Flows deterministic, mixing

Khakhar, D. V., Franjione, J. G. and Ottino, J. M., A case study of chaotic mixing in deterministic flows the partitioned pipe mixer. Chem. Eng. ScL 42,2909-2926 (1987). 

TURBULENCE is chaotic fluid flow characterized by the appearance of three-dimensional, irregular swirls. These swirls are called eddies, and usually turbulence consists of many different sizes of eddies superimposed on each other. In the presence of turbulence, fluid masses with different properties are mixed rapidly. Atmospheric turbulence usually refers to the small-scale chaotic flow of air in the Earth s atmosphere. This type of turbulence results from vertical wind shear and convection and usually occurs in the atmospheric boundary layer and in clouds. On a horizontal scale of order 1000 km, the disturbances by synoptic weather systems are sometimes referred to as two-dimensional turbulence. Deterministic description of turbulence is difficult because of the chaotic character of turbulence and the large range of scales involved. Consequently, turbulence is treated in terms of statistical quantities. Insight in the physics of atmospheric turbulence is important, for instance, for the construction of buildings and structures, the mixing of air properties, and the dispersion of air pollution. Turbulence also plays an... 

Methods accounting for mixing are most easily illustrated for steady state or stationary reactor operation, as in Fig. 12.3-1. Because of its stochastic nature, turbulent flow is in fact only statistically iiormy. The random behavior of the variables results in rapid fluctuations of their values around mean or so-called Reynolds-averaged, "steady state" values. Nevertheless, turbulent flow is governed by deterministic equations, the Navier-Stokes equations, whose terms have been explained in Chapter 7 and in which a transient term is included to account for the fluctuations around the statistically steady state values. 

However, at that time there still existed considerable healthy skepticism regarding the existence of nonperiodic behavior in well-controlled nonequilibrium chemical reactions. After all, nonperiodic behavior can arise from fluctuations in stirring rate or flow rate, evolution of gas bubbles from the reaction, spatial inhomogeneities due to incomplete mixing, vibrations in the stirring motor, fluctuations in the amount of bromide and dissolved oxygen in the feed, and so on. Any experimental data, no matter how well a system is controlled, will contain some noise arising from fluctuations in the control parameters therefore, it is reasonable to ask "Will noise, always present in experiments, inevitably mask deterministic nonperiodic behavior (chaos) "... 

Besides these stochastic interpretations, deterministic interpretations are presently developed GRAY [12], KUMPINSKY and EPSTEIN [13], propose systemic approaches, commonly used in chemical engineering several ideal reactors are coupled by conservative flows with expandable coefficients, so that by-passes or dead zones may be taken into account. NICOLIS and FRISCH 14] use a quasi-Semenov equation in the limit of large diffusion coefficients and obtain a renormalization of k , DEWEL et al. [15] use a phenomenological theory of turbulent mixing to study surface effects produced by the feed of the reactor. 

In a continuous-flow chemical reactor, the concern is not only with probabilistic transitions among chemical species but also with probabilistic liansitions of each chemical species between the interior and exterior of the reactor. Pippel and Philipp [8] used Markov chains for simulating the dynamics of a chemical system. In their approach, the kinetics of a chemical reaction are treated deterministically and the flow through the system are treated stochastically by means of a Markov chain. Shinnar et al. [9] superimposed the kinetics of the first order chemical reactions on a stochastically modeled mixing process to characterize the performance of a continuous-flow reactor and compared it with that of the corresponding batch reactor. Most stochastic approaches to analysis and modeling of chemical reactions in a flow system have combined deterministic chemical kinetics and stochastic flows. 

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