Optimal synthesis and design of catalytic distillation columns: A rate-based modeling approach

Highlights

• An MINLP rate-based model for catalytic distillation (CD) design is presented.

• The rate-based model explicitly considers multiscale and multiphase phenomena in CD.

• MINLP was efficiently solved using a new Discrete-Steepest Descent Algorithm (D-SDA).

• A case study using a rate-based CD model was intensified using D-SDA.

• Results show that rate-based CD models are essential for optimal design of CD units.

Abstract

This work presents the optimal synthesis and design of a rigorous catalytic distillation (CD) column that explicitly considers the multiscale and multiphase nature of this intensification process. A rate-based model that couples micro and macroscale events taking place inside the CD column is explicitly considered. The direct solution of such intensive optimization problems is challenging due to nonlinearities introduced by heterogeneous reactions, transport phenomena, and interactions between discrete and continuous variables. Also, combinatorial complexity involved in the selection of multiple structural decisions complicates the problem, e.g., distribution of reactive stages along the column, location of feed stages, and total number of stages. A discrete-steepest descent-based optimization framework recently developed is used to address the optimal synthesis and design of rate-based CD columns. A case study involving the production of ethyl tert-butyl ether (ETBE) has been considered. The results show that the multiscale events occurring in this process intensification (PI) unit cannot be ignored since they produce process designs different from those obtained with an equilibrium-based CD model. The results also show that neglecting multiscale phenomena may result in infeasible CD designs. The outcomes gained through this study illustrate the critical need to systematically consider multiscale and multiphase events in CD columns for the optimal design of realistic, cost-effective, and attractive process intensification (PI) systems.

Introduction

Process intensification (PI) is a rapidly evolving area that is gaining considerable attention to develop new efficient, environmentally friendly, and sustainable chemical operations that can produce valuable products at a low cost. The current trend in chemical engineering is to investigate and use the multiscale structure of processes and products to further intensify and optimize chemical systems (Charpentier, 2010, Kumar and Nigam, 2012, Kurt et al., 2016, Mansouri et al., 2016, Medina-Herrera et al., 2020, Nigam and Larachi, 2005, Singh et al., 2019, Uhlemann et al., 2020). Although the combination of multiscale considerations with PI may result in higher computational requirements, recent developments in modeling tools and algorithms make these combined problems tractable for optimization. For instance, multiscale optimization approaches have been successfully applied to the intensification of processes such as milli-scaled fixed bed reactors (Stamenić et al., 2018, Stamenić et al., 2017) as well as the implementation of radial microchannel reactors for the multiscale intensification of gas-to-liquid technologies in the oil industry (Onel et al., 2017). Multiscale frameworks have also been applied to other processes such as hydraulic fracturing (Zhang et al., 2019); crystallization (da Rosa and Braatz, 2018, Kwon et al., 2015); kraft pulping (Choi and Kwon, 2020); lithium-ion batteries (Golmon et al., 2012, Kashkooli et al., 2016); thin film deposition (Karakasidis and Charitidis, 2007); polymerization (Urrea-Quintero et al., 2019); and catalytic surface reactions (Bruix et al., 2019, Keil, 2012, Liu and Evans, 2013, Salciccioli et al., 2011, Sengar et al., 2019). These studies are motivated by the fact that coupling fine-scale events with macroscale events is critical to capture the actual behavior of systems that evolve at multiple spatial and temporal scales (Christofides et al., 2009, Kwon et al., 2015, Ricardez-Sandoval, 2011, Salciccioli et al., 2011, Vlachos, 2005, Vlachos, 1997). The development of multiscale models is not an easy task since it may require special mathematical tools that would enable accurate prediction of the system’s observables. For instance, tailored coupling and model reduction techniques may be needed to efficiently transfer information between different time domains and across different spatial dimensions. In general, multiscale models are complex; hence, low-order models are developed to capture the process behavior under various operating conditions and then used to perform control and optimization for these systems (Kimaev and Ricardez-Sandoval, 2019, Rasoulian and Ricardez-Sandoval, 2015, Rasoulian and Ricardez-Sandoval, 2014). A recent trend in this field is to develop hybrid models that couple mechanistic multiscale models with low-order models, which replace a section of the actual model that can be challenging to compute (Bangi and Kwon, 2020, Chaffart and Ricardez-Sandoval, 2018, Choi and Kwon, 2020, Ding et al., 2019, Narasingam and Sang-Il Kwon, 2018, Siddhamshetty et al., 2018, Urrea-Quintero et al., 2019).

In the context of the distillation and reaction systems, PI is accomplished by efficiently integrating processes into a single operation; this is referred to as a reactive distillation (RD) or catalytic distillation (CD)¹ processes. PI alternatives that have recently been investigated in this field include columns with multiple reactive zones, reactive dividing wall columns, and catalytic cyclic distillation (García-Sánchez et al., 2019, Huang et al., 2018, Kaur et al., 2020, Kiss, 2019, Mansouri et al., 2016, Segovia-Hernández et al., 2020); additional applications are provided in (Kiss et al., 2019). Despite the progress in this area, PI solutions for RD/CD are still emerging, but multiscale interactions are seldom considered, hence the need to advance this field. This work investigates PI of tray CD columns while considering multiscale phenomena by integrating microscale rate-based and kinetic equations with macroscale material and energy balances, hydrodynamic constraints, and discrete decisions. Particularly, the proposed model and solution strategy allow one to optimally rearrange a fixed number of reaction stages along the catalytic column, such that optimal non-consecutive distribution of reactive stages can be specified.

A catalytic distillation (CD) column is a PI unit that combines heterogeneous catalysis and multiphase separation for the cost-effective production of valuable chemical products. Despite the significant progress in the optimal synthesis and design of CD columns, most of the available optimization studies for this PI unit assume phase/thermal equilibrium (Almeida-Rivera et al., 2004, Segovia-Hernández et al., 2015, Tian et al., 2018). This assumption neglects complex micro and nanoscale heat and mass transfer events taking place in the vapor–liquid interphase within the column, which will impact the unit's overall performance. The rate-based model, also referred to as the MERSHQ model (Mass, Energy, Rate, Summation, Hydraulic, and interface eQuilibrium), allows one to perform distillation calculations without the assumption that the vapor and liquid bulks are in equilibrium (Taylor and Krishna, 1993). Thus, this modeling approach accounts for the mass and energy transfer rates from the bulks to their interphase, where thermodynamic equilibrium is assumed. To account for these microscale phenomena, two approximations are usually considered: (1) a differential approximation, where partial differential equations are directly used to perform mass and energy balances over finite elements; and (2) an integral approximation that usually results in a set of algebraic equations, with products of transfer coefficients and driving forces. These characteristics suggest that the rate-based model predicts a real column's behavior with better accuracy than the traditional equilibrium models, which have been discussed in detail in previous reports. For instance, Taylor et al. (2003) showed that the rate-based model is better than the equilibrium model at predicting composition trajectories for multicomponent systems such as water-ethanol-acetone and water-cyclohexane-ethanol. Furthermore, it has been previously reported that an equilibrium model may not provide accurate descriptions of the actual system’s behavior. Hence, rate-based models are more suitable to specify reliable CD designs (Springer et al., 2003). Similar observations have also been made for CD and RD units (Al-Janabi, 2010, Buchaly et al., 2007, He et al., 2010, Higler et al., 2000, Klöker et al., 2005, Rouzineau et al., 2003, Singh et al., 2020, Sundmacher and Hoffmann, 1996, Xu et al., 2005, Zheng et al., 2003, Zheng et al., 2001). Some of these works also highlighted the need to use comprehensive nonequilibrium models that incorporate both continuous and discrete variables to intensify both laboratory and industrial-scale units (Singh et al., 2020, Sundmacher and Hoffmann, 1996). Hence, nonequilibrium models are essential to specify CD designs that remain feasible during operation.

Optimization-based process syntheses and designs of CD columns with nonequilibrium models are required to specify realistic optimal process designs. Optimization procedures seek to replace heuristics, step-by-step procedures, and overdesign factors by automatic algorithms that rely on the objective function and the superstructure model provided by the user (Costa and Bagajewicz, 2019, Mencarelli et al., 2020). These optimization models are often posed as Mixed-Integer Nonlinear Programming (MINLP) problems, which are challenging to solve given the presence of discrete variables and nonlinear equations (Kronqvist et al., 2019). The first obstacle posed by MINLP models for CD synthesis and design is that they introduce multiple nonlinear equations required to specify multi-component diffusion, thermodynamic, hydrodynamic, and reaction models. Moreover, using a rate-based model increases the size of the problem when compared to an equilibrium model. For example, if design variables are ignored, and both equilibrium and rate-based models are expressed in terms of their input and state variables, then the rate-based model has approximately two times the number of variables of the equilibrium model (Gómez, 2005). Hence, both the model complexity and the corresponding computational costs increase when solving those problems. Accordingly, the selection of an adequate nonlinear solver is critical to guarantee convergence. The second obstacle is the introduction of discrete decisions such as the total number of stages, feeds location, and reaction stages location, which add combinatorial complexity. The interaction between discrete and continuous variables is another major obstacle that makes the optimal synthesis and design of CD columns a non-trivial task. The issue with these discrete–continuous interactions is that both convergence and optimality may be compromised if an adequate MINLP formulation is not selected, e.g., variables being forced to zero by the discrete choices can lead to numerically undefined nonlinear expressions. To address the challenges mentioned above, our group recently developed a new logic-based optimization framework that permits optimization of integrated distillation systems while considering rigorous nonlinear models and discrete decisions (Liñán et al., 2020a, Liñán et al., 2020b). A key feature that distinguishes our method from the available techniques is that optimality guarantees are independent of the approach used to formulate the discrete decisions of the MINLP. That is, the proposed strategy can explicitly deal with nonlinear interactions between continuous and discrete variables, e.g., bilinear variables.

Optimization studies for the optimal design of rate-based CD units are still lacking in the literature. The optimal synthesis and design of nonequilibrium RD/CD columns has been previously studied by Gómez et al. (2006). A decomposition strategy was used in that work, which resulted in a medium-scale master MINLP model solved using simulated annealing and nonlinear subproblems solved with successive quadratic programming. Another study addressed the rate-based design of a CD column for the synthesis of tert-amyl ethyl ether by combining a heuristic procedure with a modified differential evolution algorithm (González-Rugerio et al., 2014). More recently, the stochastic approach has been extended to intensified distillation sequences by linking a genetic algorithm to a simulator, such as Aspen Plus (Gómez-Castro et al., 2015). In contrast to stochastic techniques, there is a gap in the literature regarding the deterministic² optimal design of heterogeneously catalyzed or reactive distillation columns using rate-based models. Deterministic optimization techniques provide optimality guarantees, yield replicable results, and do not require system-dependent parameters to be tuned (Costa and Bagajewicz, 2019, Gómez et al., 2006, Segovia-Hernández et al., 2015, Smith, 2005). To the authors’ knowledge, the only relevant studies that used deterministic optimization for optimal CD design applied orthogonal collocation on finite elements, which leads to the reformulation of the original MINLP model into a large-scale continuous nonlinear problem (Dalaouti and Seferlis, 2006, Damartzis and Seferlis, 2010). The orthogonal collocation approximation may be a suitable representation of the original process if an adequate number of collocation points is considered. While orthogonal collocation is an alternative to the design of distillation processes, the relaxation of the discrete decisions is an approximation that inherently results in solutions that may not necessarily comply with the discrete nature of the corresponding decision variables (Dalaouti and Seferlis, 2006, Seferlis and Grievink, 2001). Contrary to those studies, our recently proposed logic-based optimization algorithm explicitly considers discrete choices and deterministic procedures to optimize the design of nonequilibrium CD columns (Liñán et al., 2020a). The algorithm is based on discrete convex optimality criteria and the Discrete-Steepest Descent Algorithm (D-SDA) introduced in (Murota, 2004, Murota, 2003).

Based on the above, this study aims to perform the optimal synthesis and design of staged CD columns modeled with rate-based and hydrodynamic equations that take into account the complex multiscale and multiphase behavior of this intensified process. This optimization problem can be summarized as follows: find the optimal operating variables (e.g. reflux ratio and reboiler duty), continuous design variables (e.g. diameter, stage height and weir height), and discrete design decisions (e.g. number of stages, location of feeds and distribution of reactive stages) that minimize an economic objective function for a CD column. The optimization of the objective function is subject to microscale constraints related to multicomponent mass/energy transfer, reaction kinetics and interphase equilibrium, as well as macroscale constraints associated with mass/energy balances, pressure drop, hydrodynamic inequality constraints, and a feasible region for discrete variables. The contributions of this work are summarized as follows:

• A general MINLP model for the rate-based synthesis and design of CD columns is presented. This model can be easily extended to similar case studies, as well as other stage separation processes, e.g., reactive distillation, conventional distillation, extractive distillation, among others.

• A reliable deterministic optimization methodology to solve the resulting MINLP problem is applied. This optimization strategy has already demonstrated its superiority to find optimal configurations for the design of CD columns with equilibrium models when compared to existing MINLP solvers (Liñán et al., 2020b). In the current work, we show the potential of our efficient optimization strategy to deal with more complex multiscale and multiphase models.

This study is organized as follows: In Section 2, nonequilibrium models are introduced, emphasizing their multiscale nature and their differences and similarities compared to the equilibrium MESH (Mass, Equilibrium, Summation, entHalpy) model. Next, a set of MINLP equations that combine the selected rate-based model, macroscale phenomena, and a discrete superstructure is presented in Section 3. Section 4 summarizes the D-SDA methodology. Section 5 presents the case study used in this work to test the rate-based CD model's performance. Two computational experiments are presented in this section to motivate the synthesis of CD columns with rate-based models. Also, the ETBE CD column's optimal design is performed with the D-SDA to consider discrete variables and their interaction with the multiscale model. Furthermore, state-of-the-art MINLP solvers are compared with the D-SDA at the end of this section. Concluding remarks and future research directions are outlined in the last section.