Application of a functional transformation to simulation of separation processes

Abstract

A functional transformation method has been developed to be used with different techniques for solving separation problems which are difficult to converge. Problems of this kind consist of sets of nonlinear equations with the presence of either maxima, minima, turning points, singular or near singular Jacobians in their solution paths. First, the Functional Transformation Method is used in combination with the Newton-Raphson method to trace the path of a function through a local maximum or minimum. If, in the course of searching for the value of the unknown x that makes the function f(x) = 0, an x[k] is found for which f'(x[k]) = 0, or for which the norm of the functions |f(x[k])| > |f(x[k-1])|, a new function F(x), having the same solution as f(x) but a different slope, is defined such that |F(x[k])| < |f(x[k-1])|. After having passed through the maximum or minimum, the trial procedure returns to the original function f(x). Next, the combination of this method with other forms of the Newton-Raphson method is used to solve large systems of nonlinear equations. In particular, it is shown by the solution of different types of numerical examples, that the aforementioned combination results in a significant extension of the regions of convergence of the methods studied; namely, the Newton-Raphson method, the Almost Band formulation of the Newton-Raphson method, the 2N Newton-Raphson method with the Broyden modification, the 2N Newton-Raphson with the Broyden-Bennett modification, the Almost Band Algorithm with the Broyden-Householder modification, the Almost Band Algorithm with the Schubert modification, and the Gear Parametric Continuation method. By use of a combination of the Functional Transformation method with the Broyden modified forms of the Newton-Raphson method it is possible to solve most of the problems which are either difficult or impossible to solve by the use of the Broyden forms of the Newton-Raphson method alone. The Functional Transformation method is used to reduce the number of steps required to solve problems by use of the Parametric Continuation method with automatic step size selection by G ear's method. By relaxing the control parameters of G ear's method the step size becomes larger, and the resulting corrector equations are more dificult to solve...