Space fractional kinetic model for different types of suspension profiles in turbulent flows with a neural network-based estimation of fractional orders

Abstract

Estimation of particle concentration distribution profile is important to understand the inherent mixing process, changes of river morphology, and nature of transportation of particles. In this study, the effects of non-locality are investigated to analyze, distinguish and predict different types of profiles of the suspended concentration distribution in turbulent flows through open channels, pipes, and mixing boxes. The effects of non-locality are captured through fractional derivatives and integrals. Starting from the fractional Liouville equation and considering the power-law jumps of particles in the carrier fluid, the fractional kinetic equation is derived through mathematical analysis. The kinetic model is solved and an analytical model is proposed using the Laplace transformation. The proposed solution contains the two parameters (as α the order of fractional model and μ the shape parameter) Mittag-Leffler function and is a more general one that contains several previous models as special cases. The proposed model can predict four different types of suspension concentration distributions namely type-I, type-II, plume-like, and convex type profiles under different conditions. Further, the model is validated with existing experimental data of dilute and dense flow in open channels and pipes, flows in square pipes, slurry pipelines, and fluid mud gravity currents. Satisfactory results are obtained. A detailed non-linear multivariate regression analysis is carried out with a wide range of selected data and models are proposed to compute model parmeters. To increase the efficiency of the model, an artificial neural network (ANN) model is proposed to predict the model parameters. The ANN-based models show better results than the regression analysis. Validation results show that non-locality has significant effects on the structure of the concentration profiles. From the analysis of the model parameters, it is found that type-I profile occurs under the subdiffusion process where α⩽1 and μ=1 and other three types (type-II, plume-like and convex) of profiles are associated with the superdiffusion process when α>1 and μ>1.