(430d) A Data-Driven Framework for the Design of Reactor Simulations: Exploiting Multiple Continuous Fidelities

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. 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. 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. Multi-fidelity Bayesian optimization algorithms have been developed and applied across a number of domains. All approaches all have a common characteristic in that both decision variables x, and fidelity parameters z, need to be selected at each iteration. How these decisions are made, with respect to the trade-off between information gained, and expense incurred, are key aspects of multi-fidelity Bayesian optimization algorithms.

Here we present a framework to rapidly solve this nonlinear, computationally expensive, and derivative-free problem, enabling the fast prototype of novel reactor parameterizations. Table 1 demonstrates the merits of our approach. In addition to dependency on simulation fidelity, the cost of a simulation is also dependent on inputs x in chemical reactor simulation domains. Variable step-size CFD solvers, along with early stopping criteria both contribute to how x can influence simulation cost, particularly in cases where operating conditions are optimized over. As such, two-step approaches cannot be applied, where x is selected first, and then z, as the second step where information and cost is traded off becomes dependent on x. Therefore, we follow an approach utilizing a cost-adjusted acquisition function to trade off information gain and cost in a single step. To do so, the optimization objective and cost of simulation are modeled by two separate Gaussian processes. A single optimization step consists of solving the single cost-adjusted acquisition function to obtain both the next set of inputs and next simulation fidelity, to maximise the weighted expected information gain.

In traditional Bayesian optimization, function evaluations are terminated when the computational budget is exhausted, and the evaluated data point with the highest objective is selected as a final solution. In a multi-fidelity framework, this criterion contains subtle nuances. As the underlying function of interest is where the fidelity is the ‘highest’, the optimal solution returned should be the highest evaluated function-value evaluated at the highest fidelity. Throughout optimization the underlying high-fidelity function is maximized through lower fidelity simulations, with no guarantee that simulations at the highest fidelity will be selected to be evaluated. Therefore, 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. Algorithm 1 demonstrates our framework.

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.

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 shows the optimization progress throughout the 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 2 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 3 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. The optimal geometry is 3D printed and evaluated using associated optimal pulsed-flow operating conditions, validating both the performance of the framework as well as our multi-fidelity model. We investigate the effect of hyper-parameters within our framework, demonstrating their effect of convergence.

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.