A High-Order, Flow-Alignment-Based Compartmental Modelling Method

Abstract

Industrially-relevant chemical engineering processes, such as stirred tank bioreactors in the pharmaceutical sector, inherently operate across multiple scales and involve complex, multiphysics, and multiphase interactions. Modelling of these systems is essential for their design, optimization, control, and operational troubleshooting; these processes are often too intricate for experimental approaches alone, with trial runs proving prohibitively costly or key metrics being difficult or impossible to measure. Traditionally, modelling such systems has relied on simplified design equations or idealized models, such as the continuously stirred tank reactor (CSTR). However, these approaches lack the explanatory power required to capture real-system outcomes, such as concentration gradient formation. With advancements in computational capabilities, computational fluid dynamics (CFD) simulations have become standard for investigating specific questions within these systems. Nonetheless, certain critical applications, such as extended simulations of microorganism growth or real-time predictive control, remain impractical due to their high computational demands.

Reduced Order Models (ROMs) offer a middle ground between the simplistic CSTR models and the computationally intensive CFD simulations. ROMs trade off some of the generality and accuracy of CFD simulations in exchange for a substantial reduction in computational cost, often by several orders of magnitude. This work focuses exclusively on a specific type of ROM: Compartmental Models (CMs). They are underpinned by the assumption of one-way coupling between the hydrodynamics and mass transport of reactive species. CMs are constructed through a two-step process. First, the domain is divided into non-overlapping compartments using a set of criteria; next, each compartment is represented by one or more simplified models. This network of models decouples mass transport from hydrodynamics and reduces the number of degrees of freedom on which the conservation of mass of the reactive species needs to be solved. This reduction is particularly important for bioreactors, where hundreds of coupled nonlinear reactions are common.

Current compartmental modelling methods exhibit several limitations, such as a disconnect between the criteria used for compartment identification and their subsequent modelling, an assumption that each compartment is well-mixed, a reliance on manual compartmentalization or manual intervention, and a non-prescriptive framework that is challenging to adapt to new geometries. This work introduces a novel compartmental modelling method based on flow alignment. The velocity field is analyzed and split into compartments within which the flow is unidirectional. Each compartment is then modelled as a series of 1D Plug Flow Reactors (PFRs). Benchmarking this method against the state-of-the-art method demonstrates that it yields more accurate results while achieving computational speeds that are orders of magnitude faster than traditional CFD simulations.

Further, many current CM approaches simplify three-dimensional geometries by either modelling two-dimensional cross-sections and relying on rotational symmetry or by using a uniform grids of compartments. The developed method is extended to fully three-dimensional two-phase stirred tank systems without using these assumptions. It successfully compartmentalizes the distinct recirculatory regions generated by the impellers, eliminating the manual ad hoc intervention required by past methods. Mixing time and concentration predictions at probe locations are validated against CFD simulations, other CMs, and experimental data. The proposed general method performs as well or better than past CMs which were tailor made for the stirred tank geometry. Further, the model's capability to handle complex spatially varying reactions is demonstrated by simulating oxygen dissolution into the liquid phase, accurately capturing spatial gradients in dissolved oxygen concentration.

Lastly, a significant limitation in previous compartmental modelling work is the reliance on a single velocity snapshot or a time-averaged steady-state velocity field. For instance, in the case of vortex shedding from a cylinder in the laminar flow regime, neither time-averaged velocity-based CM nor an ensemble of CMs based on discrete velocity snapshots accurately captures the impact of the inherently non-stationary flow topology. The non-stationary nature of such flow fields is addressed by employing projection mappings to cycle through a series of compartmental models, allowing dynamically updating their shape, number, location, and connections. This approach successfully captures the oscillation period of the flow and demonstrates promise in representing non-stationary flow behaviours accurately.

In summary, this work advances the field of compartmental modelling by unlocking their the application to complex, industrially-relevant systems by developing a generalized, alignment-based method. This method extends the capability of CMs to handle both time-varying and fully three-dimensional multiphase flows without requiring manual intervention. The approach is validated through benchmarking against CFD simulations, other CM approaches, and experimental data, demonstrating improvements in computational efficiency and accuracy.