Microwave-assisted process intensification is a promising technique hindered by the unavailability of design rules for optimization and scale-up. High-throughput computational models allowing the exploration of a vast design space are imperative for the rapid development and deployment of intensified microwave reactors. Recently, structured reactors, especially monoliths, are emerging as a canonical multiphase reactor setup in the microwave-heating community. The vast separation of scales in these reactors leads to a large number of mesh elements, making the simulations time-consuming and challenging to converge. To this end, we employ volume and asymptotic averaging to represent monoliths, a multiphase system consisting of a fluid and a solid phase, as a continuum or effective medium. We rigorously verify the adequacy of the averaging techniques to predict the spatiotemporal distribution of the electric field, electromagnetic power dissipation, and temperature against fully resolved monolith simulations. The developed continuum model can replicate the three-dimensional transient behavior obtained from the multiphase simulations with an order of magnitude lower computational expense. Moreover, the continuum model allows easier mesh generation and convergence of numerical solution than the multiphase model. The developed multiscale framework can be used to simulate microwave-heated monoliths and other multiphase reactors, such as packed beds and open-cell foams.
Catalytic monoliths are being explored in conventional catalytic processes for their ability to achieve process intensification. Scientific computing can play an essential role in this exploration. The high computational cost of the first-principles models has led to modeling these reactors as a porous medium. However, this modeling strategy is not performed in a mathematically rigorous manner. We use the volume averaging technique as a mathematical framework to convert the pointwise governing equations for a monolith into averaged equations for a porous domain. These averaged equations require the closure of several unclosed terms, which are neglected in the classical porous medium (CPM) assumption. We discuss these unclosed terms and their impact on model predictions. We show that except in the limit of negligible Damköhler number the treatment of the catalytic reactions in CPM leads to significant errors. We propose a technique to accurately calculate the catalytic reaction rates in the volume-averaging-based porous medium (VAPM) model developed here. This technique is valid for a wide range of Damköhler numbers for both linear and nonlinear kinetics. Moreover, to calculate the effective properties of the porous medium, such as thermal conductivity, we employ asymptotic averaging (numerical homogenization) that can be used for any arbitrary channel shape and size. Predictions of the proposed VAPM model are assessed against three-dimensional multichannel monolith simulations, resolving the solid and fluid phases, for elementary and complex kinetics. In addition, VAPM is validated against the experiments of steam methane reforming in a catalytic monolith. The developed methodology reduces the computational cost by 3 orders of magnitude while maintaining the accuracy of the detailed multichannel simulations.
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