Fluid micromixers

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). 

Each stage of particle formation is controlled variously by the type of reactor, i.e. gas-liquid contacting apparatus. Gas-liquid mass transfer phenomena determine the level of solute supersaturation and its spatial distribution in the liquid phase the counterpart role in liquid-liquid reaction systems may be played by micromixing phenomena. The agglomeration and subsequent ageing processes are likely to be affected by the flow dynamics such as motion of the suspension of solids and the fluid shear stress distribution. Thus, the choice of reactor is of substantial importance for the tailoring of product quality as well as for production efficiency. 

Baldyga, J. and Bourne, J.R., 1984b. A fluid mechanical approach to turbulent mixing and chemical reaction. Part II Micromixing in the light of turbulence theory. Chemical Engineering Communications, 28, 243-258. 

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. 

Note that we have used the fluid velocity U to describe convection of particles, which is valid for small Stokes number. In most practical applications, / is a highly nonlinear function of c. Thus, in a turbulent flow the average nucleation rate will depend strongly on the local micromixing conditions. In contrast, the growth rate G is often weakly nonlinear and therefore less influenced by turbulent mixing. 

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. 

As a preliminary consideration for these two micromixing models, we may associate three time quantities with each element of fluid at any point in the reactor (Zwietering, 1959) its residence time, t, its age, ta, and its life expectancy in the reactor (i.e., time to reach the exit), te ... 

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. 

The maximum-mixedness model (MMM) for a reactor represents the micromixing condition of complete dispersion, where fluid elements mix completely at the molecular level. The model is represented as a PFR with fluid (feed) entering continuously incrementally along the length of the reactor, as illustrated in Figure 20.1 (after Zwieter-ing, 1959). The introduction of feed incrementally in a PFR implies complete mixing... 

For non-linear chemical reactions that are fast compared with the local micromixing time, the species concentrations in fluid elements located in the same zone cannot be assumed to be identical (Toor 1962 Toor 1969 Toor and Singh 1973 Amerja etal. 1976). The canonical example is a non-premixed acid-base reaction for which the reaction rate constant is essentially infinite. As a result of the infinitely fast reaction, a fluid element can contain either acid or base, but not both. Due to the chemical reaction, the local fluid-element concentrations will therefore be different depending on their stoichiometric excess of acid or base. Micromixing will then determine the rate at which acid and base are transferred between fluid elements, and thus will determine the mean rate of the chemical reaction. 

In addition, the PFR model assumes that mixing between fluid elements at the same axial location is infinitely fast. In CRE parlance, all fluid elements are said to be well micromixed. In a tubular reactor, this assumption implies that the inlet concentrations are uniform over the cross-section of the reactor. However, in real reactors, the inlet streams are often segregated (non-premixed) at the inlet, and a finite time is required as they move down the reactor before they become well micromixed. The PFR model can be easily... 

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. 

For higher-order reactions, the fluid-element concentrations no longer obey (1.9). Additional terms must be added to (1.9) in order to account for micromixing (i.e., local fluid-element interactions due to molecular diffusion). For the poorly micromixed PFR and the poorly micromixed CSTR, extensions of (1.9) can be employed with (1.14) to predict the outlet concentrations in the framework of RTD theory. For non-ideal reactors, extensions of RTD theory to model micromixing have been proposed in the CRE literature. (We will review some of these micromixing models below.) However, due to the non-uniqueness between a fluid element s concentrations and its age, micromixing models based on RTD theory are generally ad hoc and difficult to validate experimentally. 

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. 

For non-interacting fluid elements, the RTD function is thus equivalent to the joint PDF of the concentrations. In composition space, the joint PDF would he on a one-dimensional sub-manifold (i.e., have a one-dimensional support) parameterized by the age a. The addition of micromixing (i.e., interactions between fluid elements) will cause the joint PDF to spread in composition space, thereby losing its one-dimensional support. 

Another class of micromixing models is based on fluid environments (Nishimura and Matsubara 1970 Ritchie and Tobgy 1979 Mehta and Tarbell 1983a Mehta and Tarbell 1983b). The basic idea behind these models is to divide composition space into a small number of environments that interact due to micromixing. Thus, unlike zone models, which divide up physical space, each environment can be thought of as existing at a particular... 

In Chapter 6, this is shown to be a general physical requirement for all micromixing models, resulting from the fact that molecular diffusion in a closed system conserves mass. ( a)) is the mean concentration with respect to all fluid elements with age a. Thus, it is a conditional expected value. 

Figure 5.12. The chemical source term for Y2x, will be non-zero in the triangular region bordered by the line Y2co = 0 and the two lines found from setting h = 0 and h2 = 0. The mixing line corresponds to the upper limit for y2co in the range 0 < < and results from micromixing between fluid elements at (0, 0) and ( max> Y2lmK).

Perhaps the simplest Lagrangian micromixing model is the interaction by exchange with the mean (IEM) model for a CSTR. In addition to the residence time r, the IEM model introduces a second parameter tm to describe the micromixing time. Mathematically, the IEM model can be written in Lagrangian form by introducing the age a of a fluid particle, i.e., the amount of time the fluid particle has spent in the CSTR since it entered through a feed stream. For a non-premixed CSTR with two feed streams,100 the species concentrations in a fluid particle can be written as a function of its age as... 

Figure 5.15. Two examples of age-based micromixing models. In the top example, it is assumed that fluid particles remain segregated until the latest possible age. In the bottom example, the fluid particles mix at the earliest possible age. Numerous intermediate mixing schemes are possible, which would result in different predictions for micromixing-sensitive reactions. 

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. 

As shown in Fig. 5.20, such regions normally occur only near the inlet zones where micromixing is poor. Further downstream, interaction between flamelets will become significant, and the assumptions on which the flamelet model is based will no longer apply.117 Reactors with recirculation zones are also problematic for flamelet models. For these reactors, partially reacted fluid is brought back to mix with the feed streams so that the simple non-premixed flow model no longer applies. 

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