Tear streams

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,... 

When Cagliostro s black japanned coach finally rumbled into his courtyard on rue St-Claude, olf the boulevard St-Antoine, the night was dark, the quarter in which I resided but little frequented. What was my surprise, then, to hear myself acclaimed by eight or ten thousand persons. My door was forced open the courtyard, the staircase, the rooms were crowded with people. At such a moment my heart could not contain all the feelings which strove for mastery in it. My knees gave way beneath me. I fell on the floor unconscious. With a shriek my wife sank in a swoon. Our friends pressed around us, uncertain whether the most beautiful moment in our life would not be the last. I recovered. A torrent of tears streamed down from my eyes. ... 

Natasha s face fell apart and she turned and ran, tears streaming down her cheeks and a clenching ache in her chest. She fled along the landing, thinking desperately of Tushestorovo and all its other memories Lastochka, open fields, Papa s smile, his stout figure and cheerful face as he ventured out on walks with her hand in his, the scent and colour of blossoms in the orchards. And Dusha. They had been resident at Tushestorovo when Dusha came to them. 

Tears streaming as hotly as the surrounding flames, she dashed back to the carriage and screamed urgent commands at Irena, who sobbed and pleaded - but Dusha made herself deaf to all of it. 

At this point all the units in the flowsheet are installed and converged. The last issue is to converge the recycle stream. The initial guessed values are adjusted to be close to the calculated values of flow and composition leaving the split S1. When these two streams are fairly close, the source of the recycle stream is defined as the split SI and the recycle stream is defined as a Tear stream. The flowsheet did not converge when the default convergence method... 

In Fig. 2, the recycle stream has been selected as the tear stream. A guess for xn must first be made, then the equations for units 1, 2, and 3 are solved. The output of unit 3, y3i, is then compared with the original guess for xn. The problem is solved when y3i has converged to X12 within the desired tolerance. The nonlinear algebraic equations to be solved can be written as xn = g(xn) or f(.Xn) = 12 - g(xn) = 0 and solved using the techniques discussed in Section III. Notice that the streams between units 1 and 2 or 2 and 3 could also have been chosen as the tear stream in Fig. 2. 

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

Once the tear streams have been selected, the problem is reduced to solving the simultaneous nonlinear equations ... 

Here X is the initial estimate of the tear stream and (X) is calculated from the linearized model. Alternately all streams can be torn and then be reestimated from the linearized model. 

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. 

Obviously, the minimum number of tear streams is the smaller value... 

In many process simulations, the user is responsible for structuring all computations and the computational sequence directly. In ASPEN the system is capable of complete automatic determination of the computational sequence. Alternatively, the user can select certain tear streams and can, in fact, easily specify the entire sequence. 

Another potential advancement is permitted in the ASPEN system. Tear streams can be designated as desired, so that a user might define blocks or series of blocks and simulate these sets as quasi-linear blocks. The convergence method could utilize this information and solve the material (and energy) balances explicitly. In this way, a simultaneous modular architecture could be utilized. Implementation of these programs will be for later enhancements of ASPEN, not the initial version. 

Early research in flowsheeting involved discovering automatically the better streams to use as guesses (tear streams) by selecting the order in which the unit subroutines should be called. Methods were also published for... 

One stream in each recycle loop must be chosen as the tear stream for that loop. It is the one to be assumed, checked for convergence, and iterated. There is not an unambiguous guideline for this choice, nor is it usually critical which stream is chosen. If there is one for which it is easier to make a reasonable initial estimate, probably it should be selected. The actual recycle stream is often selected as the tear stream. 

A decision must be made whether the two recycle loops, units 10 and 11, are to be converged simultaneously or separately. Simultaneous convergence, as illustrated by Fig. 4-8o, is approached by checking and reestimating both tear streams once in each iteration. Separate convergence proceeds by converging the... 

Many flow-sheeting programs perform the partitioning, solution ordering, and tearing functions discussed above and present the user with one or more choices of solution sequence and tear variables. FLOWTRAN, however, does not do this. The user must identify the recycle loops, the calculation sequence, and the tear streams. The preceding example illustrated their identification and selection. 

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. 

Repeat Example 4 for the styrene synthesis process with a different selection for the tear streams to determine a new 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. 

Clearly define, in your own words, the terms design variables and state variables, sequential modular flowsheet simulation, equation-based flowsheet simulation, tear stream, convergence block, and design specification. 

Given a description of a multiple-unit process, determine the number of degrees of freedom, identify a set of feasible design variables, and if there are cycles in the flowchart, identify reasonable tear stream variables and outline the solution procedure. Draw a sequential modular block diagram for the process, inserting necessary convergence blocks. 

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

If the assumed and calculated tear stream variables agree within a specified tolerance, the solution is complete if they do not. use the new values or some combination of the new and old values to initiate another swing around the cycle. Iterate in this manner until convergence is achieved. 

Suppose, for example, that in the process described above we choose S5 as the tear stream. 

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 ... 

Choose a tear stream variable and convert the flowchart into a block diagram for a sequential modular simulation, using blocks MIX. REACT. SEP. and a convergence block CONVC. 

The solution is to tear the cycle. We can tear it in any of three places between the mixing point and the reactor, between the reactor and the separation process, or between the separation process and the mixing point. The first choice involves the trial-and-error determination of two variables, the second one involves three variables, and the third involves only one (hi). The fewer variables you have to determine by trial and error, the more likely you are to succeed. Let us therefore choose the recycle stream as the tear stream. 

The program would take as input a guessed value for the tear stream variable hi stream S4A) and might contain the following sequence of statements ... 

The first unit called is always the one following the tear stream.) The call of SEP would result in the recalculation of h from the separation process balances (stream S4R). CONVC would compare this value with the one initially assumed. If the two values agree to within a specified tolerance, the calculation would be terminated otherwise, the new value would be used to begin another journey around the cycle. The nature of the iterative procedure performed by CONVC is discussed in greater detail in Appendix A.2. 

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