Reliable computation of equilibrium cascades with affine arithmetic

Abstract

Computing the steady state of multistage counter-current processes like distillation, extraction, or absorption is the equivalent to finding solutions for large scale non-linear equation systems. The conventional solution techniques are fast and efficient if a good estimation is available but are prone to fail, and do not provide information about the reason for the failure. This is the main motive to apply reliable methods in solving them. Topical heading: Separations

Computation of an extractive distillation column with affine arithmetic

“…The mixed AA/IA was used only at the critical parts (where otherwise division by zero or calling the logarithm function with negative argument would have occurred) in the previous work41 of the authors due to implementation design flaws. Based on the conclusions of the previous work, the affine class has been redesigned and implemented in C++.…”

“…It follows that only the first LP subproblem has to be solved from scratch; all other subproblems should use the optimal solution of the preceding subproblem as a primal feasible basis and run only Phase II of the primal simplex algorithm, thus hopefully reducing the computational efforts. The naïve sequence41 to process the x j variables would be min x 1 , max x 1 , min x 2 , max x 2 , … etc but the subproblems min x 1 and max x 1 are likely to produce completely different solutions. As a consequence, this is expected to result in a lot of simplex iterations when using the optimal solution of subproblem min x 1 as the initial primal feasible basic solution when solving the subproblem max x 1 .…”

“…The enhancements presented in this subsection will be referred to as Achterberg's heuristic. Numerical examples, suggesting the superiority of this heuristic to the previous implementation,41 will be presented after subsection Separation problem which describes the numerical examples used for comparison.…”

“…A new linearization technique, based on affine arithmetic (AA),24–27 has been proposed recently by Kolev 28–36. Numerical evidence published in the literature28–32, 34, 37–41 suggests that the new technique may be superior to the traditional linearization techniques such as the interval Newton or the Krawczyk42 method. Linear programming may be preferable as pruning technique for this new linearization in the case of the vapor‐liquid equilibrium cascades 41…”

Computing multiple steady states in homogeneous azeotropic and ideal two‐product distillation

“…One of the characteristic components of the proposed algorithm is the linearization technique. Numerical evidence published in the literature30–42 seem to indicate superiority of the linear enclosure compared with the traditional one such as the interval Newton methods [see Ref 43…”