A Novel Method To Find All Physical Solutions of Constrained Chemical Engineering Models in Polynomial Equations

Abstract

Sometimes, there are more than one solutions of a chemical process simulation. In this work, a novel strategy is proposed for finding all physical solutions of chemical engineering models described in polynomials with prescribed variable bounds. First, the Gröbner basis method is proposed to transform the original model into a lower-triangular polynomial system. Based on this triangular structure, the problem of finding solutions of a constrained multivariable polynomial system is converted to solve a set of constrained univariate polynomials sequentially. Next, a real-root-isolation algorithm is developed to compute the disjoint intervals of each univariate polynomial such that each interval contains one solution. Finally, a numerical method with convergence monitoring is proposed to locate each solution. Case studies show that the proposed method can accurately analyze the multisolution features of the systems. All physical solutions can be found directly and effectively.

SyPSE: A Symbolic Computation Toolbox for Process Systems Engineering Part II─Design for PSE Applications

“…This case was proposed by Carnahan and Wilkes and later studied by Zhao et al The problem is formulated as follows where k 45 = 0.09495165682, k 24 = 2.20290484696, and k 34 = 0.19904332800. This problem only has one physical solution, where Q 24 = 51.751546, Q 34 = 92.918111, p 4 = 57.125560, p 3 = 80.233779, and p 2 = 136.459767.…”

“…Pipe and Pump Network. This case was proposed by Carnahan and Wilkes 8 and later studied by Zhao et al 9 The problem is formulated as follows…”

SyPSE: A Symbolic Computation Toolbox for Process Systems Engineering Part I─Architecture and Algorithm Development

“…Several applications have been studied, including robust controller design, Birkhoff interpolation problem, simulation and optimization, and model predictive control, based on the QE algorithm. Zhao et al − proposed a series of hybrid numerical–symbolic strategies for structural analysis, solution space analysis, and parameter space analysis of process models. The problems of simulation, multisolution analysis, and analytical description of the flexibility space are addressed with the triangular reconstruction of process models.…”