Nonstandard singularly perturbed systems with two time scales

3 Nonstandard singularly perturbed systems with two time scales 

Clearly, the system in Equation (2.36) is not in a standard singularly perturbed form and therefore the results derived so far are not directly applicable. 

As will be seen throughout the rest of the text, typical examples include fast reactions and large heat and mass transfer rates. 

Kumar et al. (1998) analyzed the two-time-scale property of the system (2.36) and addressed the construction of nonlinear coordinate changes that would yield a standard singularly perturbed representation. 

Example 2.3. Depending on the mechanism, reacting systems with vastly different reaction rates can be modeled by either standard or nonstandard singularly perturbed systems of equations. Systems in which a reactant is involved in both slow and fast reactions belong to the latter category. Consider the reaction system in Example 2.2, with the difference that the reactant Ri also participates in the second reaction  

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The nonstandard singularly perturbed form of the model of this system potentially indicates a dynamic behavior with two time scales. This is, in effect, quite intuitive, in view of the presence of different rates of heat transfer induced by the different heat-transfer coefficients U and Ue. 

Such nested applications of single-parameter singular perturbation theory (i.e., the extension of the analysis of two-time-scale systems presented in Chapter 2 to multiple-time-scale systems) have been used for stability analysis of linear (Ladde and Siljak 1983) and nonlinear (Desoer and Shahruz 1986) systems in the standard form. However, as emphasized above (Section 2.3), the ODE models of chemical processes are most often in the nonstandard singularly perturbed form, with the general multiple-perturbation representation... 

According to the developments in Section 2.3, the model of Equation (3.10) is in a nonstandard singularly perturbed form. We thus expect its dynamics (and, consequently, the dynamics of integrated process systems with large material recycle) to feature two distinct time scales. However, the analysis of the system dynamics is complicated by the presence of the term u1, which, as we will see below, precludes the direct application of the methods presented in Chapter 2 for deriving representations of the slow and fast components of the system dynamics. 

The process response is presented in Figure 4.6. Observe that all the state variables exhibit a fast transient, followed by a slow approach to steady state, which is indicative of the two-time-scale behavior of the system, and is consistent with our observation that processes with impurities and purge are modeled by systems of ODEs that are in a nonstandard singularly perturbed form. 

From a mathematical point of view, we can see that Equation (5.10) is in a (nonstandard) singularly perturbed form. This suggests that the integrated processes under consideration will feature a dynamic behavior with at least two distinct time scales. Drawing on the developments in Chapters 2, 3, and 4, the following section demonstrates that these systems evolve in effect over three distinct time scales and proposes a method for deriving reduced-order, non-stiff models for the dynamics in each time scale. 

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