Macromixing distributions

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 ... 

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 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 .)... 

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. 

The mean concentration vector is found by assuming that the CSTR is homogeneous on large scales (i.e., well macromixed).101 Using the fact that the age distribution in a well macromixed CSTR is exponential,... 

A further important conclusion is that for a given C-curve or residence time distribution obtained from tracer studies, a unique value of the conversion in a chemical reaction is not necessarily obtainable unless the reaction is first order. Tracer measurements can certainly tell us about departures from good macromixing. However, tracer measurements cannot give any further information about the extent of micromixing because the tracer stimulus-response is a first-order (linear) process as is a first-order reaction. 

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 ... 

The experimental SCISR is the same as that used for the measurements of macromixing and residence time distribution, as shown in Fig. 10.2, while its major dimensions are shown in Fig. 10.6 and the equipment system scheme is illustrated in Fig. 10.7. 

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. 

When the fluid elements pass through the reactor, the exchange of mass between the fluid elements occurs both on a microscale as well as on a macroscale. The mixing process on a macroscale is characterized by the residence-time distribution of the fluid elements. Usually, only the macromixing is considered to have a... 

RESIDENCE-TIME DISTRIBUTION AND MODELS FOR MACROMIXING IN THE REACTORS... 

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