Recycle Convergence Methods

In the previous subsection, the successive substitution and Wegstein methods were introduced as the two methods most commonly implemented in recycle convergence units. Other methods, such as the Newton-Raphson method, Broyden s quasi-Newton method, and the dominant-eigenvalue method, are candidates as well, especially when the equations being solved are highly nonlinear and interdependent. In this subsection, the principal features of all five methods are compared. 

The Newton-Raphson second-order method can be written as 

In each iteration of the Newton-Raphson method, when the guesses are close to the true values, the length of the error vector, y, is the square of its length after the previous iteration that is, when the length of the initial error vector is 0.1, the subsequent error vectors are reduced to 0.01, 10 , 10 , . However, this rapid rate of convergence requires that n partial derivatives be evaluated at x. Since most recycle loops involve many process units, each involving many equations, the chain rule for partial differentiation cannot be implemented easily. Consequently, the partial derivatives are evaluated by numerical perturbation that is, each guess, Xj, i = 1. , is perturbed, one at a time. For each 

This requires n + 1 passes through the recycle loop to complete the Jacobian matrix for just one iteration of the Newton-Raphson method that is, for n = 10, eleven passes are necessary, usually involving far too many computations to be competitive. 

Alternatively, so-called secant methods can be used to approximate the Jacobian matrix with far less effort (Westerberg et al., 1979). These provide a superlinear rate of convergence that is, they reduce the errors less rapidly than the Newton-Raphson method, but more rapidly than the method of successive substitutions, which has a linear rate of convergence (i.e., the length of the error vector is reduced from 0.1, 0.01, 10 , 10 , 10 , ...). These methods are also referred to as quasi-Newton methods, with Broyden s method being the most popular. 

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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... 

A direct recycle configuration was then added (Figure 2.4). Depending on the type of pretreatment, the recycle stream vapor can contain water or a mixture of water and chemicals. From the convergence methods available in Aspen, the Broyden method was selected to solve the direct recycle flowsheet. Several simulations were run in order to analyze the operation cost (energy cost + chemical cost) as a function of the vapor fraction sent to the recycle the vapor fraction with the minimum cost was then identified. 

But, computational difficulties can arise due to the iterative methods used to solve recycle problems and obtain convergence. A major limitation of modular-sequential simulators is the inability to simulate the dynamic, time dependent, behaviour of a process. 

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. 

The older modular simulation mode, on the other hand, is more common in commerical applications. Here process equations are organized within their particular unit operation. Solution methods that apply to a particular unit operation solve the unit model and pass the resulting stream information to the next unit. Thus, the unit operation represents a procedure or module in the overall flowsheet calculation. These calculations continue from unit to unit, with recycle streams in the process updated and converged with new unit information. Consequently, the flow of information in the simulation systems is often analogous to the flow of material in the actual process. Unlike equation-oriented simulators, modular simulators solve smaller sets of equations, and the solution procedure can be tailored for the particular unit operation. However, because the equations are embedded within procedures, it becomes difficult to provide problem specifications where the information flow does not parallel that of the flowsheet. The earliest modular simulators (the sequential modular type) accommodated these specifications, as well as complex recycle loops, through inefficient iterative procedures. The more recent simultaneous modular simulators now have efficient convergence capabilities for handling multiple recycles and nonconventional problem specifications in a coordinated manner. 

The one level optimal control formulation proposed by Mujtaba (1989) is found to be much faster than the classical two-level formulation to obtain optimal recycle policies in binary batch distillation. In addition, the one level formulation is also much more robust. The reason for the robustness is that for every function evaluation of the outer loop problem, the two-level method requires to reinitialise the reflux ratio profile for each new value of (Rl, xRI). This was done automatically in Mujtaba (1989) using the reflux ratio profile calculated at the previous function evaluation in the outer loop so that the inner loop problems (specially problem P2) could be solved in a small number of iterations. However, experience has shown that even after this re-initialisation of the reflux profile sometimes no solutions (even sub-optimal) were obtained. This is due to failure to converge within a maximum limit of function evaluations for the inner loop problems. On the other hand the one level formulation does not require such re-initialisation. The reflux profile was set only at the beginning and a solution was always found within the prescribed number of function evaluations. 

Orbach, O. Crowe, C. M., "Convergence Promotion in the Simulation of Chemical Processes with Recycle - The Dominant Eigenvalue Method", Can. J. of Chem. Eng. (1971) 49 503-513. 

In addition, convergence calculations may be combined simultaneously with design specifications. The usual methods would be to embed the design in a convergence loop and meet the design specification in each recycle calculation. A quasi-Newton method convergence calculation in ASPEN will allow a simultaneous, more efficient solution for the more difficult problems. 

In the past, most simulation programs available to designers were of the sequential-modular type. They were simpler to develop than the equation-oriented programs and required only moderate computing power. The modules are processed sequentially, so essentially only the equations for a particular unit are in the computer memory at one time. Also, the process conditions, temperature, pressure, flow rate, etc., are fixed in time. With the sequential modular approach, computational difficulties can arise due to the iterative methods used to solve recycle problems and obtain convergence. A major limitation of sequential modular simulators is the inability to simulate the dynamic, time-dependent behavior of a process. 

The methods used to converge recycle loops in the commercial process simulation programs are similar to the methods described in Section 1.9. Most of the commercial simulation programs include the methods described below. 

Newton and quasi-Newton methods are used for more difficult convergence problems, for example, when there are many recycle streams, or many recycles that include operations that must be converged at each iteration, such as distillation columns. The Newton and quasi-Newton methods are also often used when there are many recycles and control blocks (see Section 4.8.1). The Newton method should not normally be used unless the other methods have failed, as it is more computationally intensive and can be slower to converge for simple problems. 

Frequently process plants contain recycle streams and control loops, and the solution for the stream properties requires iterative calculations. Thus efficient numerical methods for convergence must be used. In addition, appropriate physical properties and thermodynamic data have to be retrieved fi om a data base. Finally, a master program must exist that links all the building blocks, physical property data, thermodynamic calculations, subroutines, and numerical subroutines, and that also supervises the information flow. You will find that optimization and economic anafy-sis are really the ultimate goal in the use of flowsheet codes. 

Derivative-based optimization methods, where the optimization algorithm is used to simultaneously converge the recycles and determine the decision variables that improve the objective function (Biegler et al., 1999 Edgar et al., 2001). 

The calculation procedures [the 0 method, Kb method, and constant composition method] developed in Chap. 2 for conventional distillation columns are applied to complex distillation columns in Sec. 3-1. For solving problems involving systems of columns interconnected by recycle streams, a variation of the theta method, called the capital 0 method of convergence is presented in Secs. 3-2 and 3-3. For the case where the terminal flow rates are specified, the capital 0 method is used to pick a set of corrected component-flow rates which satisfy the component-material balances enclosing each column and the specified values of the terminal rates simultaneously. For the case where other specifications are made in lieu of the terminal rates, sets of corrected terminal rates which satisfy the material and energy balances enclosing each column as well as the equilibrium relationships of the terminal streams are found by use of the capital 0 method of convergence as described in Chap. 7. 

A column modular method which makes use of the capital 0 method of convergence is presented for solving problems involving systems containing both mass and energy recycle streams. On the basis of an assumed set of values of the variables for the recycle streams, one complete trial is made on each column or unit of the system by use of the most appropriate calculational procedure for the column or unit. Then the capital 0 method is used to place all terminal streams in material and energy balance. The values of the variables so obtained for the recycle streams are used to make the next trial on each column or unit of the system.6, 7... 

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