Peclet numbers, calculation

Here Pe = aU /D is the Peclet number calculated from the constrained flow velocity, which can be calculated by the formula... 

FIG. 23 The dependence of the reduced hulk transfer rate k a/D (Sherwood number) on the Peclet number calculated numerically for a uniformly accessible surface in the case of no external force, < )q = —1(P curve 1, a/Le = 2 curve 2, a/Le = 5 curve 3, ajLe = 10 curve 4, ajLe = 100. The limiting values calculated from Eq. (156) are plotted by the dashed-dotted line and these calculated from Eq. (165) are depicted by the dashed line. (From Ref. 37.)... 

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. 

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 problem of axial conduction in the wall was considered by Petukhov (1967). The parameter used to characterize the effect of axial conduction is P = (l - dyd k2/k ). The numerical calculations performed for q = const, and neglecting the wall thermal resistance in radial direction, showed that axial thermal conduction in the wall does not affect the Nusselt number Nuco. Davis and Gill (1970) considered the problem of axial conduction in the wall with reference to laminar flow between parallel plates with finite conductivity. It was found that the Peclet number, the ratio of thickness of the plates to their length are important dimensionless groups that determine the process of heat transfer. 

The bulk temperature Tb onb is close to saturation temperature Ts, when the values calculated using Eqs. (6.32), (6.33) and (6.34) do not differ significantly from unity. In Fig. 6.11 the experimental results reported by Kennedy et al. (2000) are presented as the dependence of the value /c,onb (Eq. 6.32) on the Peclet number. The data may be described by the single line of /c,onb = 0.96. In this case the bulk temperature 2b,onb, at ONE point should not differ significantly from 7s. Experimental results given in Table 6.4 support this statement. 

For gas-liquid flows in Regime I, the Lockhart and Martinelli analysis described in Section I,B can be used to calculate the pressure drop, phase holdups, hydraulic diameters, and phase Reynolds numbers. Once these quantities are known, the liquid phase may be treated as a single-phase fluid flowing in an open channel, and the liquid-phase wall heat-transfer coefficient and Peclet number may be calculated in the same manner as in Section lI,B,l,a. The gas-phase Reynolds number is always larger than the liquid-phase Reynolds number, and it is probable that the gas phase is well mixed at any axial position therefore, Pei is assumed to be infinite. The dimensionless group M is easily evaluated from the operating conditions and physical properties. 

As in Section II,B,l,b, the gas-phase Peclet number is assumed to be infinite, and the dimensionless group M is easily evaluated. The interfacial area a can be calculated with a knowledge of the holdup of the film phase... 

The interfacial heat transfer coefficient can be evaluated by using the correlations described by Sideman (S5), and then the dimensionless parameter Ni can be calculated. If the Peclet numbers are assumed to be infinite, Eqs. (30) can be applied to the design of adiabatic cocurrent systems. For nonadiabatic systems, the wall heat flux must also be evaluated. The wall heat flux is described by Eqs. (32) and the wall heat-transfer coefficient can be estimated by Eq. (33) with... 

From this discussion of parameter evaluation, it can be seen that more research must be done on the prediction of the flow patterns in liquid-liquid systems and on the development of methods for calculating the resulting holdups, pressure drop, interfacial area, and drop size. Future heat-transfer studies must be based on an understanding of the fluid mechanics so that more accurate correlations can be formulated for evaluating the interfacial and wall heat-transfer coefficients and the Peclet numbers. Equations (30) should provide a basis for analyzing the heat-transfer processes in Regime IV. 

We consider two cases, one with a higher Peclet number than the other. Disper-sivity tt[, in the first case is set to 0.03 m in the second, it is 3 m. In both cases, the diffusion coefficient D is 10-6 cm2 s-1. Since Pe L/oti., the two cases on the scale of the aquifer correspond to Peclet numbers of 33 000 and 330. We could evaluate the model numerically, but Javandel el al. (1984) provide a closed form solution to Equation 20.25 that lets us calculate the solute distribution in the aquifer... 

Fig. 20.3. Transport model of the migration of a chemical species through an aquifer, calculated for two Peclet numbers, Pe. Species is not present initially, but from t = 0 to t = 2 years recharge at the left boundary contains the species at concentration C0. After this interval, concentration in recharge returns to zero. Fine line shows result for dispersivity aL of 0.03 m, corresponding to a P6clet number on the scale of the aquifer (1000 m) of 33 000 bold line shows results for oil = 3 m, or Pe = 330.

The axial dispersion coefficient may be calculated from a knowledge of the Peclet number, where... 

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. 

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... 

Values of Dr can be calculated from the change in shape of a pulse of tracer as it passes between two locations in the bed, and a typical procedure is described by Edwards and Richardson(27). Gunn and Pryce(28), on the other hand, imparted a sinusoidal variation to the concentration of tracer in the gas introduced into the bed. The results obtained by a number of workers are shown in Figure 4.6 as a Peclet number Pe = ucd/eDL) plotted against the particle Reynolds number (Re c = ucdp//j ). 

Knowing the ascent velocity U from above, the Peclet number (Pe) is calculated,... 

Figure 5.15 shows streamlines and concentration contours calculated by Masliyah and Epstein (M6). Even in creeping flow, Fig. 5.15a, the concentration contours are not symmetrical. The concentration gradient at the surface, and thus Shjoc is largest at the front stagnation point and decreases with polar angle see also Fig. 3.11. The diffusing species is convected downstream forming a region of high concentration at the rear (often referred to as a concentration wake ) which becomes narrower at higher Peclet number.

Here, it has to be noted that for calculating the Peclet number in fixed beds, the actual velocity has to be used, i.e. the interstitial velocity, which influences the degree of mixing. In slurry bubble column reactors, the real velocity of the fluid is the bubble velocity, which is much higher than the gas superficial velocity. The mean bubble rise velocity for a batch liquid is (eq (3.201))... 

Figure 10.4 [1] shows the results for theoretical calculations [4] for the ratio n, the number of viable cells leaving the holding section of a continuous sterilizer, to Hq, the number of viable cells entering the section, as a function of the Peclet number (Pe), as defined by Equation 10.7, and the dimensionless Damkohler number (Da), as defined by Equation 10.8 ... 

Figure 22.3 One-dimensional concentration profiles at steady-state calculated from the diffusion/advec-tion/reaction equation (Eq. 22-7) for different parameter values D (diffii-sivity), x (advection velocity), and kr (first-order reaction rate constant). Boundary conditions at x = 0 and x - L are C0 and CL, respectively. Pe = 7. vx ID is the Peclet Number, Da = Dk/v] is the Damkohler Number. See text for further explanations.

The variance, calculated from the experimental data, can be used to derive the Peclet number or number of tanks in series. These numbers can then be used In their respective mathematical models. 

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