Multi-stage model predictive control for slug flow crystallizers using uncertainty-aware surrogate models

TL;DR

This work tackles the challenge of controlling a slug-flow crystallizer, a spatially distributed system with no backmixing that is difficult to model for optimization. It first develops a full dynamic model capturing slug-to-slug variability and velocity evolution, then trains uncertainty-aware surrogates (conformalized quantile regression and Bayesian last layer neural networks) from its data. The surrogates enable a multi-stage model predictive control scheme that explicitly accounts for prediction uncertainty via scenario trees, achieving robust real-time control and improved constraint handling. Simulation on an L-alanine/water system demonstrates the approach’s potential, highlighting tradeoffs between uncertainty handling and data availability, and outlining paths toward experimental validation and fouling-aware operation.

Abstract

This paper presents a novel dynamic model for slug flow crystallizers that addresses the challenges of spatial distribution without backmixing or diffusion, potentially enabling advanced model-based control. The developed model can accurately describe the main characteristics of slug flow crystallizers, including slug-to-slug variability but leads to a high computational complexity due to the consideration of partial differential equations and population balance equations. For that reason, the model cannot be directly used for process optimization and control. To solve this challenge, we propose two different approaches, conformalized quantile regression and Bayesian last layer neural networks, to develop surrogate models with uncertainty quantification capabilities. These surrogates output a prediction of the system states together with an uncertainty of these predictions to account for process variability and model uncertainty. We use the uncertainty of the predictions to formulate a robust model predictive control approach, enabling robust real-time advanced control of a slug flow crystallizer.

Figures (10)

• Figure 1: Sketch of the slug flow crystallizer system under consideration.
• Figure 2: Illustration of the new dynamic slug flow crystallizer model. The outer tempering medium is discretized into static finite volumes. The inner process medium is modeled using batch crystallizers which are advanced through the crystallizer according to their local velocity $v(z_i)$. Depicted are three different slugs. The dotted slugs represent the respective position at the next time step.
• Figure 3: Illustration of a single slug advanced for one time step. Due to the nature of the solution method, it is possible for a slug to pass several finite volumes of the external tempering medium in one time step. This must be taken into account when solving the heat balance.
• Figure 4: Relative temperature difference at different positions of the slug flow crystallizer during steady-state operation, comparing the presented case (where slugs may not coincide with actual slugs) versus the case using a correlation for the actual slug length from kufnerModelingContinuousSlug2023.
• Figure 5: Results of the proposed model for an exemplary simulation. The concentration and some characteristic diameters of the particle size distribution are plotted over the length of the crystallizer at a certain time. The model yields the full distribution at the outlet as well as the temperatures of tempering and process medium.