Number Peclet

The Peclet number is a ratio of the time for diffusion, xj D, to the time for convection, Xg/v. Typically, Pe is very large, and this presents numerical problems. 

The second form of Eq. (22) shows that the coefficient of the term in the differential equation with highest derivative goes to zero this makes the problem singular. For large but hnite Pe the problem is still difficult to solve. For a simple onedimensional problem it can be shown [9] that the solution will oscillate from node to node unrealistically unless  

This means that as Pe increases, the mesh size must decrease. Since the mesh size decreases, it takes more elements or grid points to solve the problem, and the problem may become too big. One way to avoid this is to introduce some numerical diffusion, which essentially lowers the Peclet number. If this extra diffusion is introduced in the flow direction only, the solution may still be acceptable. Various techniques include upstream weighting (finite difference [10]) and Petrov-Galerkin (finite element [11]). Basically, if a numerical solution shows imphysical oscillations, either the mesh must be refined, or some extra diffusion must be added. Since it is the relative convection and diffusion that matter, the Peclet number should always be calculated even if the problem is solved in dimensional units. The value of Pe will alert the chemist, chemical engineer, or bioengineer whether this difficulty would arise or not. Typically, is an average velocity, x is a diameter or height, and the exact choice must be identified for each case. 

Sometimes an approximation is used to neglect axial diffusion since it is so small compared with axial convection. If the flow is fully developed in a channel then one solves (for steady problems)  

This is a much simpler problem since the D can be absorbed into the length, z. 

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

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. 

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. 

Both phases are siibstantiaUy in plug flow. Dispersion measurements of the hquid phase usuaUy report Peclet numbers, Uid /D, less than 0.2. With the usual smaU particles, the waU effect is negligible in commercial vessels of a meter or so in diameter, but may be appreciable in lab units of 50 mm (1.97 in) diameter. Laboratory and commercial units usuaUy are operated at the same space velocity, LHSy but for practical reasons the lengths of lab units may be only 0.1 those of commercial units. 

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... 

Plug flow is approached at low values of the dispersion coefficient or hi values or Peclet number. A criterion developed by Mears (Chem. Eng. Sci., 26, 1361 [1971]) is that conversion will be within 5 percent of that predicted by phig flow when... 

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