Iterations Converge

In this example, you have two iterations - one because of the circular reference due to the recycle streams and one because of the nonlinear Rachford-Rice equation. Some computer programs cannot handle both of these complications together. Neither Goal Seek nor Solver worked for this example, and you iterated the vapor fraction by hand. 

Step 1 In effect you used the spreadsheet to do the molar balances, and you set v yourself. The spreadsheet shows the value oif(v ) for your choice of v, but it does the molar balance anyway. 

Repeat You keep this up until/( ) is small enough to satisfy you. Once you are close. Solver should have no problem converging to a tight tolerance. What you are doing is replacing a problem with two iteration loops with a problem in which you supply one of the numbers, and the computer solves the other iteration loop. Then you change your number until the other equation is satisfied. 

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At the conclusion of each iteration, convergence is checked by employing an approximate criterion such as... 

Convergence was achieved in 3 iterations. Converged values of temperatures, total flows, and component flow rates are tabulated in Table 13-14. Computed reboiler duty is 1,295,000 W (4,421,000 Btu/h). Computed temperature, total vapor flow, and component flow profiles, shown in Fig. 13-54, are not of the shapes that might be expected. Vapor and liquid flow rates for nC4 change dramatically from stage to stage. 

Thus, in principle, higher-order iterations converge more rapidly. In... 

It is worth noting here that the iterations converge no matter how the initial approximation y is chosen, because 0 < 1. 

Remark Condition (11) for fixed B may be viewed as a selection rule for those values of r for which the iterations converge. For example, for the explicit scheme with the identity operator B = E condition (IT) is ensured if all the eigenvalues are subject to the relation... 

Thus, the iterations converge for any r < 2/. 4 (. Let us stress here that the estimate obtained for p is too rough for determination of the total number n(e, N) of the necessary iterations and indicates mainly the true order in n as At — . 

Fig. 14. Iterative convergent-divergent preparation of oligophenylenes according to Tour et al.

The sequence of values taken by Jc is tabulated in Table 5.13. Convergence is achieved in 6-7 iterations. Convergence is towards e = 1, i.e., the initial value is entirely fortuitous. This value is an important indication that scavenging is not very efficient for Ni. The quality of the fit may be seen in Figure 5.10 <=... 

Successive iterations converge extremely fast. After the fifth step, the results hardly change (Table 5.24) which, using the Newton method outlined in Section 3.1, indicates an age of T = 2.9065 Ga. 

Certain special problems related to decomposition are also considered. One is whether the process model is determinate, that is, does the model have a solution. Section III indicates how decomposition can help in validating determinacy. Another problem is that of convergence of the iterative strategy. If certain of the equations are too sensitive to the values of the variables being iterated, convergence may not be obtained. Therefore, the decomposition procedure must be constrained so as to choose as iterates only those variables for which the system of equations has suitable sensitivity, as discussed in Section VI. 

A perturbation expansion version of this matrix inversion method in angular momentum space has been introduced with the Reverse Scattering Perturbation (RSP) method, in which the ideas of the RFS method are used the matrix inversion is replaced by an iterative, convergent expansion that exploits the weakness of the electron backscattering by any atom and sums over significant multiple scattering paths only. 

In order to circumvent this problem, one may either manipulate the initial guess or set up a constrained optimization, where the SCF iterations converge to predefined (constrained) properties. The latter was achieved in a protocol by Van Voorhis and coworkers (133-135). This approach suffers from the fact that the constrained Slater determinant may not represent a local energy minimum of the unconstrained potential energy surface. 

As illustrated in Fig. 15.5, the initial iterate (point 0) is within the domain of convergence of Newton s method. As a result the iteration converges rapidly. However, imagine the behavior of the algorithm if the starting iterate (initial guess at the solution) were just... 

Table 2.4 shows the SAS NLIN specifications and the computer output. You can choose one of the four iterative methods modified Gauss-Newton, Marquardt, gradient or steepest-descent, and multivariate secant or false position method (SAS, 1985). The Gauss-Newton iterative methods regress the residuals onto the partial derivatives of the model with respect to the parameters until the iterations converge. You also have to specify the model and starting values of the parameters to be estimated. It is optional to provide the partial derivatives of the model with respect to each parameter, b. Figure 2.9 shows the reaction rate versus substrate concentration curves predicted from the Michaelis-Menten equation with parameter values obtained by four different... 

Mathematical modeling of physical processes in fuel cells inevitably involves some assumptions that may or may not be valid under all circumstances. Furthermore approximations have to be introduced to make the computational models robust and tractable. These approximations in the mathematical models lead to the so called modeling errors . That is if the equations posed are solved exactly, the difference between this exact solution and the corresponding true but usually unknown physical reality is known as the modeling error. However, it is rarely the situation that the solution to the mathematical models is exact due to the inherent numerical errors such as round off errors, iteration convergence and discretization errors, among oth-... 

One way of estimating the iteration convergence error is done by way of monitoring the normalized residual, Ruer, i.e. some norm (such as the L2 norm) over the computational cells of the remainder after the numerical solution is substituted into the discretized counter part of the PDE. An example of residual monitoring is depicted in Figure A. 1. It is seen in this case that the residual of each equation reduces to machine accuracy. 

However this may not always be possible and then the residuals by themselves may not be a good indicator of the magnitude of iteration convergence errors in the variables being solved for. In such situations, the difference in the solution between... 

Consider for example the variation of the axial velocity as a function of iteration number shown in Figure A.2. If the calculations were stopped at iteration number ii = 50, there would be an iteration error of about 200%. The fully converged solution in this case is givenby u = 0.160495 m/s. Since in cases of incomplete iterative convergence the fully converged solution is not known an estimate for this error is necessary. The approximate relative iteration error and the relative true iteration error are defined respectively by... 

According to Ferziger and Peric (1996) the iteration convergence error should be approximated by... 

There are two "do loops" in each of these equilibrium calculations. When the unknown is either temperature or pressure, the inner "do loop" of 30 iterations converges on the unknown phase compositions and the outer "do loop" of 50 iterations... 

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