Dispersion models, mixing Peclet number, axial

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. 

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. 

Other models to characterize residence time distributions are based on fitting the measured distribution to models for a plug flow with axial dispersion or for series of continuously ideally stirred tank reactors in series. For the first model the Peclet number is the characteristic parameter, for the second model the number of ideally stirred tank reactors needed to fit the residence time distribution typifies the distribution. However, these models should be used with care because they assume a standard distribution in residence times. Most distributions in extruders show a distinct scewness, which could lead to erroneous results at very short and very long residence times. The only exception is the co-kneader the high amount of back mixing in this type of machine leads to a nearly perfect normal distribution. 

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 main parameter in this model characterizing the quality of the flow is the axial dispersion coefficient. The term axial is used to distinguish mixing in the direction of flow from mixing in the radial direction. Then, based on this parameter, the particle Peclet number is introduced ... 

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 liquid flowing inside a MWPB can be described with a one-parameter dispersion flow model. As we show in Section 3.3, the axial mixing coefficient or, more correctly, the axial dispersion coefficient is the specific parameter for this model. Relation (3.112) contains the link between the variance of the residence time of liquid elements and the Peclet number. We can rewrite this relation so as to particularize it to the case of a MWPB. Here, we have the possibility to compute the variance of the residence time of the liquid through the stochastic model for the liquid flow developed previously in order to obtain the value of the axial dispersion coefficient ... 

Thus, the correct axial Peclet number for defining the equivalent length of a perfect mixing cell is 2. This, in turn, provides the relationship between the number of mixing-cell parameters of the mixing-cell model and the corresponding axial dispersion parameters of the dispersion model. 

The Peclet numbers also bear further scrutiny. At several points we have said, without much justification, that axial dispersion is a factor that is physically important only in shallow beds. We should be a little more quantitative about this. In Chapter 5 the relationship between axial dispersion and CSTR models was discussed. For a packed bed of length L and packing of diameter dp, the effective number of CSTR mixing stages, is... 

Unlike in MASRs, where liquid mixing is always considered complete, in this case allowance must be made for partial mixing. Thus it may often be necessary to use the dispersion model given by Equation 17.25. The liquid-phase axial diffusion coefficient for estimating the Peclet number in this equation may be calculated from the correlations of Hikita and Kikukawa (1975) or Mangartz and Pilhofer (1981). 

Equation 3.329 is the axial dispersion model equation and the Peclet number Pe is the model parameter. Pe = for an ideal PFR and Pe = 0 for an ideal CSTR. Pe is a finife value greater than 0 for any non-ideal PFR wifh axial mixing. 

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