Content area
A functional transformation method has been developed for solving separation problems which are difficult to converge. These problems 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 x that makes the function f(x) = 0, an $x\sb{k}$ is found for which $f\sp\prime(x\sb{k})$ = 0, or for which the norm of the functions $\vert f(x\sb{k})\vert$ $>$ $\vert f(x\sb{k-1})\vert$, a new function $F(x)$, having the same solution as $f(x)$ but a different slope, is defined such that $\vert F(x\sb{k})\vert$ $<$ $\vert F(x\sb{k-1})\vert$. After having passed through the maximum or minimum, the 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 systems of nonlinear equations (distillation columns and absorbers). In particular, it is shown 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. The algorithms describing the application of the Functional Transformation method to the above methods are given in detail.