(56f) Improved Formulations for Superstructure Optimization: The Case of Carbon-Capture Columns

Process synthesis is central to the conceptualization of new chemical processes that can meet the manifold constraints of a circular economy. Process synthesis is the activity of identifying an optimal flowsheet which entails choosing a) process units (e.g., unit operations) from a set of alternatives, b) the connectivity of selected process units, and c) the degrees of freedom of selected units such that a design objective is optimized and all process constraints are satisfied.

Superstructure optimization is a mathematical programming approach to process synthesis. In this paper, we focus our attention on the State Operator Network¹ (SON) representation of the process superstructure. A key feature of the SON is that each allowed process unit is described by its rigorous model which includes MESH equations and equipment sizing and costing correlations. The SON relies on a network of conceptual mixers and splitters that enable up to full connectivity between the set of selected process units. A mixer and a splitter are located at each inlet and outlet, respectively, of each process unit.

The optimization of the SON is a challenging mixed-integer nonlinear programming problem (MINLP). A particular issue in the optimization of the SON, which is the subject of this paper, is the fate of a process unit that is deselected, that is, excluded from the flowsheet. Naturally, when a process unit is deselected, mass flowrates at each inlet of the unit must be set to zero. However, constraints of several process units are implicitly well defined only at strictly positive mass flows. At zero-valued inlet flows, several numerical singularities (including undefined behaviour) in the constraint functions that describe the unit and/or their derivatives can occur. For example, consider an isobaric-isenthalpic flash unit. At zero valued flows, a two-phase solution to the phase-equilibrium problem does not exist. Further, the Jacobian of the mass-balance constraints of the unit is rank-deficient². Several costing and sizing correlations that depend on the flowrates may become numerically singular. As a result, the optimization of the SON may fail to converge.

To overcome this challenge, one may exactly Disjunctive Programming (GDP)³. However, the application of GDP to simulation-based superstructure optimization is limited and computationally expensive⁴. In this paper, we build upon the recently developed Modified State Operator Network (MSON)⁵. The MSON modifies mixers by introducing fictitious inlet streams with strictly positive flowrates and intensive properties that guarantee simulation (evaluation) of a deselected unit. The MSON modifies splitters to reject any flows at the outlets of units that arise due to these fictitious inlet flows, thus, resulting in an exact reformulation.

The SON¹ and MSON⁵ are only applicable to the synthesis of isobaric processes. We present the first extension of the MSON to enable the synthesis of processes at variable pressures. We illustrate the use of the extended MSON on a carbon-capture process for the separation of carbon-dioxide and methane.

We simultaneously find all degrees of freedom of a carbon-capture column, namely, the number of stages, the column pressure and the solvent flowrate, such that the total annualized cost of the column is minimized and column performance constraints are satisfied. The extended MSON identifies an optimal column with only 4 stages and an operating pressure that is 65% lower and a total annualized cost that is 37% lower than the MSON, in which the column pressure was fixed. The extended MSON is robust and averts numerical singularities due to zero flows. The extended MSON enables the design of a variable-pressure process on the basis of detailed thermodynamic models and simulation-based optimization.

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   Manuscript in preparation. https://zenodo.org/records/10946545