Dispersion Peclet number

When a number of competing reactions are involved in a process, and/or when the desired product is obtained at an intermediate stage of a reaction, it is important to keep the residence-time distribution in a reactor as narrow as possible. Usually, a broadening of the residence-time distribution results in a decrease in selectivity for the desired product. Hence, in addition to the pressure drop, the width of the residence-time distribution is an important figure characterizing the performance of a reactor. In order to estimate the axial dispersion in the fixed-bed reactor, the model of Doraiswamy and Sharma was used [117]. This model proposes a relationship between the dispersive Peclet number ... 

Stanton Number = a /puCp) mass Peclet number = dpu/D) axial heat dispersion Peclet number = (dpu/Xa) radial heat dispersion Peclet number = dpu/A,.)... 

The relative contribution of mechanical dispersion and diffusion to solute transport is evaluated using Peclet numbers. A Peclet number is a dimensionless number that relates the effectiveness of mass transport by advection to the effectiveness of mass transport by diffusion or dispersion. Peclet numbers have the general form... 

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. 

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 ... 

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... 

Figure 3.2.1 illustrates the mixing in packed beds (Wilhelm 1962). As Reynolds number approaches the industrial range Rep > 100, the Peclet numbers approach a constant value. This means that dispersion is influenced by turbulence and the effect of molecular diffusion is negligible. 

Peclet number independent of Reynolds number also means that turbulent diffusion or dispersion is directly proportional to the fluid velocity. In general, reactors that are simple in construction, (tubular reactors and adiabatic reactors) approach their ideal condition much better in commercial size then on laboratory scale. On small scale and corresponding low flows, they are handicapped by significant temperature and concentration gradients that are not even well defined. In contrast, recycle reactors and CSTRs come much closer to their ideal state in laboratory sizes than in large equipment. The energy requirement for recycle reaci ors grows with the square of the volume. This limits increases in size or applicable recycle ratios. 

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 ... 

Estimate die dispersion coefficient Dg[ from the definition of the Peclet number. 

Equations 8-148 and 8-149 give the fraction unreacted C /C o for a first order reaction in a closed axial dispersion system. The solution contains the two dimensionless parameters, Np and kf. The Peclet number controls the level of mixing in the system. If Np —> 0 (either small u or large [), diffusion becomes so important that the system acts as a perfect mixer. Therefore,... 

The dispersed plug flow model has been successfully applied to describe the flow characteristics in the Kenics mixer. The complex flow behavior in the mixer is characterized by the one-parameter. The Peclet number, Np, is defined by ... 

A breakthrough curve with the nonretained compound was carried out to estimate the axial dispersion in the SMB column. A Peclet number of Pe = 000 was found by comparing experimental and simulated results from a model which includes axial dispersion in the interparticle fluid phase, accumulation in both interparticle and intraparticle fluid phases, and assuming that the average pore concentration is equal to the bulk fluid concentration this assumption is justified by the fact that the ratio of time constant for pore diffusion and space time in the column is of the order of 10. ... 

The Peclet numbers decrease when the dispersion coefficients increase. In the Reynolds number range of 10-200, in a packed bed of pellets, Peii = 2 and PeH = 0.5 (119, 120). The dispersions in the transverse... 

Naturally, there are two more Peclet numbers defined for the transverse direction dispersions. In these ranges of Reynolds number, the Peclet number for transverse mass transfer is 11, but the Peclet number for transverse heat transfer is not well agreed upon (121, 122). None of these dispersions numbers is known in the metal screen bed. A special problem is created in the monolith where transverse dispersion of mass must be zero, and the parallel dispersion of mass can be estimated by the Taylor axial dispersion theory (123). The dispersion of heat would depend principally on the properties of the monolith substrate. Often, these Peclet numbers for individual pellets are replaced by the Bodenstein numbers for the entire bed... 

N = L/d, or that each row of pellets amounts to a mixing cell. This is intuitively reasonable. However, the parallel dispersion for heat has a Peclet number equal to 0.5, which would argue that four rows of pellets should be considered to be a mixing cell. The heat balance equation for cell i in a cascade of N cells is... 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

The form of the solution of the dispersion equation (11.61) depends on the sign of the determinant D = q + Pl, i.e., on the values of the characteristic parameters g and P. The latter are determined by the physical properties of the liquid and its vapor, as well as the values of the Peclet number. This allows us to use g and P as some general characteristics of the problem considered here. 

---