Tearing algorithms

Pho, T. K. Lapidus, L., "Topics in Computer Aided Design. Part I - An Optimum Tearing Algorithm for Recycle Streams", AIChE J(1973) 19, No. 6 ... 

EQUATION-TEARING PROCEDURES USING THE TRIDIAGONAL-MATRIX ALGORITHM... 

Tridiagonal-Matrix Algorithm Both the BP and the SR equation-tearing methods compute hqnid-phase mole fractions in the same way by first developing linear matrix equations in a manner shown by Amundson andPontinen [Jnd. ng. Ch m., 50, 730 (1958)]. Equations (13-69) and (13-68) are combinedto eliminate yjj and yij + i (however, the vector yj stiU remains imphcitly in K j) ... 

Steward (S3) proposed an algorithm based on tearing a variable from only one equation at a time and evaluating each tear on the basis of the size of the resulting subsystems of simultaneous equations in the torn system and numerical considerations of the particular equations. Each variable is torn successively from each equation in which it appears and the effectiveness of the tear evaluated. 

The algorithm is executed on the adjacency matrix of a block. In order to determine how many subsystems of simultaneous equations will remain after a tear, one must first enumerate all of the loops of information flow in the block and record which equations are included in each loop. The loops are found... 

Steward s algorithm is simple and very well suited to hand calculations for smaller systems. However, the computer storage and time requirements needed to tear large systems are prohibitive. 

Ledet (LI) proposed a somewhat different algorithm for tearing based on tearing a variable from all the equations except one in a block. He evaluated the effectiveness of the tears in terms of finding the minimum number of tears to reduce the torn block to subsystems of single equations. In this method no initial output set in a block is chosen, and the tearing and assignment of an output set are carried out simultaneously. For example, consider the block of Eqs. (14), which must be solved simultaneously. We could tear x3 from equations /j and... 

Ledet s algorithm involves two phases. In the first phase an ordering of the rows and columns of the occurrence matrix takes place according to certain optimality criteria. The second phase involves reordering the occurrence matrix to reduce the number of torn variables. The first phase carries out an initial tearing, and the second phase improves the results of the first phase. However, instead of tearing individual variables as in Steward s algorithm, Ledet s method systematically reorders the occurrence matrix as described below. 

The second phase of Ledet s algorithm is carried out on the reordered occurrence matrix resulting from the first phase of the algorithm described immediately above. Because the second phase is quite complex, it will not be discussed here, but the details can be found in the original reference. Ledet shows that in most cases the first phase of the algorithm will obtain the minimum number of tears, or very nearly the minimum number of tears, and that the extra effort of executing the second phase is seldom justified. 

Lee (L2) proposed a different algorithm for tearing, which also obtains the tears for the minimum number of variables associated with the torn streams. The procedure is completely different from that of Sargent s in that the loops of material flow are determined first and tears made so that all the loops are broken. The algorithm requires a previous knowledge of the loops of material flow and which units and streams are included in each loop. The loops can be found by forming the adjacency matrix for the process, and determining the loops in the matrix by Steward s method as described in Section IV. 

In the residual occurrence matrix comprised solely of independent columns, if a row contains only one nonzero element, the stream corresponding to the column in which the nonzero element appears is the only stream that can be torn to break the loop corresponding to that row. The next step of the algorithm is (1) to remove the columns in which a lone nonzero element in a row appears and (2) to remove all of the rows that have nonzero elements in that column. Each column removed in this step corresponds to tearing the stream represented by the column, and all of the rows removed represent loops that are broken by the tear. 

Second, the adjacency matrix would require n2 words of storage where n is the number of equations in the system. In finding the loops by either the method described here or by the other method proposed by Steward (S3), all of the loops are found more than once, which tends to reduce the efficiency of the procedure. Finally, since no exact criteria was given by Steward for evaluating the effectiveness of each tear, all possible tears must be performed and the best tear chosen by inspection of all the tears. Steward s algorithm, however, is simpler than Ledet s algorithm, and therefore better suited for decomposition of small systems by hand. 

Ledet s algorithm has the advantage that a realistic tearing can be accomplished because the output variables chosen by the algorithm can be controlled to correspond with the accustomed information flow for the solution of each... 

Execution Times for Tearing by the First Phase of Ledet s Algorithm... 

C SUBROUTINE TEAR EXECUTES THE INITIAL PHASE OF LEDET S ALGORITHM. CALL TFAR mIN,MAX.L)... 

C A THIS SUBROUTINE PERFORMS THE TEARING ON A PARTITION BY IEDET S c algorithm. 

The m variables of x- can be regarded as tearing variables and clearly the smaller m is, the more effective the algorithm. Since computer programs for selection of minimal sets are readily available (Ledet and Himnelblau (5)), this problem will not be discussed. 

For the methanol synthesis process illustrated in Fig. 4-1, Example 1, assume that there are algorithms for calculating the outputs of each process unit from the inputs. Determine how many stream variables must be specified and decide what these should be so that a unique solution exists for the mass and energy balances. Identity all recycle loops, tear streams for these loops, and a calculation sequence. 

For a process flow sheet obtained in Problem 3, assume that algorithms are available to calculate the outputs from each process unit from known inputs. Determine the number of stream variables that must be specified, decide what they should be, identify all recycle loops, select tear streams for these loops, and establish a calculation sequence. 

Murphree efficiencies are easily incorporated within simultaneous convergence algorithms (something that is not always easy, or even possible, with some tearing methods). (As an aside, note that vaporization... 

Fig. 4. Rubbery tear energy master curves for Epon 878/diaminodiphenylmethane networks of different reactant ratios O A/E = 0.65 A/E = 1.00 A A/E - 1.60. Reference temperature is Tg. Curves constructed using a best fit algorithm. (After Swetlin...

Compute a new set of values of the TJ tear variables by solving simultaneously the set of N energy-balance equations (13-72), which are nonlinear in the temperatures that determine the enthalpy values. When linearized by a Newton iterative procedure, a tridiagonal-matrix equation that is solved by the Thomas algorithm is obtained. If we set gj equal to Eq. (13-72), i.e., its residual, the linearized equations to be solved simultaneously are... 

The application of this procedure to a single tear stream variable is tantamount to solving an equation of the form x = f(x), where f(x) is the function that generates a new value of the tear stream variable x by working around the cycle. Techniques described in Appendix A.2—successive substitution and Wegsiein s algorithm—can be used to perform this calculation. 

Most simulation programs have a block, called a convergence block, that performs such calculations using the Wegstein algorithm. The output stream from this block contains the assumed set of tear stream variables, and the input stream contains the values calculated by working around the cycle. The block diagram for the simulated process would appear as follows ... 

You are to write the code for a convergence module that can deal with one to three tear stream variables using the Wegstein algorithm, as outlined in Appendix A.2. The object is to determine the values of one, two, or three of the variables x, xi, and X2 that satisfy the relations... 

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