The degrees of freedom approach

The degrees of freedom of any process constitute the number of independent variables that must be specified in order to define the process completely. Thus it is only possible to achieve the control of a process when all the degrees of freedom have been fixed. By considering the degrees of freedom involved in a process, it is possible to obtain a fuller understanding of the controllability of that process and 

If the process illustrated in Fig. 7.9 is a binary distillation with N stages, then 4N+ 5 relationships can be established as in Table 7.1. The corresponding number of independent variables are listed in Table 7.2. A fuller description of the relationships involved can be found in Stephanopoulos( ), Example 4.13. 

We have not included every possible independent variable or relationship (e.g. it is assumed that the vapour holdup on each tray is negligible, that the molar latent heats of vaporisation of both components are approximately equal, that there are no heat losses to the surroundings, etc.). Generally, such relationships are omitted in an attempt to simplify the process model. 

Liquid flow relationships Mass balances around the 1th tray (where 1 = 2. N - 1, i f) N 

This is equivalent to an assumption that any variations in these quantities have a negligible effect upon the output variables of interest. Care must be taken when 

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It can be seen that the heuristic and the degrees-of-freedom approaches lead to similar conclusions. Both methods may be usefully employed—the former to give a common-sense preliminary view of a suitable control strategy, and the second to check that the strategy is neither over-controlling nor under-controlling the plant. 

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