Robust process simulation using interval methods

Abstract

Ideally, for the needs of robust process simulation, one would like a nonlinear equation solving technique that can find any and all roots to a problem, and do so with mathematical certainty. In general, currently used techniques do not provide such rigorous guarantees. One approach to providing such assurances can be found in the use of interval analysis, in particular the use of interval Newton methods combined with generalized bisection. However, these methods have generally been regarded as extremely inefficient. Motivated by recent progress in interval analysis, as well as continuing advances in computer speed and the availability of parallel computing, we consider here the feasibility of using an interval Newton/generalized bisection algorithm on process simulation problems. An algorithm designed for parallel computing on an MIMD machine is described, and results of tests on several problems are reported. Experiments indicate that the interval Newton/generalized bisection method works quite well on relatively small problems, providing a powerful method for finding all solutions to a problem. For larger problems, the method performs inconsistently with regard to efficiency, at least when reasonable initial bounds are not provided.

Phase stability analysis using interval Newton method with NRTL model

“…The interval method is a general-purpose computational method to solve nonlinear equations to find all the solutions lying within the variable bounds [12]. It uses interval vectors and matrices starting with a specified initial box of intervals, and search all the roots by solving the linear interval equation system for a new interval N (k) :…”

“…The method is a generalized bisection algorithm with some modifications so that it is relatively less sensitive to initial values, and should provide all the roots including global optimum [11]. The method has been tested with equation of states and activity coefficient models [9], [12] and [13], and for process design calculations, such as mixed flow reactor and a reaction kinetics model [14]. This study further tests the reliability of phase stability analysis for 10 binary mixtures and 2…”

Reliable phase stability analysis for cubic equation of state models

“…Efficient techniques for implementing IN/GB are a relatively recent development, and thus such methods have not yet been widely applied. Schnepper and Stadtherr (1990) have suggested the use of this method for solving chemical process modeling problems, and recently described an implementation (Schnepper and Stadtherr, 1996). Balaji et al (1995) have also successfully applied the method to chemical engineering problems.…”

“…The technique used here for computing N (k) from (5) is the preconditioned Gauss-Seidel-like technique developed by Hansen and Sengupta (1981). A detailed step-by-step description of the IN/GB algorithm used here is given by Schnepper and Stadtherr (1996).…”

Phase Behavior and Reliable Computation of High-Pressure Solid−Fluid Equilibrium with Cosolvents

“…These procedures are outlined in more detail by Xu et al 7 , and further details are given by Hua et al 34,35 and Schnepper and Stadtherr. 36 Properly implemented, the IN/GB technique provides the power to find, with mathematical and computational certainty, enclosures of all solutions of a system of nonlinear equations, or to determine with certainty that there are none, provided that initial upper and lower bounds are available for all variables. 31,33 This is made possible through the use of the powerful existence and uniqueness test provided by the interval-Newton method.…”