Mixing macromixing

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

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. 

Notice the sensitivity to the heat transfer cooling time tjj. Solid lines perfect mixing. Dashed lines imperfect mixing (macromixing time tj4 = 0.1, micromixing time t = 0.01). J = 200 K ... 

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 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. 

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. 

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

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 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. 

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 the statistical theory of fluid mixing presented in Chapter 3, well macromixed corresponds to the condition that the scalar means () are independent of position, and well micromixed corresponds to the condition that the scalar variances are null. An equivalent definition can be developed from the residence time distribution discussed below. 

An alternative method to RTD theory for treating non-ideal reactors is the use of zone models. In this approach, the reactor volume is broken down into well mixed zones (see the example in Fig. 1.5). Unlike RTD theory, zone models employ an Eulerian framework that ignores the age distribution of fluid elements inside each zone. Thus, zone models ignore micromixing, but provide a model for macromixing or large-scale inhomogeneity inside the reactor. 

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 composition PDF thus evolves by convective transport in real space due to the mean velocity (macromixing), by convective transport in real space due to the scalar-conditioned velocity fluctuations (mesomixing), and by transport in composition space due to molecular mixing (micromixing) and chemical reactions. Note that any of the molecular mixing models to be discussed in Section 6.6 can be used to close the micromixing term. The chemical source term is closed thus, only the mesomixing term requires a new model. 

Meticulous observation of this mixing process (the slow disappearance of the Schlieren patterns as result of the disappearance of density differences), reveals that macromixing is quickly accomplished compared to the micromixing. This time-consuming process already takes place in... 

The fluidised bed will be considered as a continuous stirred tank reactor in which ideal macromixing of the particles occurs. As shown in the section on mixing (Chapter 2, Section 2.1.3), in the steady state the required exit age distribution is the same as the C-curve obtained using a single shot of tracer. In fact the desired C-curve is identical with that derived in Chapter 2, Fig. 2.3, for a tank containing a liquid with ideal micromixing, but now the argument is applied to particles as follows ... 

---