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The main purpose of operational flexibility analysis is to determine and describe the flexibility region. The existing methods are mainly developed by numerical calculation methods to estimate the contour of the flexibility region. In this article, a novel method is proposed for solving operational flexibility models described by polynomials. The proposed method can accurately describe the flexibility region and explicitly express the functional relationships between uncertain parameters and control variables. First, the original flexibility analysis model is represented as an existential quantifier formula. Then, a technique of quantifier elimination is introduced to deduce the formula to a series of explicitly triangular quantifier‐free formulas. Last, a logical combination of the quantifier‐free formulas can be used to depict the complete flexibility region. The case studies show that the deduced explicit expressions can accurately and effectively describe the flexibility region and guide the steady‐state control operations, regardless of convex or nonconvex systems. © 2018 American Institute of Chemical Engineers AIChE J, 64: 3894–3911, 2018

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Flexibility index is an effective indicator for characterizing the operational flexibility of a process design model. In this work, the concept of δ‐space projection is proposed and defined from the view of solution space, and the flexibility index can be quantified without solving any optimization problems. First, the flexibility index problem is reformulated as an existential quantifier model. Then, the projection operator in the cylindrical algebraic decomposition (CAD) method is introduced to project the solution space onto the one‐dimensional feasible space of the flexibility index. Subsequently, all the candidate values of the flexibility index, which may refer to the locations of the critical points, that is, vertexes and tangency points, are extracted. Last, two δ‐checking rules based on the traditional and improved CAD methods are proposed for finding the flexibility index. The case studies show that the proposed method can exactly find the flexibility index, regardless of convex or nonconvex systems.

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The cylindrical algebraic decomposition (CAD) method has been proposed for flexibility analysis to derive analytical expressions of a feasible region. Due to the heavy computational burden caused by symbolic computation, this method can only handle small-scale problems currently. To overcome this limitation, a novel method is proposed for high-dimensional systems with a number of equalities and limited inequalities. A surrogate model is first built to correlate the inequality constraints based on an initial sample set. Then, the flexibility region is obtained with explicit expressions via the CAD method. Next, for any violation, a refinement will be activated by taking an iterative process of boundary check, surrogate modeling, region deriving, and underestimation check, until the termination condition is satisfied. The case studies show the proposed method can effectively describe the flexibility region for both the convex and nonconvex systems.

Facilitating the exposure of the active crystal facets on the surfaces of composite catalysts is a representative route to promote catalytic activity. Based on a tailored galvanic replacement reaction, herein, a self-assembly route is reported to prepare Pt−WC/CNT with Pt (200) preferential orientation and well-dispersed structure, which are capable of substantially boosting electrocatalysis in hydrogen evolution reaction (HER). Formation mechanism reveals that the (200)-dominated Ptbased catalysts form in galvanic replacement reaction through selective anchored on WC, and the multistep galvanic replacement process plays a critical role to realize the Pt (200)-dominated growth in higher Pt loading catalyst. These unique structural features endow the Pt−WC/CNT with a high turnover frequency of 94.18 H 2 •s −1 at 100 mV overpotential, 7-fold higher than that of commercial Pt/C (13.55 H 2 •s −1 ), ranking it among the most active catalysts. In addition, this method, which combines with gas−solid reaction and galvanic replacement reaction, paves the way to scalable synthesis as Pt facets-controllable composite catalysts to challenge commercial Pt/C.