Information flow maximal loop

Neither equation can be solved independently because both equations contain the same variables. In terms of information flow, regardless of the output set chosen, the first equation must feed information to the second equation and the second to the first, constituting a loop of information flow. For any number of equations, as long as each equation feeds information to the next equation in sequence, and the last equation feeds information to the first equation, the whole system of equations has to be solved simultaneously. If a set of equations comprises part of a larger set of equations, which themselves form a larger loop of information flow, the subset must be solved together with the bigger system. Thus, any set of equations in which each equation is included in a maximal loop of information flow must be solved simultaneously. 

Previously we have termed the largest loop of information flow a maximal loop, and indicated that it is not tied into other loops, by definition. One might wonder, because the choice of an output set is not unique, whether the maximal loops of information flow in one adjacency matrix will differ from those of another adjacency matrix, i.e., one formed from a different output set. It is shown in the following paragraphs that the maximal loops will be the same and therefore any output set will suffice for accomplishing the partitioning. 

For any maximal loop, which is by definition not connected to a larger loop, it is clear that the. information flow into and out of the set of equations... 

The column of matrix P identified in step 1 and the row corresponding to the same maximal loop are removed from P to obtain a reduced matrix that contains the information flow among the remaining maximal loops. 

From a computational viewpoint, the presence of recycle streams is one of the impediments in the sequential solution of a flowsheeting problem. Without recycle streams, the flow of information would proceed in a forward direction, and the cal-culational sequence for the modules could easily be determined from the precedence order analysis outlined earlier. With recycle streams present, large groups of modules have to be solved simultaneously, defeating the concept of a sequential solution module by module. For example, in Figure 15.8, you cannot make a material balance on the reactor without knowing the information in stream S6, but you have to carry out the computations for the cooler module first to evaluate S6, which in turn depends on the separator module, which in turn depends on the reactor module. Partitioning identifies those collections of modules that have to be solved simultaneously (termed maximal cyclical subsystems, loops, or irreducible nets). 

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