Reduced-order modeling of particle-fluid flows with heat transfer via a curriculum learning approach

Reduced-order models (ROM) exhibit promising potential for accelerating large-scale numerical simulations. Many studies have developed ROMs of particle-fluid flow systems, but those that also include heat transfer phenomena have only been used to forecast temperatures over time. In this work, we developed non-intrusive proper orthogonal decomposition (POD)-based ROMs for forecasting and parametrizing the fluid and particle temperatures in a cooling fluidized bed. A physics-based full-order model (FOM) of the fluidized bed is set up using the coupled computational fluid dynamics and discrete element method (CFD-DEM) and an interphase heat transfer model. Afterward, snapshots of the FOM are decomposed into modes and coefficients via Lanczos-based POD. The number of modes in the ROMs are determined via energy truncation, while the coefficients are predicted by long-short term memory (LSTM) and Bahdanau attention-based neural networks via recursive single-step forecasting in both temporal and parametric spaces. Neural network training is accomplished via a curriculum learning approach, and ablation tests showed that this is necessary for learning the nonlinear characteristics of the high-order modal coefficients associated with particle-fluid dynamics. The results show that both the time forecasting and parametric ROMs can accurately predict temperature fluctuations and macroscopic particle-fluid dynamics while reducing computational costs by several orders of magnitude compared to the FOM. The POD-based ROM in this work is the first to parametrize the temperatures from a CFD-DEM simulation, which is a substantial contribution to the modeling and simulation of heat transfer systems for chemical engineering applications.