Peclet

Figure 2.25 Comparison of the analytical and finite element results for a low Peclet number problem...

Figure 2.26 Comparison of the finite element solution of low and high Peclet mimber problems...

The thermal conductivity of polymeric fluids is very low and hence the main heat transport mechanism in polymer processing flows is convection (i.e. corresponds to very high Peclet numbers the Peclet number is defined as pcUUk which represents the ratio of convective to conductive energy transport). As emphasized before, numerical simulation of convection-dominated transport phenomena by the standard Galerkin method in a fixed (i.e. Eulerian) framework gives unstable and oscillatory results and cannot be used. 

PEC-1000 Pechmann reaction Peclet numbers PE coatings Pectic acid Pectic acids [9046-40-6] Pectic materials Pectic substances Pectin... 

Fig. 18. Peclet numbers in large scale gas—Hquid contactors using 2.54-cm Bed saddles (---) or 2.54 cm (---) or 5.08 cm (----) Raschig rings (51).

Equation 74 is shown graphically ia Figure 19a for a given set of conditions. Curves such as these cannot be directly used for design, however, because the Peclet number contains the tower height as a characteristic dimension. Therefore, new Peclet numbers are defined containing as the characteristic length. These relate to the conventional Pe as... 

Fig. 28. Relationship between Fqq and at different degrees of Hquid backmixing (4). Curves represent different Peclet numbers. From top to bottom...

Peclet number specifically defined for phase /, i = x or jy, see equation 77... 

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. 

FIG. 14-39 Mixing curves. Pe = Peclet niimher (Npe). (Biihhle-Tray Design Manual, Ameiican Institute of Chemical Engineers, New Yoth, 1958. )... 

From these the continuous and dispersed phase Peclet numbers can be determined from the relationships ... 

Pe) = Peclet number, continuous phase (Pe) = Peclet number, dispersed phase... 

A flow reac tor with some deviation from plug flow, a quasi-PFR, may be modeled as a CSTR battery with a characteristic number n of stages, or as a dispersion model with a characteristic value of the dispersion coefficient or Peclet number. These models are described later. 

Peclet number for dispersion Pe = uUD where t/ is a Bnear velocity, L is a hnear dimension, and is the dispersion coefficient. In packed beds, Pe = udp/De, where u is the interstitial velocity and dp is the pellet diameter. 

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. 

FIG. 23-10 Residence time distributions of pilot and commercial reactors. <3 = variance of the residence time distribution, n = number of stirred tanks with the same variance, Pe = Peclet number. 

Dispersion Model An impulse input to a stream flowing through a vessel may spread axially because of a combination of molecular diffusion and eddy currents that together are called dispersion. Mathematically, the process can be represented by Fick s equation with a dispersion coefficient replacing the diffusion coefficient. The dispersion coefficient is associated with a linear dimension L and a linear velocity in the Peclet number, Pe = uL/D. In plug flow, = 0 and Pe oq and in a CSTR, oa and Pe = 0. 

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 ... 

The solution of this partial differential equation is recorded in the literature (Otake and Kunigata, Kngaku Kogaku, 22, 144 [1958]). The plots of E(t ) against t are bell-shaped, resembling the corresponding Erlang plots. A relation is cited later between the Peclet number,... 

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. 

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