Slow dynamics of the impurity levels

in order to obtain a description of the slow dynamics, we define the slow, compressed, time scale 0 = 21, in which the model of the process becomes 

On dividing Equation (5.24) by 2, and considering the same limiting case under the constraints above, we obtain a description of the slow dynamics of the system. Note that, in this limit, the term f( , us)/e2 becomes indeterminate. On defining z = I i 1 n,.2, 0 f( , 11s)/to, z e ]RC+TO 1, the slow dynamics of process (5.10) takes the form 

In the DAE system (5.25), the variables z IRC+TO 1 are implicitly fixed by the algebraic constraints, rather than specified explicitly, and thus the index of the system is again nontrivial (i.e., higher than 1). Also, note that, as in the previous reduction step, the index of (5.25) is well-defined only if the flow rates us are specified as a function of the state variables (in this case, expressed in the new coordinates ), i.e., us = us( ). Specifying these flow rates via feedback control laws allows z to be determined through differentiation of the algebraic constraints in Equation (5.25). Differentiating these constraints once yields 

Dynamics and control of generalized integrated process systems 

This is the dynamics associated with the small amount of feed impurity removed by the small purge stream. 

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