Please cite this article in press as: Golshan, S., et al. A hybrid deterministic–stochastic model for spouted beds. Particuology (2018), https://doi.org/10.1016/j.partic.2018.05.005 ARTICLE IN PRESS G Model PARTIC-1166; No. of Pages 10 Particuology xxx (2018) xxx–xxx Contents lists available at ScienceDirect Particuology journal homepage: www.elsevier.com/locate/partic A hybrid deterministic–stochastic model for spouted beds Shahab Golshan, Reza Zarghami ∗ , Navid Mostoufi Process Design and Simulation Research Centre, School of Chemical Engineering, College of Engineering, University of Tehran, Tehran, Iran a r t i c l e i n f o Article history: Received 18 November 2017 Received in revised form 15 May 2018 Accepted 21 May 2018 Available online xxx Keywords: CFD–DEM Monte Carlo Spouted bed Hydrodynamics Solid circulation rate a b s t r a c t A new hybrid deterministic–stochastic model is developed and used to simulate a slot-rectangular spouted bed. The model includes deterministic and stochastic steps that are executed in turn. The simula- tion starts with the deterministic part of the model, in which the computational fluid dynamics–discrete element method (CFD–DEM) equations are solved for 1 s to give the initial velocity distribution of the particles. The stochastic part is then executed, with the hydrodynamics of the bed taken from the velocity distributions acquired in the first step through Monte Carlo sampling. A full deterministic (CFD–DEM) simulation of the bed is also conducted for comparison with the proposed hybrid model. Additionally, the proposed hybrid is validated using experimental data from the literature. These validations are based on the axial and lateral velocity distributions of the particles and the bed voidage. The effects of the cell size and number of sampling steps on the accuracy of the model are also investigated. The performance of the proposed model is compared with the CFD–DEM results in terms of the computation time and the rate of solid circulation in the bed. The hybrid model is found to have shorter runtimes than the CFD–DEM approach. © 2018 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved. Introduction Spouted beds were first proposed by Mathur and Gishler (1955) as an alternative to fluidized beds when handling coarse parti- cles. Spouted beds exhibit special flow characteristics that make them highly efficient in terms of heat and mass transfer (Dabhade, Saidutta, & Murthy, 2008; Du, Bao, Xu, & Wei, 2006a, 2006b; Esmailpour, Mostoufi, & Zarghami, 2018; Rahimi & Azizi, 2011), and so these beds are convenient for investigating the contact between gas and solids in many applications, such as gasification, drying, chemical vapor deposition, coating, combustion, and gran- ulation (Golshan, Zarghami, & Mostoufi, 2017; Golshan, Zarghami, Mostoufi, Koksal, & Kulah, 2016). In recent years, spouted beds have also been used to examine the pyrolysis of different types of wastes, such as biomass and plastics (Alvarez et al., 2015; Arabiourrutia, Elordi, Olazar, & Bilbao, 2017). Because of these advantages, several experimental and modeling studies have been conducted to inves- tigate the hydrodynamics of spouted beds (Golshan et al., 2018). The increasing capabilities of computational facilities have led to an increase in the number of simulation studies on the hydrodynamics of spouted and fluidized beds (Loha, ∗ Corresponding author. E-mail address: rzarghami@ut.ac.ir (R. Zarghami). Chattopadhyay, & Chatterjee, 2013). Generally, two different sim- ulation approaches are used: the Eulerian–Eulerian approach (two-fluid model, TFM) (Du et al., 2006a, 2006b; Golshan, Esgandari, & Zarghami, 2017; Huilin et al., 2004; Lan, Xu, Gao, & Al-Dahhan, 2012; Loha et al., 2013; Zhou, Gao, Xu, & Lan, 2013) or the Eulerian–Lagrangian approach (computational fluid dynamics–discrete element method, CFD–DEM) (Alobaid & Epple, 2013; Deb & Tafti, 2014; Golshan, Zarghami, & Mostoufi, 2017; Golshan et al., 2016; Goniva, Kloss, Deen, Kuipers, & Pirker, 2012; He et al., 2012; Limtrakul, Boonsrirat, & Vatanatham, 2004; Ren et al., 2012; Takeuchi, Wang, & Rhodes, 2004; Wu, Ouyang, Yang, Li, & Wang, 2012; Zhao, Li, Liu, Song, & Yao, 2008; Zhao, Li, Liu, Yao, & Marshall, 2008). In the TFM approach, the solid and gas phases are treated as interpenetrating continua for which the momentum and mass conservation equations are written separately, and inter- phase forces (lift, drag, and virtual mass) are used to couple the two phases (Loha et al., 2013). In CFD–DEM simulations, the solid phase is treated as a discrete phase and the equation of motion is solved for all particles in each time step, with the Navier–Stokes equation used to describe the motion of the gas phase. In CFD–DEM, cou- pling is accomplished by transferring the drag force and voidage distributions between the phases (Golshan, Zarghami, & Mostoufi, 2017). Although simulations using CFD–DEM provide accurate pre- dictions of the hydrodynamics of gas–solid contactors, the main https://doi.org/10.1016/j.partic.2018.05.005 1674-2001/© 2018 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.