Termination Convergence rate

On the subject of comparing iterative methods a word of caution is in order. Clearly in any quantitative comparison, the termination criteria should be comparable and the benchmark problems should be run on the same computer. Yet even for simple problems and methods, these two requirements prove to be difficult to enforce and insufficient to ensure meaningful comparisons. To allow for the fact that different methods do not terminate at exactly the same point even when the same termination criterion is used, Broyden (B13) introduced a mean convergence rate, R, which is... 

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. 

In the column modular approach, the equations for each column are solved by use of the most efficient procedure for each column. After one trial has been made on each column, the terminal flow rates are placed in component-material balance and in agreement with the specified values of the terminal flow rates by use of the capital 0 method for systems. The entire calculational process is repeated until convergence has been achieved. 

As the feed composition approaches a plait point, the rate of convergence of the calculation procedure is markedly reduced. Typically, 10 to 20 iterations are required, as shown in Cases 2 and 6 for ternary type-I systems. Very near a plait point, convergence can be extremely slow, requiring 50 iterations or more. ELIPS checks for these situations, terminates without a solution, and returns an error flag (ERR=7) to avoid unwarranted computational effort. This is not a significant disadvantage since liquid-liquid separations are not intentionally conducted near plait points. 

The Taylor series expansion is an approximation, and therefore the shift vector 5k is an approximation as well. However, the new parameter vector k + 5k will generally be better than the preceding k. Thus, an iterative process should always move toward the optimal rate constants. As the iterative fitting procedure progresses, the shifts, 5 k, and the residual sum of squares, ssq, usually decrease continuously. The relative change in ssq is often used as a convergence criterion. For example, the iterative procedure can be terminated when the relative change in ssq falls below a preset value ji, typically ji = 10 4. 

Thus, before the rate of coke formation can build up to a steady level, the coke precursors must reach some suitable concentration. At this point, the most logical candidates for coke precursors are the / -resins. Plotting the amount of coke formed as a simple function of the quantity of / -resins produces the curve shown in Figure 5. The / -resins/coke data obtained at 800°, 825°, and 850°F (430°, 440°, and 450°C, respectively) lie approximately on the same curve, while the data obtained at 980°F (530°C) follow a separate curve. At relatively low levels of / -resin formation the coke concentration increases only slowly, but as the /3-resin concentration approaches 14-16%, the amount of coke formed rises rapidly and the two curves converge. In the pyrolysis runs where the reactions were terminated before the / -resin concentration had risen much above 10-12%, a substantial portion of the / -resin producing reaction follows first-order kinetics. However, if the pyrolysis reaction is allowed to continue, the concentration of / -resins levels off at about 16-17%, regardless of any further reaction, and the first-order relationship no longer... 

Templeton 117) has examined the rate of convei ence of Bertaut s series for various functions /(r) and the errors obtained by termination after varying numbers of terms. Values of g and < (h) are given in Table I for various functions /(r). From Templeton s, results it is clear that the assumption of a linear charge distribution gives the most rapid convergence down to an accuracy of 0.1% in the Madelung constant. 

In this application of the 0 method of convergence, it is used to weight the mole fractions which are employed in the Kb method for computing a new temperature profile. The corrected product rates are used as weight factors in the calculation of improved sets of mole fractions. The corrected terminal rates are selected such that they are both in overall component-material balance and in agreement with the specified value of D, that is,... 

The calculational procedure recommended by Nartker et al.12 consists of applying the 6 method of convergence (for conventional and complex distillation columns) one time to each column of the system in succession. The terminal rates so obtained for each column of the system are called the calculated values for the system, and denoted by the subscript ca. To initiate the calculational procedure for the system, the compositions for the minimum number of streams are selected as the independent variables for the system. For example, to initiate the calculational procedure for column 1 of the system shown in Fig. 3-9 the composition of the stream B2 is assumed. One complete column trial is made on column 1. A complete column trial consists of the application of the 6 method for single columns, the calculation of a new set of temperatures by use of the Kb method, and the calculation of a new set of total-flow rates for use of constant-composition method. These temperatures and total-flow rates are stored for use in the next trial calculation for the system. The set of bit s obtained by the... 

The FCC was developed during a study of the influence of gas convection into the base of the arc column near the cathode of the arc (11). The conical tip of the cathode is surrounded by an annular nozzle which terminates upstream from the tip and which directs the gas in a converging high-speed layer into the column of the arc close to the point where it originates on the cathode. It was found that if this were done so that the gas impinged on the arc column in the contraction zone, the gas would preferentially enter the column. Further, the gas could be so injected at 10-20 times the natural aspiration rate. If an attempt is made to force the gas into the arc column elsewhere, the degree of penetration is far less and the injected gas tends to unstabilize and blow out the arc. 

A Taylor expansion is an approximation to the real function by a polynomial terminated at order N. For a given (fixed) N, the Taylor expansion becomes a better approximation as the variable x approaches Xo. For a fixed point x at a given distance from Xo, the approximation can be improved by including more terms in the polynomial. Except for the case where the real function is a polynomial, however, the Taylor expansion will always be an approximation. Furthermore, as one moves away from the expansion point, the rate of convergence slows down, i.e. more and more terms are... 

Also the group of Olaj have performed quite a number of simulation studies, mainly on the kinetics of PLP. In a recent series of papers [172-174, 230] they have used a simulation method based on an iterative procedure [239-241]. The time interval is divided in small, equal intervals, each of duration kp [M]. The radical profile is then converged until a pseudo-steady-state profile is obtained, from which relevant kinetic data can then be calculated such as the rate of polymerization and / or the MWD. Olaj et al. [172-174, 230] used these kinetic data to explore the possibilities of recovering chain-length dependent termination rate coefficients from PLP experiments. In their simulations they compared the exact value of (which can be calculated in a simulation, but is not experimentally accessible) with the value of that they obtained from analyzing output data of their simulations, which are experimentally accessible parameters. In the case of using MWD data [172], the second moment of this distribution, according to [242-244] ... 

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