Machine learning based filtered drag force model

Gas-particle flows in industrial vessels such as circulating and turbulent fluidized beds are inherently unstable and they exhibit structures at various temporal and spatial scales. These flows are commonly studied via Euler-Euler (EE) approach, where the gas and solid phases are modeled as inter-penetrating continua and interactions between phases are accounted for with empirical laws. However, resolving all the structures in large-scale systems via EE simulations is impractical, and it has led to the development of the filtered Euler-Euler approach [1,2].

The filtered equations require constitutive models to account for the consequences of sub-grid inhomogeneities, and among the terms requiring modification, the sub-grid contribution to the drag force term is known to be the most significant [3]. Failure to account for the drag force correction is shown to lead to inaccurate description of the flow features such as bed expansion [3]. In the previous attempts to model the sub-grid contribution of drag force, researchers have used, as markers to estimate the extent of corrections, the filtered solid volume fraction, the filtered slip velocity, the filter size, as well as sub-grid quantities such as the drift velocity and scalar variance of the solid volume fraction [2,3,4].

In the present study, we develop a neural network model for filtered drag correction using additional markers such as the filtered gas pressure gradient, its Laplacian, and the gradients and Laplacian of filtered solid volume fraction. The potential role of these additional markers was recognized by deriving and analyzing a transport equation for the drift velocity, which has previously been shown [3, 4, 5] to be a good marker for modeling the drag correction. Budget analysis of this drift velocity transport equation further revealed that a quasi steady-state assumption, leading to an algebraic model for the drift velocity in terms of the various markers, is adequate for fluidized beds. In our study, we have trained a neural network model for the drift velocity using fine-grid EE simulation results. A priori testing of this neural network model revealed good, with correlation a Pearson correlation coefficient of 0.99. A posteriori test of the filtered model supplemented with this neural network model for drift velocity model confirmed consistency with fine-grid simulation results.