Fast dynamics at the unit level

Let us define the new time variable r = 1/e i, which is of the order of magnitude of the residence time in an individual process unit. In this fast ( stretched ) time scale, the model of Equation (5.10) becomes 

Dynamics and control of generalized integrated process systems 

Following the procedure devised in Chapter 3, we consider the limit e — 0, corresponding to the recycle flow rate becoming infinite. This results in the following description of the fast dynamics of the system  

Equation (5.12) effectively corresponds to the dynamics of the individual process units that are part of the recycle loop. The description of the fast dynamics (5.12) involves only the large flow rates u1 of the recycle-loop streams, and does not involve the small feed/product flow rates us or the purge flow rate up. As shown in Chapter 3, it is easy to verify that the large flow rates u1 of the internal streams do not affect the total holdup of any of the components 1. C — 1 (which is influenced only by the small flow rates us), or the total holdup of I (which is influenced exclusively by the inflow Fjo, the transfer rate Af in the separator, and the purge stream up). By way of consequence, the differential equations in (5.12) are not independent. Equivalently, the quasi-steady-state condition 0 = G (x)u corresponding to the dynamical system (5.12) does not specify a set of isolated equilibrium points, but, rather, a low-dimensional equilibrium manifold. 

As in the previous chapters, we will assume that the linearly independent quasi-steady-state conditions in G (x)u can be isolated, i.e., the vector function G1 can be reformulated as 

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