(422b) Multi-Fidelity Data-Driven Design and Analysis of Reactor and Tube Simulations

The development of new manufacturing techniques such as 3D printing have enabled the creation of previously infeasible chemical reactor designs. These novel reactor configurations have increasingly been considered for chemical synthesis, for example, microfluidic reactors, and mesoscale reactors. Microfluidic reactors can enable finer control over local conditions resulting in increased product selectivity, and improved heat transfer resulting in more sustainable processes. 3D printed mesoscale reactors have been proposed as next-generation alternatives to traditionally manufactured designs, lending to their large potential design space. 3D printed reactors for biodiesel production have been shown to provide high yields and helical coil-in-coil reactors, unable to be manufactured otherwise, have been shown to demonstrate `excellent plug-flow behaviour'.

The modeling and design of traditional chemical reactors has largely been considered an art, with small improvements in performance resulting in wider impacts on product yield, sustainability, and economic costs. However, with the promise of new reactors comes the need for new analytical techniques to model and optimize in increasingly complex design spaces. Chemical reactors have been investigated through computational fluid dynamics (CFD) simulations, where systems of partial differential equations (PDEs) with large degrees of freedom are solved iteratively, resulting in large computational costs. In addition to being expensive, gradient information is practically unavailable.

Derivative-free optimization has found significant application in domains where mathematical expressions or gradients are unavailable. Examples include the optimization of proprietary chemical process software, chemical reaction optimization, real time optimization, and topology optimization of two-dimensional chemical reactor channels. With the advent of new technologies in reactor design, reactor geometries are becoming highly-parameterized, resulting in higher-dimensional, more complex derivative-free optimization problems. As such, there exists significant scope for a robust, domain-specific approach for the optimization of simulated chemical reactors to support the next generation of sustainable chemical processes.

In many real-world and simulated engineering systems, differing quality evaluations of quantities of interest exist. Reactor performance can be quantified by a correlation of dimensionless numbers, a CFD simulation, or a pilot-scale experiment. These all attempt to capture the true underlying performance of a system, with differing accuracies and associated costs. Taking the view that only the industrial-scale reactor in its intended setting will provide a true evaluation of performance: any approximation to this, including pilot and lab-scale experiments, simulations, and basic calculations all become valid lower-fidelity evaluations which may be used simultaneously for design and optimization. For CFD simulations of a reactor, fidelities are most often associated with the number of finite element cells in a simulation as they dictate the accuracy and computational cost. By motivating the notion that all predicted, unmeasured quantities derive from a lower-fidelity approximation to a desired high-fidelity function, it becomes pertinent to investigate methodologies that apply these approximations to learn about the true system of interest.

Here we present a framework to rapidly solve this nonlinear, computationally expensive, and derivative-free problem, enabling the fast prototype of novel reactor parameterizations. We take advantage of Gaussian processes to adaptively learn a multi-fidelity model of reactor simulations across a number of different continuous mesh fidelities.The search space of reactor geometries is explored through an amalgam of different, potentially lower, fidelity simulations which are chosen for evaluation based on weighted acquisition function, trading off information gain with cost of simulation. Within our framework we derive a novel criterion for monitoring the progress and dictating the termination of multi-fidelity Bayesian optimization, ensuring a high-fidelity solution is returned before experimental budget is exhausted.

We demonstrate our framework by investigating the optimization of helical-tube reactors under pulsed-flow conditions, which have demonstrated outstanding mixing characteristics, have the potential to be highly parameterized, and are easily manufactured using 3D printing. A helical-tube reactor is parameterized by a coil radius, coil pitch, and inversion. Coil pitch denoted by φ controls how extended the helical tube is, coil radius denoted by ρ controls how tight the coils are within the helical tube, and the inversion parameter is denoted by δ controls the change in coil direction. Inversions within helical-tube reactors have been shown to provide effective mixing properties. δ takes a value between 0 and 1, and specifies where along the coil the inversion takes place. The length of the coil is maintained as fixed, resulting in all parameterized coils having the same volume. Within the parameterization, we include a fixed-length inlet and outlet to the coil.The inlet and outlet are horizontal, and a smooth interpolation is used to ensure that the transition from inlet to coil and coil to outlet, is smooth.

In addition to geometric design and oscillatory parameters, the output of a simulation is also influenced by one or more fidelities. A typical fidelity used within CFD simulations is the number of discrete finite elements that are contained within the mesh, known as cell count. Fluid behaviour within pulsed-flow helical-tube reactors is complex due to the transient formation and unravelling vortex structures affecting axial and radial mixing. Therefore, it is important to capture simulation resolution both axially, through the length of the tube, as well as radially throughout the cross-section of the tube for a high-quality simulation. In this work, as opposed to applying cell count as a single scalar fidelity parameter, we instead maintain the ability to adapt the resolution of a simulation both axially and radially resulting in two independent fidelities. We implement a custom meshing procedure that allows for axial and radial fidelity to be varied independently. The simulation that serves as the highest fidelity corresponds to when both the axial and radial fidelities are greatest. Whilst both axial and radial fidelities are discrete due to their meaning within a finite element context, we treat them as continuous parameters. During mesh creation values of both fidelities are rounded to the nearest integer. Within the solution to CFD problems, a number of solver options may be considered as valid fidelities. However, as many of these take on non-continuous or Boolean values, we leave their integration for future work. Figure 1 demonstrates the effect of δ, ρ and φ on helical coil tube geometry on the final mesh, and Figure 2 demonstrates the impact each fidelity parameter on final mesh.

Figure 3 shows the optimization progress throughout 64 hours. The time to generate, and objective values of initial solutions generated are denoted by using negative iteration and wall-clock time values. Therefore, optimization itself it denoted as starting at iteration 0, and at a wall-clock time of 0. Figure 4 shows the different simulation fidelities evaluated throughout optimisation.

The optimal coil geometry has a pitch of 1.04cm, radius of 1.25cm, and an inversion that occurs 66% along the coil length. The associated optimal operating conditions are pulsed-flow with an amplitude of 1mm, frequency of 2 Hz and a Reynolds number of 50. Figure 4 highlights the specific behaviour of fluid within the optimal reactor throughout a single oscillatory cycle. Streamlines are coloured with tracer concentration to show the movement through the coil. For the positive part of the oscillatory cycle, the tracer moves in the forward direction in a streamlined manner towards the outlet. The important features start to occur at the negative part of the oscillation cycle where the secondary flow starts to emerge. Cross-sectional flow streamlines transitioning to Dean-type vortices due to the centrifugal forces at the coil turns are shown. These counter-rotating Dean vortices promote radial mixing of the tracer with the water medium and close to the walls of the computational domain, no tracer dead zones are left behind. Additionally, along with the reverse flow, the swirling motion developed during the negative oscillation cycle, redirects the flow in the tangential direction which limits the axial dispersion of the tracer. This combination of promoting radial mixing and inhibiting axial mixing results in high plug flow performance. This flow cycle is periodic in nature for the amplitude of 1 mm and frequency of 2 Hz, and the simulation is continued until the tracer leaves the computational domain.

The optimal reactor geometry was exported into an STL file format and modified into a 3D printable model by adding a bounding box to give the part volume. The straight sections at the inlet and outlet were given 8 mm and 10 mm OD tube fittings for connection to the experiment apparatus, and the bounding box was then trimmed to reduce the amount of resin needed for the print. The inlet region was also extended by 20 mm to provide additional development length to ensure the experiment matched as closely as possible the parabolic inlet velocity used in the simulations. The optimal geometry was printed using a FormLabs Form3+ using the Clear V4 resin using the default settings. Post-processing involved washing in IPA for 20 min, drying for 24 hours, and post-curing in a FormCure at 60°C for 30 min. Figure 5 shows the raw STL file, modified geometry, and printed geometry.

In obtaining a robust and explainable solution, so we demonstrate our design framework to be extensible to a broad variety of expensive simulation-based optimization problems, supporting the design of the next generation of highly parameterized chemical reactors.