Recently, mechanochemical processes have emerged as a promising and sustainable technology for reactive milling of solid feedstocks, since transient impact areas of high dissipated energies and elevated local temperatures can obviate the need for the costly bulk heating and large solvent volumes associated with many traditional chemical processes. Among a growing list of application areas, this class of reactors has displayed its potential in depolymerizing polystyrene [4], poly(ethylene terephthalate) [5], [6], and poly(methyl methacrylate) [7] for plastics recycling applications, synthesizing carbon fibers and other commercial goods from lignin feeds [8], and even providing a mild-conditioned ammonia synthesis pathway [9]. In order for mechanochemical reactors to approach industrial maturity, the development of mechanistically informed PBMs to predict process outcomes and product properties remains an indispensable tool towards the design, integration and adoption of this technology.
The modeling of reactive milling systems is uniquely challenging since there is mechanistic and kinetic uncertainty in the mechanical collisions that form reactive hot spots, and in the molecular reactions that take place in them. This work considers a case study of the mechanochemical depolymerization of polystyrene in a lab-scale vibratory reactor that exhibits experimental molecular weight distributions (MWDs), which are notably distinct from those characterized in traditional depolymerization literature [4]. To aid in the development of the mechanochemical PBM kernels, we develop a model identification routine based on candidate kinetic pathways and mechanochemical assumptions, which are encoded into the PBM kernels and model parameters. The PBM kernels parameterize the reaction rates and product distributions of discretized molecular weight classes, with the capacity to additionally represent nonlinear particulate interactions such as reaction inhibition [10]. However, the consideration of nonlinear interactions requires each molecular weight reactivity to become dependent on the mass fractions of all other molecular weights, thereby introducing an additional nested summation into the PBM ODE system. We show the necessary quantity of PBM ODEs to be on the order of 50-100 to sufficiently model the MWD process data across standard milling conditions, and at this limit, parameter estimation of nonlinear PBM kernels becomes computationally challenging with traditional methods. Thus, we develop a physics-informed neural network (PINN) approach to PBM parameter estimation for multi-parameter systems with limited data availability and several candidate PBM kernels, extending initial literature implementations of PINNs for non-reactive, linear PBMs [11]. Once trained on a subset of experimental data, the PINN can reduce the computational expense of parameter estimation by estimating ODE evaluations as backpropagated values [12], thereby reducing reliance on repeated forward solutions of the PBM at varied parameter sets.
We compare the computational and data requirements of the novel PINN approach to stochastic and deterministic global optimization routines involving repeated forward ODE simulations at varied parameter sets, as well as form-specific methods developed for PBMs. Specifically, we identify and map the most appropriate parameter estimation methods for varying levels of PBM complexity, model uncertainty, and data availability, defining clearly the use cases of each methodology as it relates to mechanochemical and general particulate processes.
In conclusion, the novelty of this work is multifaceted – first, we unify milling and depolymerization kinetic literature by developing a mechanistic PBM that models polystyrene’s evolving MWD during reactive milling. Second, to aid in the model identification and parameter estimation of the PBM, we develop a PINN approach that encodes the physical constraints of mechanochemical kinetic assumptions into a computationally inexpensive surrogate model, which can consider a wide range of potential mechanistic PBM forms. Lastly, we provide a comprehensive comparison of the PBM-PINN approach towards model identification and parameter estimation for particulate process assumptions of varying complexity to map the types of systems that benefit most from this ML-based approach.
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