Dispersion models, mixing residence-time distribution

A model of a reaction process is a set of data and equations that is believed to represent the performance of a specific vessel configuration (mixed, plug flow, laminar, dispersed, and so on). The equations include the stoichiometric relations, rate equations, heat and material balances, and auxihaiy relations such as those of mass transfer, pressure variation, contac ting efficiency, residence time distribution, and so on. The data describe physical and thermodynamic properties and, in the ultimate analysis, economic factors. 

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. 

Another problem arising from the results of the investigation on residence time distribution is the strong mixing of the materials in dispersed phase in the impingement zone. The fact that the model of RTD derived above fits well the concentration of the tracer in the out stream of the device indicates that the assumption of perfect mixing of... 

FIGURE 7.13 Residence time distribution for various extents of back mixing as predicted by the dispersion model. From Levenspiel [9]. Copyri t 1972 by John Wiley Sons, Inc. Reprinted by permission of John Wiley Sons, Inc. 

The determination of volumetric mass transfer coefficients, kLa, usually requires additional knowledge on the residence time distribution of the phases. Only in large diameter columns the assumption is justified that both phases are completely mixed. In tall and smaller diameter bubble columns the determination of kLa should be based on concentration profiles measured along the length of the column and evaluated with the axial dispersed plug flow model ( 5,. ... 

These expressions demonstrate that the normalized mean residence time and variance of the normalized residence time distribution increase with increased values of the axial dispersion number Dj. In the limit of = 0, the signal is convected and behavior corresponding to the parallel tube model is approximated the normalized residence time fi= 1 and Act = 0. For very large values of D, the behavior corresponds to a single well-mixed compartment. 

The coefficient representing axial dispersion, E, is measured using a tracer by pulse, sinusoidal, or step-change residence time distribution tests, or by measuring backflow. Sometimes the phenomenon is represented by a cefl model, in which the number of well-mixed cells fits dispersion. The coefficient representing radial dispersion, is determined by measuring the radial spread of a tracer from the centerline toward the wall. 

The dispersion model assumes that the residence-time distribution of a real tubular reactor can be regarded as the superimposition of the plug flow that is characteristic of the ideal tubular reactor and diffusionlike axial mixing, characterized by an axial dispersion coefficient which has the same dimensions as, but can be much larger than, the molecular diffusion coefficient. The following effects can contribute to the axial mixing ... 

Roes and van Swaaij [29] used tracer technique to determine the residence time distribution and the extent of axial mixing in both flowing phases. The axially dispersed model was used to describe the degree of axial mixing. 

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. 

In turbulent flow regime (J Cp>I00), the fluid entering each void is considered to be fully mixed and overall dispersion phenomenon can be well described by a tanks-in-series model, where the residence time in a tank is equated with the residence time of flowing fluid in a void of the length Pidf, Pidfl u. Then the tanks-in-series model gives the residence time distribution of the Poisson type (Aris and Amundson, 1957), which can be approximated by the impulse response of the dispersion model by equating... 

Shallow bed model. Fan and coworkers (87,88) have treated the case of shallow reactors (bed height low relative to diameter or length for rectangular vessels). While mixing is rapid in the vertical direction, it tends to be limited in the lateral direction so that the residence time distribution for solids can differ considerably from the perfect mixing assumption adopted in most of the models covered in Sections 7.1 and 7.2. Lateral mixing of both solids and gas is described in terms of lateral dispersion coefficients. Unsteady state terms are retained in order to provide dynamic solutions. In other respects the models are "conventional in their treatment of two phases (bubbles and dense phase). It is shown that lateral... 

Models of BCR can be developed on the basis of various view points. The mathematical structure of the model equations is mainly determined by the residence time distribution of the phases, the reaction kinetics, the number of reactive species involved in the process, and the absorption-reaction regime (slow or fast reaction in comparison to mass transfer rate). One can anticipate that the gas phase as well as the liquid phase can be either completely backmixed (CSTR), partially mixed, as described by the axial dispersion model (ADM), or unmixed (PFR). Thus, it is possible to construct a model matrix as shown in Fig. 3. This matrix refers only to the gaseous key reactant (A) which is subjected to interphase mass transfer and undergoes chemical reaction in the liquid phase. The mass balances of the gaseous reactant A are the starting point of the model development. By solving the mass balances for A alone, it is often possible to calculate conversions and space-time-yields of the other reactive species which are only present in the liquid phase. Heat effects can be estimated, as well. It is, however, assumed that the temperature is constant throughout the reactor volume. Hence, isothermal models can be applied. 

The dispersion model is another way of characterizing the residence time distribution of a real reactor. In this model it is assumed that axial mixing is super-... 

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