Maximal loop

Correspond to the Maximal Loops in the Adjacency Matrix and are Invariant of the Output Set. 200... 

One method of locating these maximal loops is to compute the reachability matrix, R (H1), which is the element by element Boolean union of all of the powers of the adjacency matrix up to the nth, where n is the number of rows of R. An element of the reachability matrix is defined as... 

The maximal loops in the graph can be found from the reachability matrix by finding those sets of vertices that satisfy the following conditions (1) r j = rfi = 1, where i and j take on all possible combinations of the vertex numbers in the set (2) no other vertices, not included in the set, satisfy condition (1). The first condition requires that each vertex in the set is reachable by some path from every other vertex in the set. The second condition requires that there is no path from a vertex in the set to a vertex outside the set... 

We have shown thus far how the system of equations representing a process can be related to a linear diagraph and its associated Boolean adjacency matrix. In the Section IV, we show how the location of the maximal loops in this adjacency matrix leads to identification of the subsystems of equations that must be solved simultaneously. 

We now demonstrate how the subsystems of equations that must be solved simultaneoulsy correspond to the maximal loops found in the related adjacency matrix. Consider the following set of two equations each containing common variables ... 

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... 

One method of partitioning the system equations is to compute the maximal loops using powers of the adjacency matrix as discussed in Section II. Certain modifications to the methods of Section II are needed in order to reduce the computation time. The first modification is to obtain the product of the matrices using Boolean unions of rows instead of the multiplication technique previously demonstrated to obtain a power of an adjacency matrix. To show how the Boolean union of rows can replace the standard matrix multiplication, consider the definition of Boolean matrix multiplication, Eq. (2), which can be expanded to... 

Actually, only one matrix need be stored if the adjacency matrix is stored initially and thereafter multiplied by itself. Matrix elements are replaced by the resulting product elements as they are computed. The product matrix obtained in this manner for the fcth power may contain some nonzero elements which correspond to paths longer than k steps instead of strictly k step paths, but this will not affect the final matrix obtained corresponding to the nth power, since these paths would eventually be identified in any case. All of the modifications to the methods of Section II mentioned above simplify the calculations needed to obtain the reachability matrix. The procedure for identifying the maximal loops given in Section II remains the same. 

In addition to identifying the maximal loops, it is necessary to order them into a sequence such that the equations of each maximal loop feed information only to the equations of maximal loops appearing thereafter in sequence. One scheme of precedence ordering of the maximal loops is accomplished by first forming an adjacency matrix of loops, P, in which the rows and columns correspond to the maximal loops (S3). The elements of the matrix P are either 1 or 0 according to the rule... 

The nonzero elements of a column, j, of the matrix P indicate the loops that are fed information by the equations of loop j, and the nonzero elements of any row, i, indicate the loops that feed information to loop i. The ordering procedure for the irreducible sets of equatins (maximal loops) is as follows ... 

Choose the first column of the matrix P that contains all zeros. This column corresponds to a maximal loop, which is not fed information by any of the other maximal loops hence the set of equations that make up the maximal loop can be solved independently of the remaining equations in the system. A maximal loop so identified is placed first in the ordering sequence for calculation. 

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. 

Steps 1 and 2 are repeated on the reduced matrix and each maximal loop that is isolated from the others as it is detected is placed sequentially in the ordered sequence for solving the subsystems of simultaneous equations. The procedure is continued until all of the columns of the matrix P are removed. 

When a loop is found by the above procedure, it may or may not be a maximal loop and it may or may not contain the set of equations that should be solved next in sequence. If the loop is maximal and contains the set of equations to be solved next in the sequence, it will not be fed information by any of the other equations of the reduced system, and, in the reduced matrix, the row corresponding to the loop will contain all zero elements. Therefore, when a loop is found and the reduced matrix is formed, the procedure returns to the first phase and removes the rows without nonzero elements. When a row corresponding to a loop is removed from the matrix, the set of equations that correspond to all of the rows that were combined to form the composite row of the loop are placed next in the precedence order. If no rows have all zero elements, the procedure continues with the second phase by tracing a new path starting with any row of the reduced matrix and considering only... 

Following Le Chatelier s principle (or Carnot s principle), the global heat demand of the section decreases when the maximal loop temperature increases. However, no sensible decrease is pointed out above 840°C. Thus, increasing the maximal temperature in the cycle is not useful beyond ca. 840°C. Two antagonist phenomena can explain the levelling off ... 

The only interest in increasing the maximal loop temperature is increasing the reaction kinetics. Effect of HTR inlet temperature... 

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