Convergence by Dynamic Iteration

In this method, the component material balance, enthalpy balance, and phase equilibrium equations are written for component i on stage j at unsteady state  

The method has been applied to absorbers (Khoury, 1980), where all heat duties, side draws, and all feeds except the liquid feed at the top of the column and the vapor feed at the bottom are set to zero. The rates of change of the component molar holdup and total energy on a stage may be broken down into liquid and vapor contributions (subscripts j and i are dropped in the rates of change terms for simplicity)  

Equations 13.56 and 13.57, applied to the absorber model, are rearranged as follows  

If two successive iterations are made to simulate two successive points in time in the approach to steady state and if quantities with superscript k represent iteration k, the following expressions will hold  

As in other methods, the calculations are started with an assumed temperature 

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In problems concerning dynamic NMR spectra the relationship between f and the parameters g is non-linear and therefore the determination of the minimum must be carried out by an iterative procedure. This is often a complicated computational problem and the procedure need not converge. 

Several simulation runs follow for the two cases. The runs are represented by their initial conditions and a few selected profiles in order to show the progress of the dynamics. The spatial domain was divided into 100 segments so that A.x = 1 x 10. The dimensionless time increment for these runs was At = 1 x 10. At each time interval, the linearized response converged within 5 iterations to the nonlinear response. Figure 8.23 shows the dynamics of a startup from a cold reactor with a single steady state. For this run, the system exhibits uniform asymptotic stability. 

Thus, at each iteration of (5.8), system (5.14) should be solved. The rate of convergence of this procedure depends on the correct choice of initial conditions. The method of differential approximation refers to universal approaches in the function approximation theory to the analysis of dynamic systems. Under remote monitoring conditions, the use of this method can be justified by allowing aircraft and satellite measurements to be spaced in time with respect to the objects to be monitored and, hence, in processing the readings from measuring instruments it is necessary to take into account possible changes in the object between moments of measurement. 

In addition to corrections tising an appropriate spectral density function, in principle one also needs to consider an ensemble of structures. Bonvin et al. U993) used an ensemble iterative relaxation matrix approach in which the NOE is measured as an ensemble property. A relaxation matrix is built from an ensemble of structures, using averaging of contributions from different structures. The needed order parameters for fast motions were obtained fi um a 50-ps molecular dynamics calculation. The relaxation matrix is then used to refine individual structures. The new structures are used again to reconstruct the relaxation matrix, and a second new set of structures is defined. One repeats the process until the ensemble of structures is converged. The caveat espressed earlier that the accuracy of the result is limited by the accuracy of the spectral density function applies to all calculations of this typ . 

Iterative minimization techniques for ab initio total energy calculations molecular dynamics and conjugate gradients , by M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias and J. D. Joannopoulos, Rev. Mod. Phys., 64, 1045 (1992). Despite the title which makes it appear highly specialized, this article has much to offer on everything from the EDA formalism itself, to issues of convergence and the use of EDA calculations (rather than classical potentials) as the basis of molecular dynamics. 

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