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Systems equations tearing

This section treats the partitioning of the system equations into the smallest irreducible subsystems, that is, the smallest groups of equations that must be solved simultaneously. Partitioning represents the first and easiest of the two phases of decomposition tearing (which is discussed in Section VI) is more difficult. [Pg.198]

Networks of recycle loops are commonly encountered in large processes, and a suitable choice of a tear stream may minimize the number of iterations required to solve the balance equations of such systems. For example, consider the block diagram shown below. There are two cycles in this process 52-S3-S4 and S3-S5-S7. To solve the system equations you could, for example, tear both S4 and S7, which would require the inclusion of two convergence blocks and hence the simultaneous solution of two iterative loops however, you can instead tear one stream common to both cycles (S3), probably decreasing the computation time required to achieve the solution. [Pg.520]

There are three cycles S2-S3-S4-S5, S7-S9-S11, and S3-S4-S6-S7-S8, and no single stream that if torn would permit the solution of all the system equations. For instance, if you tear S3 you could work your way around the first cycle to unit M2 but you would be stuck there for lack of knowledge of S8, and you would be similarly stuck at unit M3 in the third cycle since you would not know Sll. [Pg.520]

B. Tearing Methods for Large Scale Systems of Equations. 212... [Pg.185]

Once the complete system of equations has been partitioned into the irreducible subsystems of simultaneous equations, it is desirable to decompose further these irreducible blocks of equations so that their solution can be simplified. The decomposition of the irreducible subsystems is called tearing. In the remainder of this section the subsystems of irreducible equations found by partitioning will be referred to as blocks to distinguish them from the smaller subsystems of simultaneous equations obtained within a block after the tearing is accomplished. [Pg.211]

In tearing, the objective is to wind up with less computation time required to solve the torn system compared with the time required to solve the entire block of equations simultaneously. However, the criteria for evaluating the effectiveness of the tearing are by no means so well defined as those for partitioning, where the objective is clearly to obtain the smallest possible subsystems of irreducible equations. There is no general method for determining the time needed to effect a solution of a set of equations it is necessary to consider the particular equations involved. Any feasible method of tearing, then, must be based on criteria that are related to the solution time. Some of the more obvious criteria are ... [Pg.211]

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. [Pg.212]

In the adjacency matrix a tear is accomplished by removing the nonzero element in the column corresponding to the equation in which the variable is the output and the row corresponding to the equation from which the variable is torn. For example, in Fig. 14 removal of the nonzero element of row 1 and column 5 corresponds to tearing the output variable ofA from/,. When this tear is made there is no longer any information flow directly from fs to /j in the torn block, and loop A (/i,/5) is broken only loops B, C, and D are retained in the torn block. If the element of row 3 and column 4 is torn instead of the element of the first row and fifth column, loop D is broken, and only loops A, B, and C remain in this torn system. [Pg.214]

Steward s method for choosing the tears breaks the most loops and therefore minimizes the number of loops remaining in the torn system. It is easily seen that if two loops contain two or more equations in common and in the same sequence, the path of information flow between these equations is part of both loops, and if the path is broken by a tear, both loops will be broken, For example, the path from f5 to fx is contained both in loops A and D of Fig. 15. If the element in the fifth row and first column of the adjacency matrix is removed both loops A and D are broken. [Pg.215]

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. [Pg.225]

A number of variations are possible with such two tiered sytems. Tearing can take place in the conventional way and the torn streams can be estimated. Each module in turn can be calculated as in the sequential modular systems. A linearized model of each module can then be generated which in turn can be used in the linearized flowsheet model. From Equation (1)... [Pg.31]

Kehat and Shacham ( 6) used split fraction models to estimate the Jacobian when the Newton-Raphson method is used to solve Equation (1). The authors concluded that their method is very efficient for systems with more than one tear stream and when there is only a weak interaction between variables in the tear stream. [Pg.33]

Class I Methods. The methods of the first class are based on tearing and partitioning the system so that subsets of the primitive variables are paired with subsets of the equations through which they typically show their greatest effect. [Pg.137]

The solution of a sparse system of equations can be carried out in three stages 1. Partitioning, 2. Reordering or "tearing", and 3. Numerical solution. Stages 1 and 2 contain only logical operations and their objective is to obtain a system which can be solved faster and/or with smaller round-off error propagated. [Pg.267]

The concept of "tearing11 has been developed in connection with the iterative methods. First an output set for the system of equations is chosen. Then one or more tearing variables are selected. These variables are the iterates that need to be chosen to obtain a solution of the system. The number of tearing variables is usually much smaller than the number of the equations. An accepted criteria for selecting tearing variables is the minimum number of such variables which will make it possible to solve the whole system. The ordered set of equations that results is then solved using an iterative method. [Pg.268]

Solve the system balance equations, working around the cycle from unit to unit until the tear stream variables are recalculated. [Pg.515]

Models of the steady-state behavior of a chemical plant consist of a system of algebraic equations that can be solved by well-established algebraic numerical methods. These can be implemented on computer clusters, which can achieve supercomputing execution times. Recycle streams require the use of tear variables... [Pg.1957]

The algorithmic treatment depends on the architecture of the flowsheeting system. In Equation-Oriented mode, the approach consists of solving all the equations describing the problem simultaneously. In Sequential-Modular approach the mathematical solution must take into account the convergence of units and tear streams, as well as of all design specifications. Supplementary equations must be added, so that the general formulation of the optimisation problem (3.10) becomes ... [Pg.107]

As a result, certain numerical techniques, such as tearing and partitioning, were developed in the past to automatically rearrange the system to minimize the number of equations to be solved simultaneously. Unfortunately, the following problems should not be underestimated ... [Pg.237]

Steward, D. V., Partitioning and Tearing Systems of Equations, J. SIAM Numerical Analysis Series B 2, 345 (1965). [Pg.124]


See other pages where Systems equations tearing is mentioned: [Pg.33]    [Pg.1467]    [Pg.1341]    [Pg.1464]    [Pg.277]    [Pg.216]    [Pg.223]    [Pg.225]    [Pg.32]    [Pg.33]    [Pg.541]    [Pg.1466]    [Pg.1467]    [Pg.296]    [Pg.237]    [Pg.1463]    [Pg.1464]    [Pg.352]    [Pg.19]    [Pg.124]    [Pg.585]   
See also in sourсe #XX -- [ Pg.211 , Pg.212 , Pg.213 , Pg.214 , Pg.215 , Pg.216 , Pg.217 , Pg.218 , Pg.219 ]




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