Dispersion Peclet number correlation

Dispersion In tubes, and particiilarly in packed beds, the flow pattern is disturbed by eddies diose effect is taken into account by a dispersion coefficient in Fick s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of oo. Some rough correlations of the Peclet number uL/D in terms of Reynolds and Schmidt numbers are Eqs. (23-47) to (23-49). There is also a relation between the Peclet number and the value of n of the RTD equation, Eq. (7-111). The dispersion model is sometimes said to be an adequate representation of a reaclor with a small deviation from phig ffow, without specifying the magnitude ol small. As a point of superiority to the RTD model, the dispersion model does have the empirical correlations that have been cited and can therefore be used for design purposes within the limits of those correlations. 

The value of n is the only parameter in this equation. Several procedures can be used to find its value when the RTD is known experiment or calculation from the variance, as in /i = 1/C (t ) = 1/ t C t), or from a suitable loglog plot or the peak of the curve as explained for the CSTR battery model. The Peclet number for dispersion is also related to n, and may be obtainable from correlations of operating variables. 

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 ... 

Comparison of Models Only scattered and inconclusive results have been obtained by calculation of the relative performances of the different models as converiers. Both the RTD and the dispersion coefficient require tracer tests for their accurate determination, so neither method can be said to be easier to apply The exception is when one of the cited correlations of Peclet numbers in terms of other groups can be used, although they are rough. The tanks-in-series model, however, provides a mechanism that is readily visualized and is therefore popular. 

Axial Dispersion and the Peclet Number Peclet numbers are measures or deviation from phig flow. They may be calculated from residence time distributions found by tracer tests. Their values in trickle beds are fA to Ve, those of flow of liquid alone at the same Reynolds numbers. A correlation by Michell and Furzer (Chem. Eng. /., 4, 53 [1972]) is... 

The inverse of the Bodenstein number is eD i/u dp, sometimes referred to as the intensity of dispersion. Himmelblau and Bischoff [5], Levenspiel [3], and Wen and Fan [6] have derived correlations of the Peclet number versus Reynolds number. Wen and Fan [6] have summarized the correlations for straight pipes, fixed and fluidized beds, and bubble towers. The correlations involve the following dimensionless groups ... 

From the Peclet number of dispersion when the Peclet number can be found from other correlations. The relationships are brought out in problems P5.0B.04 and P5.08.14. 

A significant merit of the dispersion model is some experimental correlations for the Peclet number. There are no such direct correlations for the parameters of the Gamma or Gaussian or other similar models. 

Rough correlations of Peclet numbers for dispersion are given by Wen (in Petho Noble, Residence Time Distribution in Chemical Engineering, 1982)... 

The equations have previously been derived in Section 4.4.4 in a form suitable for programming with MADONNA. Correlations for the column Peclet number are taken from the literature and used to calculate a suitable value for the dispersion coefficient for use in the model. 

The experimental results for dispersion coefficients in gases show that they can be satisfactorily represented as Peclet number expressed as a function of particle Reynolds number, and that similar correlations are obtained, irrespective of the gases used. However, it might be expected that the Schmidt number would be an important variable, but it is not possible to test this hypothesis with gases as the values of Schmidt number are all approximately the same and equal to about unity. 

Die Reynolds and Peclet numbers are based on the superficial liquid velocity, whereas c/p and au are expressed in cm and cm2/cm3, respectively. For gas-phase dispersions, the Hochman-Effron correlation is available (Satterfield, 1975) ... 

Deviation from the ideal plug flow can be described by the dispersion model, which uses the axial eddy diffusivity (m s ) as an indicator of the degree of mixing in the flow direction. If the flow in a tube is plug flow, the axial dispersion is zero. On the other hand, if the fluid in a tube is perfectly mixed, the axial dispersion is infinity. For turbulent flow in a tube, the dimensionless Peclet number (Pe) deflned by the tube diameter (v dlE-Q is correlated as a function of the Reynolds number, as shown in Figure 10.3 [3] dz is the axial eddy diffusivity, d is the tube diameter, and v is the velocity of liquid averaged over the cross section of the flow channel. 

Although it also is subject to the limitations of a single characterizing parameter which is not well correlated, the Peclet number, the dispersion model predicts conversions or residence times unambiguously. For a reaction with rate equation rc = fcC , this model is represented by the differential equation... 

The assessment of the role of kf during protein adsorption in a fluidized bed may be performed with the help of a dimensionless transport number. Slater used the correlations provided by Rodrigues to simulate film transport limited adsorption of small ions to fluidized resins [54], In this study dimensionless groups were used to describe the influence of the system parameters particle side transport, liquid dispersion, and fluid side transport. Dispersion was accounted for by the column Peclet number analogous to Bo as introduced above and mass transport from the bulk solution to the resin was summarized in a fluid side transport number NL. 

The model is referred to as a dispersion model, and the value of the dispersion coefficient De is determined empirically based on correlations or experimental data. In a case where Eq. (19-21) is converted to dimensionless variables, the coefficient of the second derivative is referred to as the Peclet number (Pe = uL/De), where L is the reactor length and u is the linear velocity. For plug flow, De = 0 (Pe ) while for a CSTR, De = oo (Pe = 0). To solve Eq. (19-21), one initial condition and two boundary conditions are needed. The closed-ends boundary conditions are uC0 = (uC — DedC/dL)L=o and (dC/BL)i = i = 0 (e.g., see Wen and Fan, Models for Flow Systems in Chemical Reactors, Marcel Dekker, 1975). Figure 19-2 shows the performance of a tubular reactor with dispersion compared to that of a plug flow reactor. 

Numerous empirical correlations for the prediction of residual NAPL dissolution have been presented in the literature and have been compiled by Khachikian and Harmon [68]. On the other hand, just a few correlations for the rate of interface mass transfer from single-component NAPL pools in saturated, homogeneous porous media have been established, and they are based on numerically determined mass transfer coefficients [69, 70]. These correlations relate a dimensionless mass transfer coefficient, i.e., Sherwood number, to appropriate Peclet numbers, as dictated by dimensional analysis with application of the Buckingham Pi theorem [71,72], and they have been developed under the assumption that the thickness of the concentration boundary layer originating from a dissolving NAPL pool is mainly controlled by the contact time of groundwater with the NAPL-water interface that is directly affected by the interstitial groundwater velocity, hydrodynamic dispersion, and pool size. For uniform... 

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. 

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... 

Woodburn55 obtained gas-phase axial dispersion data at very high irrigation rales. He found that, over the range 15 < ReG < 500 and 126 < ReL < 1,321 (Reynolds numbers are defined in Sec. 8-3), the gas-phase Peclet number increased with the superficial gas rate. The data indicated that the gas-phase axial dispersion coefficient ZG was proportional to the gas pore velocity i.e., EZG cc C/G, where n > 1 for loading conditions and 0 < n 1 for subloading conditions. The data in the ranges 600 < ReG < 2,200 and 0 = ReL < 375 were well correlated by a correlation of Dunn ct al.,16 namely,... 

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... 

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