Loops, tearing

Figure 5. Tearing loops more than once...

A loop tack (Fig. 2c) test consists of allowing a tear-shaped loop of conditioned tape to drape into contact with a test surface of specified area (usually 25.4 x 25.4 mm), with the force of contact limited to the weight of the tape itself (ASTM Ref. D-6195). The ends of the loop are held in a tensile tester. After a momentary contact time the tester is engaged and the tape is removed at a specified speed. The maximum in the removal force is ordinarily observed just at the point where the two peel fronts Join. The value is reported in a force per area of tape width, or lb in. -. While this tack test has some popularity, it is perhaps more of a very short dwell time peel test, and it has variables more associated with that test, especially backing effects, since heavier backings lead to higher tack values. 

The ocular surface and the tear-secreting glands of the eye are now known to function as an integrated unit. This unit refreshes the tear supply and clears used tears. An autonomic neural reflex loop stimulates secretion of tear fluid and proteins by the lacrimal glands. The sensitivity of the ocular surface decreases as aqueous tear production and tear clearance decreases. This results in a decrease in sensory-stimulated reflex tearing which exacerbates dry eye.29,30 Over time, wearing contact lenses also desensitizes the cornea by constant stimulation.12... 

A well-known class of techniques for reducing the number of iterates is the use of tearing (L4). We shall illustrate this procedure by way of an example taken from Carnahan and Christensen (C3). Let us consider the two-loop network shown in Fig. 5 and assume that formulation A is used. To abbreviate the notation let us denote the material balance around vertex i [Eq. (35)] by fi = 0 and the model of the element [Eq. (36)] by fu — 0. Then assuming all external flows and one vertex pressure, p, say, are specified, we have a set of 12 equations that must be solved simultaneously. But if we now assume a value for ql2, the remaining equations may be solved sequentially one at a time to yield the variables in the following... 

To execute a sequential solution for a set of modules, you have to tear certain streams. Tearing in connection with modular flowsheeting involves decoupling the interconnections between the modules so that sequential information flow can take place. Tearing is required because of the loops of information created by recycle... 

If the objective in selecting streams to tear is to minimize the number of the tear variables (Pho and Lapidus, 1973) subject to the constraint that each loop be broken at least once, this problem is an integer programming problem known as the covering set problem. Refer to Biegler et al. (1997) and Section 8.4. 

For modular-based process simulators, the determination of derivatives is not so straightforward. One way to get partial derivations of the module function(s) is by perturbation of the inputs of the modules in sequence to calculate finite-difference substitutes for derivatives for the tom variables. To calculate the Jacobian via this strategy, you have to simulate each module (C + 2) nT + nF + 1 times in sequence, where C is the number of chemical species, nT is the number of tom streams, and nF is the number of residual degrees of freedom. The procedure is as follows. Start with a tear stream. Back up along the calculation loop until an unperturbed independent variable xI t in a module is encountered. Perturb the independent variable,... 

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... 

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. 

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. 

In order to find those tears that will break more than one loop, one must find the paths of information flow that are included as part of more than one loop. This can be done by first forming a new occurrence matrix in which the rows correspond to the loops and the columns to the equations. A nonzero entry is made in a row, i, and column, j if equation j is included in loop i. Figure 16a shows the occurrence matrix of the equations involved in the loops... 

If an equation appears in more than one loop, one tear can break all of the loops in which the equation appears if in each of these loops the equation included in both loops has exactly the same equation feeding it in the loop. For example, column 2 of the occurrence matrix in Fig. 16a contains nonzero elements in rows 2 and 4. Upon inspection of the corresponding loops, B and D, in the tree of Fig. 15 it is seen that f2 is fed by f3 in both loops (/3 appears immediately after f2 in paths involved in loops B and D). Therefore, loops B and D can be broken by tearing the output variable of f3 from the equation f2. Now instead of putting l s as entries in row i and column j of the occurrence matrix to indicate that equation j appears in loop i, we insert instead the number of the equation that feeds equation j in loop i as in Fig. 16b. Then if the same nonzero number appears more than once in the same column, all of the loops represented by the rows in which that number appears... 

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. 

B was just about to tear a strip off the edge of his pad and demonstrate when A surprised him with Oh, I know what a Mobius strip is. It s made by putting half a twist in a long strip before making it into a loop. It has only one side and only one edge. I once saw that well-known line from the Pervigilium Veneris... 

One of the basic tools of analysis for tearing is the cycle matrix. This consists of a matrix of streams (the row) and the loop in which they are contained in columns. For example, consider Figure 3 (the Cavett problem). The cycle matrix is found by placing a 1 in the loop column if a stream appears in a loop or 0 if it does not. This is shown in the cycle matrix of Figure 4. The total number of loops in which a stream is included is calculated and placed in the loop total column. We shall use this number later. Note that the loop must pass through a module (node) only once. 

Criterion (4) says, if possible, tear each loop only once. This was deduced by Upadhye and Grens (69) and separately by Barehers (70). 

Figure 5 gives a system where it is not possible to tear each loop only once and Figure 6 gives its cycle matrix. 

S4, S6 tears all the loops but the sum of the loops torn is 6 rather than 5, the number of loops present. However, there is no way to tear that can reduce this number of loop tears. 

The selection of tear streams which tear each loop at least once can often be done by inspection of... 

Evidence that tearing each loop once, if possible, is the "best" tearing criterion has been presented by Gros, Kaijaluoto and Mattsson ( 6) independent of the diagonal method (see below) used for convergence. 

It may be commented that the loop identification methodology and tearing criterion does not include control loops. With control loops present, one ordering deduced from a minimum loop tear may be vastly more efficient than an equivalent solution ordering. Just how to incorporate control loops in the tearing cri-terions does not appear to be addressed in the literature. 

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