Segregated flow model, mixing

In the segregated flow model the contents of the volume elements of the fluid do not mix with one another as they move through the reactor. Each element may be considered as a small closed system that moves through the reactor. The different systems spend varying amounts of time in the reactor, giving rise to the measured residence time distribution func-... 

To illustrate the nature of the limits that the segregated flow and mixing models place on the expected conversion level, it is useful to examine what happens to two elements of fluid that have the same volume V, but that contain different reactant concentrations C1 and C2. We may imagine two extreme limits on the amount of mixing that may occur. 

The basic premise of the segregated flow model is that the various fluid elements move through the reactor at different speeds without mixing with one another. Consequently, each little fluid element will behave as if it were a batch reactor operating at constant pressure. The conversions attained within the various fluid elements will be equal to those in batch reactors with holding times equal to the residence times of the different fluid elements. The average conversion level in the effluent is then given by... 

Given the reaction stoichiometry and rate laws for an isothermal system, a simple representation for targeting of reactor networks is the segregated-flow model (see, e.g., Zwietering, 1959). A schematic of this model is shown in Fig. 2. Here, we assume that only molecules of the same age, t, are perfectly mixed and that molecules of different ages mix only at the reactor exit. The performance of such a model is completely determined by the residence time distribution function,/(f). By finding the optimal/(f) for a specified reactor network objective, one can solve the synthesis problem in the absence of mixing. 

The segregated-flow model described by (P2) forms a basis to generate an AR. We now develop conditions for the closure of this space with respect to the operations of mixing and reaction by means of a PFR, a CSTR, or a recycle PFR (RR). Consider the region depicted by the constraints of (P2). Our aim is to develop conditions that can be checked easily for the reaction system in question so that, if these conditions are satisfied, we need to solve only (P2) for the reactor targeting problem. We will analyze these conditions based on PFR trajectories projected into two dimensions. Here, a PFR, which is an n-dimen-sional trajectory in concentration space and parametric in time, is generated by the solution of the initial value differential equation system in (PI). Figure 3 illustrates a PFR trajectory and its projections in three-dimensional space, where the solid line represents the actual PFR trajectory and the dotted lines represent the projected trajectories. 

In general, the larger the breadth of the distribution of residence times, the greater the discrepancy between the conversion levels predicted on the basis of the segregated flow model and those predicted by the various mixing models. For narrow distribution functions, the conversions predicted by both models will be in good agreement with one another. 

Thus, the segregated flow model is based on the fundamental assumption that the fluid elements are independent or do not mix (macromixing model). Until now, we considered a perfect mixture in the mass balance and concentration uniform, no interaction between the fluid elements (micromixing), called unsegregated model. 

The segregated-flow reactor model (SFM) represents the micromixing condition of complete segregation (no mixing) of fluid elements. As noted in Section 19.2, this is one extreme model of micromixing, the maximum-mixedness model being the other. 

Novosad and Thyn [Coll. Czech. Chem. Comm. 31 3,710-3,720 (1966)] solved the maximum mixedness and segregated flow equations (fit with the Erlang model) numerically. There are few experimental confirmations of these mixing extremes. One study with a... 

Many important pharmaceutical unit operations handle and process powders in various ways, and these unit operations may be analyzed and modeled by using computational mechanics. Examples include powder compression and compaction, powder flow, fluidization, mixing and segregation, packing, and milling. 

The statistical description of multiphase flow is developed based on the Boltzmann theory of gases [37, 121, 93, 11, 94, 58, 61]. The fundamental variable is the particle distribution function with an appropriate choice of internal coordinates relevant for the particular problem in question. Most of the multiphase flow modeling work performed so far has focused on isothermal, non-reactive mono-disperse mixtures. However, in chemical reactor engineering the industrial interest lies in multiphase systems that include multiple particle t3q)es and reactive flow mixtures, with their associated effects of mixing, segregation and heat transfer. 

If the deviations are small then they can be described by the dispersion model (additional dispersive flow is is superimposed on the plug flow) or cell model (cascade of ideal stirred tanks). For larger deviations the calculation of nonideal reactors is generally difficult. A more simply treated special case occurs when the volume elements flowing through the reactor are macroscopically but not microscopically mixed (segregated flow). This case can be solved by the Hofmann-Schoenemann method (see below). 

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