Precedence ordering

Sequential modular. Refers to the process simulator being based on modules, and the modules solved in a sequential precedence order imposed by the flowsheet information flow. 

When the modules are connected to one another as represented in a flowsheet, a long train of units may become coupled together for calculations. Thus, a set of modules may require a fixed precedence order of solution so that convergence of the calculations may be slower than in equation-based codes. 

To complete the search and add more modules to the precedence order, start on path II ... 

All of the modules from the block diagram have been included in the tracing, and no more paths have to be searched. The procedure identifies all the nested and outer loops. The overall precedence order is (10) —> (3) — (45621) —> (78) —> (9). 

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). 

When Timing of specific operations, run lengths Operational steps Precedence order Resource utilization... 

It must indicate the precedence order in which the equations or subsystems are to be solved. 

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... 

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... 

C REAOING. THE MATRIX OUTPUT SET IS ON THE MAIN DIAGONAL. AND THE ROWS C ARE IN THE FINAL PRECEDENCE ORDER. THE ELEMENTS IN A MATRIX ROW ARE CM THOSF VARIABLES ASSOCIATED WITH THE EQUATION GIVEN UNDER THE C COLUMN HEADED EONS. THE COLUMN HEADED VARS. IDENTIFIES THE OUTPUT c VARIABLE FOR THAT EQUATION. BY TRANSPOSING THE VECTOR OF NUMBERS C UNDER THE COLUMN VARS. AND PLACING IT ACROSS THE TQP OF THE MATRIX(X.X) C M YOU CAN IDENTIFY EACH OF THE VARIABLES BY NUMBER. EACH COLUMN WITH C M NONZERO ELEMENTS ABOVE THE MAIN DIAGONAL REPRESENTS AN ITERATED C m VARIABLE. 

The induced motions in the surroundings are estimated by examining the effects of a systematic velocity v imposed on one of the molecules, called here the central molecule. The couples of the central molecule lead to JV, other molecules which move in response to the motion of the central molecule. Likewise, the Ni first order partners are coupled to N2 second order partners, and so on through a hierarchy of orders order / contains Nj molecules, coupled for the first time to the central molecule through a sequence of / couples. The induced velocity of each order is assumed to be a constant fraction s of the velocity of the preceding order. The formal expression for the force resisting the motion of the central molecule becomes therefore ... 

Partitioning and the Cycle Matrix. Sequential modular systems require an order of calculation (precedence order) be given to the modules. There are generally four steps taken to determine this ordering. 

Precedence ordering of the modules (nodes) in the recycle loops. 

Venkatesh, C. K., "Computational Precedence Ordering in Modular Cascade Systems", M. S. Thesis in Chemical Engineering, University of Houston,... 

The next step in the problem solving procedure is to outline a solution procedure for the Equations listed in Table 3.2.2. There are algorithms available for determining in what order to solve a set of algebraic equations, which is called the precedence order. See, for example, Rudd and Watson [17] and Myers and Seider [18] for a discussion of some of these algorithms. Sometimes, we can develop a procedure by inspection of an equation set, as in the procedure given in Table 3.2.3. 

Post-mortem cost accounting, 145 Power factors for plant capacity, 185-188 Power requirements for gas compressors, 523-525 for pumping, 479492 Practical considerations in design, 9-11 Precedence ordering in design, 383n. 

The modules may require a fixed precedence order of solution (i.e., the output of one module must become the input of another) hence, convergence may be slower than in an equation-solving code, and the computational costs may be high. 

A method for precedence ordering so as to partition a model into a sequence of smaller models containing sets of irreducible equations (equations that have to be solved simultaneously as illustrated in Fig. 5.6). 

Figure E5.6 An example system containing recycle and bypass streams, precedence order ...

Steps 7 and 8 Next we determine the sequence of solution of the system equations. If you used a computer code, you might be asked to provide such a sequence, or possibly the code would select a sequence for you if you wished. As indicated by Fig. 5.6, we first prepare an occurrence matrix and then rearrange the matrix (here by inspection since it is a small matrix) to give the precedence order for solution. Figure E5.7b is the occurrence matrix, and Fig. E5.7c is the precedence matrix with the encircled numbers and associated... 

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