Convergence of recycles

Figure 13.14 Convergence of recycle loops using the sequential modular approach.

The convergence of recycle calculations is almost always better if a good initial estimate of the tear stream is provided. 

The liquid separation section may be simulated initially as a black-box. Rigorous units can be considered later after convergence of recycles. 

Most flow sheets have one or mote recycles, and trial-and-ettot becomes necessary for the calculation of material and energy balances. The calculations in a block sequential simulator ate repeated in this trial-and-ettot process. In the language of numerical analysis, this is known as convergence of the calculations. There ate mathematical techniques for speeding up this trial-and-ettot process, and special hypothetical calculation units called convergence, or recycle, units ate used in calculation flow diagrams that invoke special calculation routines. 

The result, in either case, is a large number of equations, usually nonlinear, that must be solved simultaneously. The next step is to devise a method of solving these equations that converges rapidly to the answer. Usually some optimization techniques must be employed. An improper choice of procedures can result in a program that takes so long to obtain an answer that it is too expensive to run. This is especially probable when there are a number of recycle streams that interact. 

Park, J. T. (1996). The convergence of murein recycling research with beta-lactamase research, Microb. Drug Resist. Springer, 2, 105-112. 

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... 

Consider for example the problem of converging a recycle loop within a control loop as shown in Figure 8. If units 1, 2 and 3 were, say, flash units and it was desired to adjust the temperature in unit 2 to achieve a specified component flow in stream S5, then the recycle loop (S4) could be solved within the control loops... 

Barchers, D., "Optimal Convergence of Complex Recycle Process Systems", Ph.D. Thesis in Chemical Engineering, Oregon State University, 1975. 

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. 

Simultaneous, dynamic simulators require appreciably more computing power than steady-state simulators to solve the thousands of differential equations needed to describe a process, or even a single item of equipment. With the development of fast, powerful computers, this is no longer a restriction. By their nature, simultaneous programs do not experience the problems of recycle convergence inherent in sequential simulators however, as temperature, pressure, and flow rate are not fixed and the input of one unit is not determined by the calculated output from the... 

Alternatively, closed-form model representations require a modular simulation approach, where each closed-form model is computed using the internal solver of the software tool the model is implemented in. The algorithm sets the model inputs, performs control over the simulation, and retrieves the outputs of each model through the commonly defined interface of the closed-form model representation, independently of the specific implementation. These outputs are propagated to the inputs of downstream units, and the simulation continues until all the units are computed. If the flowsheet contains recycles, an iterative strategy is performed until convergence of the flowsheet variables in tear streams is achieved. 

The countercurrent and cocurrent flow models may also be approximated by a series of perfect mixing blocks as described for the cross-flow model. The counter-current flow model would require an additional iterative loop to converge the recycle created by the counterflowing permeate stream. 

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