Correlation coefficient, axial dispersion

Figure 10.3 Correlation for axial dispersion coefficient In pipe flow, (Pe) versus (Re).

Figure 8-33. Correlation of axial dispersion coefficient for flow of fluids through pipes in laminar flow region (NRe < 2,000). (Source Wen, C. Y. and Fan, L. T, Models for Flow Systems and Chemical Reactors, Marcel Dekker Inc., 1975.)...

Gunn correlation The axial dispersion coefficient may be estimated using a correlation given by Gunn [77]... 

Figure 5.8 (a) Correlation of axial dispersion coefficient for flow of fluids through pipes in... 

Flow in microchannels with diameters between 10 and 1000 pm is mostly laminar and has a parabolic velocity profile. Therefore, the molecular diffusion in axial and radial directions plays an important role in RTD. The diffusion in the radial direction tends to diminish the spreading effect of the parabolic velocity profile, while in the axial direction the molecular diffusion increases the dispersion [7,8]. With the so-called Taylor-Aris correlation the axial dispersion coefficient can be predicted based on the molecular diffusion coefficient D, the mean velocity of the stratified flow, the hydraulic diameter of the microchannel, and the geometry... 

Baird and Rice first applied the isotropic turbulence theory to correlate the axial dispersion coefficient in Newtonian fluids [39]. Their successful approach has been widely quoted to predict design parameters in bubble columns (Kawase and Moo-Young [40]). It was extended to non-Newtonian fluids by Kawase and Moo-Young [32]. The resulting equation may be written as... 

Correlations for axial dispersion coefficients in empty pipes and in packed beds are given in the Sections 4.10.6.3 and 4.10.6.4, respectively, so we can also calculate Dax without the need of an experiment (or prove the results of measurements, respectively). 

The recommended correlation for the gas-phase axial-dispersion coefficient is given by Field and Davidson (loc. cit.) ... 

TABLE 16-10 Coefficients for Axial Dispersion Correlations in Packed Beds Based on Eq (16-79)... 

FIG. 16 11 Axial dispersion coefficient correlations for well-packed beds for e = 0.4. 

The dispersion coefficient is orders of magnitude larger than the molecular diffusion coefficient. Some rough correlations of the Peclet number are proposed by Wen (in Petho and Noble, eds.. Residence Time Distribution Theory in Chemical Tngineeiing, Verlag Chemie, 1982), including some for flmdized beds. Those for axial dispersion are ... 

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]. 

Taylor (T4, T6), in two other articles, used the dispersed plug-flow model for turbulent flow, and Aris s treatment also included this case. Taylor and Aris both conclude that an effective axial-dispersion coefficient Dzf can again be used and that this coefficient is now a function of the well known Fanning friction factor. Tichacek et al. (T8) also considered turbulent flow, and found that Dl was quite sensitive to variations in the velocity profile. Aris further used the method for dispersion in a two-phase system with transfer between phases (All), for dispersion in flow through a tube with stagnant pockets (AlO), and for flow with a pulsating velocity (A12). Hawthorn (H7) considered the temperature effect of viscosity on dispersion coefficients he found that they can be altered by a factor of two in laminar flow, but that there is little effect for fully developed turbulent flow. Elder (E4) has considered open-channel flow and diffusion of discrete particles. Bischoff and Levenspiel (B14) extended Aris s theory to include a linear rate process, and used the results to construct comprehensive correlations of dispersion coefficients. 

The values of the axial dispersion coefficients in trickle beds are 1/3 - l/6th those of the liquid flow alone at the same Reynolds numbers. A correlation by Michell and Furzer is available (Satterfield, 1975 Perry and Green, 1999) ... 

The following, well-acceptable assumptions are applied in the presented models of automobile exhaust gas converters Ideal gas behavior and constant pressure are considered (system open to ambient atmosphere, very low pressure drop). Relatively low concentration of key reactants enables to approximate diffusion processes by the Fick s law and to assume negligible change in the number of moles caused by the reactions. Axial dispersion and heat conduction effects in the flowing gas can be neglected due to short residence times ( 0.1 s). The description of heat and mass transfer between bulk of flowing gas and catalytic washcoat is approximated by distributed transfer coefficients, calculated from suitable correlations (cf. Section III.C). All physical properties of gas (cp, p, p, X, Z>k) and solid phase heat capacity are evaluated in dependence on temperature. Effective heat conductivity, density and heat capacity are used for the entire solid phase, which consists of catalytic washcoat layer and monolith substrate (wall). 

Fig. 10.16. Axial dispersion coefficients (solid line correlation, points CFD and experimental data).

Sullivan and Treybal (1972) suggested the following correlation for the liquid-phase axial dispersion coefficient for a 0.15 m i.d., 12-stage column ... 

The pore diffusivity used in this analysis was determined by the Renkin equation4, the axial dispersion coefficient calculated by assuming a constant Peclet number of 0.2, and the mass transfer coefficient from the bulk to the particle surface calculated by the correlation of Wakao and Kaguei. The product of the heat capacity and density of the solid phase was taken to be the same as that used by Raghavan and Ruthven17. The density of the fluid phase was assumed to be that of pure C02 and was calculated from data provided by the Dionix Corporation in their AI-450 SFC software. Constant pressure heat capacities for the mobile phase were also assumed to be that of pure C02 and were taken from Brunner3. 

The elimination or estimation of the axial dispersion contribution presents a more difficult problem. Established correlations for the axial dispersion coefficient are notoriously unreliable for small particles at low Reynolds number(17,18) and it has recently been shown that dispersion in a column packed with porous particles may be much greater than for inert non-porous particles under similar hydrodynamic conditions(19>20). one method which has proved useful is to make measurements over a range of velocities and plot (cj2/2y ) (L/v) vs l/v2. It follows from eqn. 6 that in the low Reynolds number region where Dj. is essentially constant, such a plot should be linear with slope Dj, and intercept equal to the mass transfer resistance term. Representative data for several systems are shown plotted in this way in figure 2(21). CF4 and iC io molecules are too large to penetrate the 4A zeolite and the intercepts correspond only to the external film and macropore diffusion resistance which varies little with temperature. 

Methods for evaluating the axial dispersion coefficient from RTD data As mentioned earlier, the one-parameter axial-dispersion model is widely used to correlate RTD data. The nature of the RTD depends upon the nature of the tracer input and the nature of the. flow, characteristics. For the RTD shown in Fig. 3-4 o), the axial dispersion coefficients for the liquid and solid phases can be obtained by fitting the equation... 

Sachova and Sterbacek89 evaluated the axial dispersion coefficient from a response to a random-pulse input by using a general analytical expression for the random-pulse input. The random-pulse input was correlated as... 

Three-parameter PDE model (Van Swaaij et aL106) This model is largely used to correlate the RTD curves from a trickle-bed reactor. The model is based on the same concept as the crossflow or modified mixing-cell model, except that axial dispersion in the mobile phase is also considered. The model, therefore, contains three arbitrary parameters, two of which are the same as those used in the cross-flow model and the third one is the axial dispersion coefficient (or the Peclet number in dimensionless form) in the mobile phase (see Fig. 3-11). 

The experimental studies have shown that, in gas-liquid trickle-bed reactors, significant axial mixing occurs in both gas and liquid phases. When the RTD data are correlated by the single-parameter axial dispersion model, the axial dispersion coefficient (or Peclet number) for the gas phase is dependent upon both the liquid and gas flow rates and the size and nature of the packings. The axial dispersion coefficient for the liquid phase is dependent upon the liquid flow rate, liquid properties, and the nature and size of the packings, but it is essentially independent of the gas flow rate. 

Here, PeL = LJLdp/EZL, Reu = dppLUL/pLi GaL = dlgpl/pl, UL is the interstitial liquid velocity, and EZL is the liquid-phase axial dispersion coefficient. Furzer and Michell28 correlated the Peclet number to the dynamic holdup by a relation... 

Stiegel and Shah35 found that the Peclet number was somewhat dependent upon the bed height. Unlike the unpacked bubble-column, the above correlation indicates that the axial dispersion coefficient in a packed bubble-column is dependent upon the liquid velocity. 

Stiegel and Shah34 measured the liquid-phase axial dispersion coefficient in a packed rectangular column. Some details of system conditions used in this study have been described earlier, in Sec. 7-3. The axial dispersion coefficient and the liquid-phase Peclet number were correlated to the gas and liquid Reynolds numbers by the expressions... 

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