Macromixing

The experimental procedure is as follows Input a certain amount of tracer, mostly KC1 solution in impulse, into a device filled with process liquid and then immediately measure and record the variations of electro-conductivity at various positions inside the device. The electro-conductivity inside the device gradually approaches uniformity 

A number of experimental methods and mathematical models for micromixing have been proposed to date, 

Bourne et al. LI61-163] presented a chemical method for micromixing measurement which has been widely accepted and applied. The core of the method is the use of the following parallel-competition reactions as the detection system  

In addition, Fournier et al [167,168] and Villermaux et al [169] proposed several new chemical systems for investigation on micromixing, including that employing the reducing reaction of iodic acid etc. 

These ideas have important consequences when a chemical reaction is now considered. In the case where the products are mixed at the molecular scale, they are in contact with each other at any point in the reactor, and the concentrations of products A and B become Cao a/( a+Fb) and CaoVb/(Va+Vb), respectively, due to dilution. In non-mixing situations, the products are only in contact at the interfaces between the two phases, and the concentrations remain equal to the initial values. 

Dealing with mixing raises two prehminary questions  

What the mechanisms producing the mixing The answer to this question needs to be provided with regard to chemical reactions. 

How can the degree of mixing of constituents in a fluid be characterized This entails introducing tools for its characterization. 

We have to answer these questions before we presume to describe the dynamics of mixing and predict, for example, the durations required to mix two products. We shall hereafter exclusively discuss mixing produced by a developed turbulence. We will not deal with mixing in a laminar flow, although the basic concepts are identical. 

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Macromixing vs Micromixing. Mixing in an agitated tank is considered to occur at two levels, macromixing and micromixing. 

Macromixing is estabflshed by the mean convective flow pattern. The flow is divided into different circulation loops or zones created by the mean flow field. The material is exchanged between zones, increasing homogeneity. Micromixing, on the other hand, occurs by turbulent diffusion. Each circulation zone is further divided into a series of back-mixed or plug flow cells between which complete intermingling of molecules takes place. 

Validation and Application. VaUdated CFD examples are emerging (30) as are examples of limitations and misappHcations (31). ReaUsm depends on the adequacy of the physical and chemical representations, the scale of resolution for the appHcation, numerical accuracy of the solution algorithms, and skills appHed in execution. Data are available on performance characteristics of industrial furnaces and gas turbines systems operating with turbulent diffusion flames have been studied for simple two-dimensional geometries and selected conditions (32). Turbulent diffusion flames are produced when fuel and air are injected separately into the reactor. Second-order and infinitely fast reactions coupled with mixing have been analyzed with the k—Z model to describe the macromixing process. 

Macromixing The phenomenon whereby residence times of clumps are distributed about a mean value. Mixing on a scale greater than the minimum eddy size or minimum striation thickness, by laminar or turbulent motion. 

In a continuous reaction process, the true residence time of the reaction partners in the reactor plays a major role. It is governed by the residence time distribution characteristic of the reactor, which gives information on backmixing (macromixing) of the throughput. The principal objectives of studies into the macrokinetics of a process are to estimate the coefficients of a mathematical model of the process and to validate the model for adequacy. For this purpose, a pilot plant should provide the following ... 

Influence of back mixing (macromixing) on the degree of conversion and in continuous chemical reaction operation. 

As the flow of a reacting fluid through a reactor is a very complex process, idealized chemical engineering models are useful in simplifying the interaction of the flow pattern with the chemical reaction. These interactions take place on different scales, ranging from the macroscopic scale (macromixing) to the microscopic scale (micromixing). 

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. 

The term macromixing refers to the overall mixing performance in a reactor. It is usually described by the residence time distribution (RTD). Originally introduced by Danckwerts (1958), this concept is based on a macroscopic lumped population balance. A fluid element is followed from the time at which it enters the reactor (Lagrangian viewpoint - observer moves with the fluid). The probability that the fluid element will leave the reactor after a residence time t is expressed as the RTD function. This function characterises the scale of mixedness in a reactor. 

Baldyga and Bourne (1999) present a comprehensive overview and comparison of macromixing models available in the literature for use in chemical reaction engineering. 

Zauner and Jones (2000b) presented a computationai metiiodoiogy for predicting precipitate particie properties and mean size based on a hybrid SFM coupied with the popuiation baiance that can simuitaneousiy account for micro-, meso- and macromixing effects, and by way of iiiustration wiii be described here in detaii. 

This scale-up criterion is based on achieving a constant pumping rate per unit volume with scale-up and therefore leads to similar macromixing on different scales, as the circulation time in the reactor remains constant. 

In order to account for both micromixing and mesomixing effects, a mixing model for precipitation based on the SFM has been developed and applied to continuous and semibatch precipitation. Establishing a network of ideally macromixed reactors if macromixing plays a dominant role can extend the model. The methodology of how to scale up a precipitation process is depicted in Figure 8.8. 

Bourne, J.R. and Dell Ava, P., 1987. Micro- and macromixing in stirred tank reactors of different sizes. Chemical Engineering Research and Design, 65, 180-186. 

FIGURE 15.14 Macromixing versus micromixing—a schematic representation of mixing space. 

Micromixing comprises the mechanisms of stretching and shrinking of slabs discussed above, accompanied by molecular diffusion, which finally lead to homogenization at the molecular level. Contrary to turbulent macromixing it depends on viscosity. This has been proven experimentally by Bourne et al. (1989). 

The reaction can be slow compared to micromixing but fast or instantaneous compared to meso-or macromixing. 

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 reactor is perfectly macromixed if the mean concentration at every point... 

The classical CRE model for a perfectly macromixed reactor is the continuous stirred tank reactor (CSTR). Thus, to fix our ideas, let us consider a stirred tank with two inlet streams and one outlet stream. The CFD model for this system would compute the flow field inside of the stirred tank given the inlet flow velocities and concentrations, the geometry of the reactor (including baffles and impellers), and the angular velocity of the stirrer. For liquid-phase flow with uniform density, the CFD model for the flow field can be developed independently from the mixing model. For simplicity, we will consider this case. Nevertheless, the SGS models are easily extendable to flows with variable density. 

Thus, the reactor will be perfectly mixed if and only if = at every spatial location in the reactor. As noted earlier, unless we conduct a DNS, we will not compute the instantaneous mixture fraction in the CFD simulation. Instead, if we use a RANS model, we will compute the ensemble- or Reynolds-average mixture fraction, denoted by ( ). Thus, the first state variable needed to describe macromixing in this system is ( ). If the system is perfectly macromixed, ( ) = < at every point in the reactor. The second state variable will be used to describe the degree of local micromixing, and is the mixture-fraction variance (maximum value of the variance at any point in the reactor is ( )(1 — ( )), and varies from zero in the feed streams to a maximum of 1/4 when ( ) = 1/2. 

In Fig. 7, the mixture fractions in each environment and l2 are shown. By definition of the inlet conditions, in the inlet tubes — 0 and 2 = 1. The variations away from the inlet values represent the effect of micromixing. For example, if we set y — 0 in Eqs. (36) and (37) to eliminate micromixing, then and 2 would remain at their inlet values at all points in the reactor. Note that the spatial distributions of 1 and %2 are antisymmetric with respect to the vertical axis (as would be expected from the initial conditions.) In the outlet tube, and 2 are very near the perfectly micromixed value of 1/2. Finally, by comparing Fig. 6 and Fig. 7, we can observe that macromixing occurs slightly faster than micromixing in this reactor (i.e.,pn are closer to their outlet values than are .)... 

Complete segregation any fluid element is isolated from all other fluid elements and retains its identity throughout the entire vessel. No micromixing occurs, but macromixing may occur. 

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. 

In addition to these two macromixing reactor models, in this chapter, we also consider two micromixing reactor models for evaluating the performance of a reactor the segregated flow model (SFM), introduced in Chapters 13 to 16, and the maximum-mixedness model (MMM). These latter two models also require knowledge of the kinetics and of the global or macromixing behavior, as reflected in the RTD. 

In the following sections, we first develop the two macromixing models, TIS and DPF, and then the two micromixing models, SFM and MMM. 

Segregated flow occurs when the fluid is macromixed and all molecules that enter together leave together. A State of segregation is associated with every RTD. Each aggregate of molecules reacts Independently of every other aggregate, thus as an individual batch reactor. 

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