(74e) A Trust-Region Algorithm for the Optimization of PSA Processes Using Reduced Order Modeling

The last few decades have seen a considerable increase in the applications of adsorptive gas separation technologies, such as pressure swing adsorption (PSA); the applications range from bulk separations to trace contaminant removal. PSA processes are based on solid-gas equilibrium and operate under periodic transient conditions [1]. Bed models for these processes are therefore defined by coupled nonlinear partial differential and algebraic equations (PDAEs) distributed in space and time with periodic boundary conditions that connect the processing steps together and high nonlinearities arising from non-isothermal effects and nonlinear adsorption isotherms. As a result, the optimization of such systems for either design or operation represents a significant computational challenge to current nonlinear programming algorithms. Model reduction is a powerful methodology that permits systematic generation of cost-efficient low-order representations of large-scale systems that result from discretization of such PDAEs. In particular, low-dimensional approximations can be obtained from reduced order modeling (ROM) techniques based on proper orthogonal decomposition (POD) and can be used as surrogate models in the optimization problems. In this approach, a representative ensemble of solutions of the dynamic PDAE system is constructed by solving a higher-order discretization of the model using the method of lines, followed by the application of Karhunen-Loéve expansion to derive a small set of empirical eigenfunctions (POD modes). These modes are used as basis functions within a Galerkin's projection framework to derive a low-order DAE system that accurately describes the dominant dynamics of the PDAE system. This approach leads to a DAE system of significantly lower order, thus replacing the one obtained from spatial discretization before and making optimization problem computationally efficient [2].

The ROM methodology has been successfully applied to a 2-bed 4-step PSA process used for separating a hydrogen-methane mixture in [3]. The reduced order model developed was successfully used to optimize this process to maximize hydrogen recovery within a trust-region. We extend this approach in this work to develop a rigorous trust-region algorithm for ROM-based optimization of PSA processes. The trust-region update rules and sufficient decrease condition for the objective is used to determine the size of the trust-region. Based on the decrease in the objective function and error in the ROM, a ROM updation strategy is designed [4, 5]. The inequalities and bounds are handled in the algorithm using exact penalty formulation, and a non-smooth trust-region algorithm by Conn et al. [6] is used to handle non-differentiability. To ensure that the first order consistency condition is met and the optimum obtained from ROM-based optimization corresponds to the optimum of the original problem, a scaling function, such as one proposed by Alexandrov et al. [7], is incorporated in the objective function. Such error control mechanism is also capable of handling numerical inconsistencies such as unphysical oscillations in the state variable profiles.

The proposed methodology is applied to optimize a PSA process to concentrate CO₂ from a nitrogen-carbon dioxide mixture. As in [3], separate ROMs are developed for each operating step with different POD modes for each state variable. Numerical results will be presented for optimization case studies which involve maximizing CO₂ recovery, feed throughput or minimizing overall power consumption.

References

[1] Ruthven, D. M., Farooq, S., and Knaebel, K. S., Pressure Swing Adsorption. VCH Publishers: New York, 1994.

[2] Kunisch, K., and Volkwein, S., Control of the Burgers Equation by a Reduced-Order Approach Using Proper Orthogonal Decomposition. J. Opt. Theory Applic., 1999, 102(2), 345.

[3] Agarwal, A., Biegler, L. T., and Zitney, S. E., Simulation and Optimization of Pressure Swing Adsorption Systems Using Reduced-Order Modeling. Ind. Eng. Chem. Res., 2009, 48(5), 2327.

[4] Fahl, M., Trust-region Methods for Flow Control based on Reduced Order Modelling. Doctoral Dissertation, Trier University, 2000.

[5] Toint, P. L., Global convergence of a class of trust-region methods for nonconvex minimization in Hilbert space. IMA J. Numer. Anal., 1988, 8(2), 231.

[6] Conn, A. R., Gould, N. I. M., and Toint, P. L., Trust-Region Methods. SIAM: Philadelphia, 2000.

[7] Alexandrov, N. M., Dennis, J. E., Jr., Lewis, R. M., and Torczon, V., A trust-region framework for managing the use of approximation models in optimization. Struct. Multidisc. Optim., 1998, 15, 16.