Nonlinear wave propagation

Another nonlinear type of dynamic behavior found in many chemical processes is spatiotemporal pattern formation of the concentration and temperature profiles [66], This phenomenon is also termed as nonlinear wave propagation. Nonlinear waves are particularly useful to understand and predict the qualitative dynamic behavior without any tedious calculations. 

In this section some basic features of nonlinear wave propagation in non-reactive and RD processes will be illustrated and compared with each other. The simulation results presented are based on simple equilibrium or non-equilibrium models [51, 65] for non-reactive separations. In the reactive case, similar models are used, assuming either kinetically controlled chemical reactions or chemical equilibrium. We focus on concentration (and temperature) dynamics and neglect fluid dynamics. Consequently, for equimolar reactions constant flows along the column height are assumed. However, qualitatively similar patterns of behavior are also displayed by more complex models [28, 57, 65] and have been confirmed in experiments [41, 59, 89, 107] for non-reactive multi-component separations. First experimental results on nonlinear wave propagation in reactive columns are presented subsequently. 

The analysis can be extended to the multi-component case [43, 44, 51, 65, 66]. The number of the fronts is directly related to the number of components in the mixture. For ideal and moderately non-ideal mixtures the concentration and temperature profiles consist of n,. - 1 fronts connecting two pinch points. Again, constant pattern waves occur for ideal and moderately non-ideal mixtures. Addi- 

Under steady state conditions, at most one of these wave fronts can be located in the middle of a column section, whereas the others are located at either one of the boundaries, where they can overlap and interact. A multicomponent example [51] is shown in Fig. 10.19. Initially, two distinct constant pattern waves can be identified in the concentration profiles of both components traveling at the same velocity to the top of the rectifying section. Close to the upper boundary, the waves start to interact and form some combined steady state pattern. 

Similar behavior can also be observed for RD processes. First investigations in this direction have been made by Marquardt and coworkers [25]. They considered the reversible reaction A + B o C for different Damkdhler numbers Da. Fig. 10.20 shows the variation of the wave front pattern. A single front is obtained in the limit for large Da. A similar trend can be observed for the propagation velocity, which 

---

A. Kienle, Low-order dynamic models for ideal multicomponent distillation processes using nonlinear wave propagation theory. Chem. 

Extensions When more than two conservation equations are to be solved simultaneously, matrix methods for eigenvalues and left eigenvectors are efficient [Jeffrey and Taniuti, Nonlinear Wave Propagation, Academic Press, New York, 1964 Jacob and Tondeur, Chem. Eng. /., 22,187 (1981), 26,41 (1983) Davis and LeVan, AlChE33, 470 (1987) Rhee et al., gen. refs.]. 

In this chapter, an overview on the present knowledge of nonlinear dynamics and control of RD columns was given. First, focus was on open-loop dynamics. It was shown that these processes can show a distinct nonlinear behavior including multiple steady states, selfsustained oscillations, and nonlinear wave propagation. Different patterns of behavior were identified depending on the properties of the reaction system and the operating conditions. 

MODELING NONLINEAR WAVE PROPAGATION AND PATTERN FORMATION AT GEOCHEMICAL FIRST ORDER PHASE TRANSITIONS... 

The last step is to analyse how the induced nonlinear polarization creates a new wave or interferes with the existing ones to generate the nonlinear Raman signal. The nonlinear wave propagation equation (taking = 0) is... 

The theory may now be applied to specific flow problems. We have used the model for studies of weakly nonlinear waves propagating through the mixture. Some interesting problems concerning wave modulation must be treated to complete these results. 

Once we know threshold values, dispersion relations and necessary conditions to be fulfilled in order to excite and eventually sustain oscillations we must show that the nonlinear terms saturate the exponential growth thus stabilizing the interfacial waves. This is a formidable task and,for simplicity,we now restrict consideration to capillary-gravity (Kelvin-Laplace) transverse waves at an a/r-liquid interface.In this case we know that in order to have sustained oscillations the elasticity Maiangoni number must be negative When due consideration is taken of the nonlinear terms -that we have omitted in Section 2-we have shown that, indeed, past the instability threshold the evolution corresponds to a nonlinear wave propagation. 

Nonlinear Wave Propagation. It is assumed that most acoustic measurements are made in the linear wave propagation region, ie, where Hooke s law applies. Polymers as a class, however, are more nonlinear than other solids. Nonlinear wave propagation is therefore significant in some cases. 

---