Time-scale decomposition and nonlinear model reduction

Equation (5.9) can be written in a more compact and mathematically convenient form as 

Clearly, the general model (5.10) contains terms of very different magnitudes, corresponding, respectively, to the process input and output flows and to the chemical reactions, to the presence of the large material recycle stream, and to the presence of the impurity and the purge stream used for its removal. While (as we have argued in the previous chapters of the book) the presence of these terms is purely a steady-state, design feature of the process, it is intuitive that their impact on the process dynamics will also be very different. 

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. 

3 Time-scale decomposition and nonlinear model reduction 

---