. 

The gas flow velocity through the emulsion phase is close to the minimum fluidization velocity When the particles are spherical and have diameters of several tens of microns, this flow condition gives a quite small particle Peclet number, dpUmf/Dc. For example, the Peclet number is estimated as 0.1-0.01 when 122-/Lim-diam. cracking catalyst is fluidized by gas, with Umt = 0.73 cm/sec and Dq = 0.09 cmVsec and it is estimated as 0.001-0.01 for 58-/u.m-diam. particles, with Umt = 0.16 cm/sec. The mechanism of mass transfer between fluid and particles in packed beds is controlled by molecular diffusion under such low Peclet numbers, and the particle Sherwood number kfdp/Dc, is well over 10 (M24). Consequently with intraparticle diffusion shown to be negligible (M21), instantaneous equilibrium is established to be a good approximation [see Eq. (6-24)]. 

Chung and Wen (1968) and Wen and Fan (1975) have proposed a dimensionless equation using the dependency of the dispersion coefficient on the (particle) Reynolds number Re (Eq. 6.169) for fixed and expanded beds. It is an empirical correlation based on published experimental data and correlations from other authors that covers a wide range of Re. Owing to two different definitions of the Reynolds number, the actual appearance varies in the literature. Since the particle diameter dp, is the characteristic value of the packing, Eq. 6.168 based on the (particle) Peclet number Pe (Eq. 6.170) is used here ... 

The presentation of numerical solutions of Eq. 2.18 and their discussion are greatly simplified if some reduced variables are introduced at this stage. These variables are the reduced axial position (x)> the reduced radial position (p), the reduced time (t), the axial (Pea) and the radial (Per) column Peclet numbers (note that these two Peclet numbers are different from the conventional particle Peclet number or reduced velocity, v = udp / D ), and the column aspect ratio (0), which are defined as follows... 

The particle Peclet number (also called the reduced velocity of the mobile phase). Pep = udp / Dp, where dp is the particle size and Dp the diffusion coefficient inside the particles,... 

More precise relationships for /Jdisp were discussed earlier, in the previous section (see Eqs. 6.91a, 6.91b, and 6.91f). A and B in the equation above are characteristic of the packing material and Pe — udp/Dm is the particle Peclet niunber, with u the interstitial velocity, hi the chromatographic literature, the particle Peclet number is frequently named the reduced velocity. Typical values of A and B in a well-packed column are 1.5 and 1.6, respectively [56]. For such a column, used at a moderate to high Peclet munber, hdisp is a small contribution to the overall reduced plate height of the column. 

Reduced velocity of the mobile phase, v Name given by chromatographers to the particle Peclet number. 

The second-order differential equation is solved with a numerical differential equation solver. The dispersion number is estimated by first predicting the particle Peclet number, Pe, from the equation above. Then, the value of the reactor Peclet number, Pe, is predicted from the particle Peclet number Pe by multiplying by the ratio of the particle diameter, dp, to the reactor length, L. Pe is the only parameter required to solve the dispersion model equation. 

Later workers correlated their data in a similar form using a particle Peclet number, with ux/ D , widi X r resmting a mean peaiicle size, in the second term [Greenkom,1982], ie. ... 

The interaction between gas and liquid is limited at low Re numbers. Neither holdup nor dispersion is strongly affected by the gas flow rate, and molecular diffusion coefficient has a rather small effect on the particle Peclet number [20]. 

The experimentally measured dispersion coefficients are usually represented on plots of the particle Peclet number against the particle Reynolds number, defined as follows ... 

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