Macromixing parameters

Determination of macromixing parameters from tracer experiments... 

In what follows, both macromixing and micromixing models will be introduced and a compartmental mixing model, the segregated feed model (SFM), will be discussed in detail. It will be used in Chapter 8 to model the influence of the hydrodynamics on a meso- and microscale on continuous and semibatch precipitation where using CFD, diffusive and convective mixing parameters in the reactor are determined. 

If the feed time of a concentrated fluid is short the reaction will often be completed within the circulation zone, outside the impeller zone. Macromixing can then be important and the blend time will be an important scale-up parameter. For long feeding times and low concentrations in the feed all the important mixing processes could be completed almost immediately in the vicinity of the outlet of the feed pipe. 

The TIS and DPF models, introduced in Chapter 19 to describe the residence time distribution (RTD) for nonideal flow, can be adapted as reactor models, once the single parameters of the models, N and Pe, (or DL), respectively, are known. As such, these are macromixing models and are unable to account for nonideal mixing behavior at the microscopic level. For example, the TIS model is based on the assumption that complete backmixing occurs within each tank. If this is not the case, as, perhaps, in a polymerization reaction that produces a viscous product, the model is incomplete. 

Another Lagrangian-based description of micromixing is provided by multienvironment models. In these models, the well macromixed reactor is broken up into sub-grid-scale environments with uniform concentrations. A four-environment model is shown in Fig. 5.16. In this model, environment 1 contains unmixed fluid from feed stream 1 environments 2 and 3 contain partially mixed fluid and environment 4 contains unmixed fluid from feed stream 2. The user must specify the relative volume of each environment (possibly as a function of age), and the exchange rates between environments. While some qualitative arguments have been put forward to fit these parameters based on fluid dynamics and/or flow visualization, one has little confidence in the general applicability of these rules when applied to scale up or scale down, or to complex reactor geometries. 

The tank-in-series (TIS) and the dispersion plug flow (DPF) models can be adopted as reactor models once their parameters (e.g., N, Del and NPe) are known. However, these are macromixing models, which are unable to account for non-ideal mixing behavior at the microscopic level. This chapter reviews two micromixing models for evaluating the performance of a reactor— the segregrated flow model and the maximum mixedness model—and considers the effect of micromixing on conversion. 

In addition to these macromixing characteristics, many authors have determined turbulence parameters and their spatial distribution within the tank volume by measuring velocity and concentration fluctuations(144-147, 19, 158). In a typical investigation (19) concerning a semi-industrial tank (0.15 m2) and aqueous medium, the following spatial variations were found uf =5 to 30 % of jTNd, Lf = 4 to 150 mm, Af = 1 to 5 mm, e/e 0.2 to 2.5, c /C = 2 to 10 x 10 4 (for eddies > 100 pm). This shows that a stirred tank is far from being the homogeneous and uniform system assumed in many academic papers. 

So far, only the axial dispersion model has been used for scaleup purposes. Very little knowledge on the effects of reactor configuration and flow conditions on the parameters of more complex macromixing models (e.g., the two-parameters model, etc.) is available. Since these complex models are more realistic, more information on the relation between their parameters and the system conditions, such as packing size, fluid properties, and flow rates, needs to be obtained. At present, complex models are not very useful for scaleup purposes. 

As discussed in Chapter 3, with LES, the smallest scale to be resolved is chosen to lie in the inertial sub-range of the energy spectrum, which means the so-called sub-grid scale (SGS) wave numbers are not resolved. As LES can capture transient large-scale flow structures, it has the potential to accurately predict time-dependent macromixing phenomena in the reactors. However, unlike DNS, a SGS model representing interaction of turbulence and chemical reactions will be required in order to predict the effect of operating parameters on say product yields in chemical reactor simulations. These SGS models attempt to represent an inherent loss of SGS information, such as the rate of molecular diffusion, in an LES framework. Use of such SGS models makes the LES approach much less computationally intensive than the DNS approach. DNS... 

Macromixing time is a parameter characterizing the time required for the distribution of a compound (added instantly) throughout the entire volume of the tank. It is calculated as the time required to reduce the maximum difference of local concentrations of the added substance to approximately 1% of its final average value (under batch mixing conditions). 

Single-component reactions can occur throughout the bulk of the material. At the start of the reaction only one eomponent, monomer or prepolymer, is present, or the components used are well miseible and premixed. For this group of reactions the temperature of the mixture plays an important role, as well as macromixing over the length of the extruder, which is related to the residence time distribution. Both parameters determine the progress of the reaction. 

Now that we know how to estimate the size fx of the smallest scales of turbulence, it is simple to conclude by summing up the principles that govern mixing and chemical reactions in a flow with homogenous turbirlence. Table 11.1 shows, for a chemical reaction, for micromixing (molecular diffusion), and for macromixing (dispersion at different scales), the characteristic times and the scales with which these various processes are associated. For a stirred reactor, the phenomenon involves six parameters of different nature ... 

These parameters influence five processes (chemical reaction, micromixing, and the three scales of macromixing), which also bring forth five characteristic times. It can be seen that the four dimensionless numbers can be expressed as a ratio of the time scales associated with the processes of micromixing, macromixing, and chemical reaction. 

This poses the problem of prediction and scale-up of mixing parameters (especially micromixing times). Macromixing is closely related to circulation times, which vary as N (5-1). 

The inherent problem of all compartment models is the poor spatial resolution. By significantly increasing the number of compartments (up to 40000 zones) the network-of-zone (NOZ) models could overcome this problem [23-25]. However, the problem of identification and physical interpretation of the model parameters remains unchanged. The macromixing effects are reflected in the choice... 

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