Research Article Flow Channel Boundaries in Silos The critical slip planes at the silo filling state are compared with the flow channel boundary during silo discharge for semi-mass flows. The static critical slip planes are determined by using the dynamic programming method based on the stress field of granular solids stored in silos at the filling state. The flow channel bound- ary is estimated through the finite element analysis of the silo discharge. The re- sults indicate that the critical slip line lies above the flow channel boundary. This characteristic can be attributed to the changeover of major principal stress direc- tions of granular solids from the silo filling to the silo discharge. The analysis de- monstrates that the silo wall friction tends to lift up the critical slip plane and flow boundary. A simple correlation is developed between the positions of critical slip planes and flow boundaries and is experimentally verified. Keywords: Dynamic programming method, Finite element method, Flow channel boundary, Plasticity, Slip plane Received: January 26, 2011; revised: March 16, 2011; accepted: April 06, 2011 DOI: 10.1002/ceat.201100049 1 Introduction In general, there are three main flow patterns in silos, which are categorized into mass flow, semi-mass flow, and internal funnel flow. In the mass flow all particles are in motion. In the semi-mass flow (or mixed flow) and the internal funnel flow (or pipe flow) some granular solids are stationary and there is a flow channel boundary separating the flowing and static granular solids. For the internal funnel flow, the flowing parti- cles do not touch the silo walls. For the semi-mass flow, the flow boundary intersects with the silo wall at the level called the effective transition. The wall pressure at the transitional level is the most critical with regard to the structural stability of the silo. The transitional wall pressure at the onset of flow can be several times the static wall pressure. Identification of the effective transitional level is important for the optimization of silo designs and study of overall silo flow behaviors since pre- defined flowing blocks are assumed in numerous studies [1–6]. The study of silo flow is a challenging task and many numer- ical approaches have been attempted to analyze the silo flow concerning flow boundaries and wall pressure distributions. They include the boundary element method [7], the pure kine- matic approach [8], the double shearing method [9, 10], and the meshless method [11, 12]. Compared with these approach- es, the discrete element method (DEM) and the finite element method (FEM) with elastoplastic constitutive models of gran- ular solids are the two main-streamed approaches. There are both advantages and disadvantages for these two approaches. Disadvantages of the DEM are limitations of the number of particles used and proper characterization of particle proper- ties [13, 14]. On the other hand, the FEM has difficulties in simulations of dynamic flowing processes. Concerning finite element analyses, numerous constitutive models of granular solids have been developed, including the Druck-Prager mod- el, Mohr-Coulomb model, Cam-Clay model, Lade’s model, viscoplastic model, nonlocal polar hypoplastic model, etc. [15–19]. Different finite element computational schemes have also been attempted, consisting of Lagrangian models with remeshing and rezoning, Eulerian models, and arbitrary Lagrangian-Eulerian models. Depending on the computational schemes used, the flow boundaries are predefined, or the dis- placement or velocity at the outlet are prescribed [6, 20]. It should be noted that these different computational schemes and constitutive models of granular materials can lead to con- siderably different predictions even for the same well-defined problems, which represents another difficulty in the use of finite element analysis in silo flow simulations [15]. Among the numerous finite element analyses of silo dis- charge, there is an important and widely used approach, in which flow channel boundaries are assumed a priori. The pre- defined flow boundary divides the granular mass into a flow- ing zone and a stationary zone. Interface elements are used to link these two zones, and the interface frictional angle is the internal frictional angle of granular solids. This scheme can facilitate simulations of granular solid flow at a large scale. In some studies, the inclination angle of flow channel boundary is assumed to be 45° + f/2 to the horizontal direction based Chem. Eng. Technol. 2011, 34, No. 8, 1295–1302 © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.cet-journal.com Yunming Yang 1 Michael Rotter 2 Jin Ooi 2 Yin Wang 2 1 Chinese Academy of Sciences, Institute of Rock and Soil Mechanics, State Key Laboratory of Geomechanics and Geotechnical Engineering, Wuhan, China. 2 University of Edinburgh, School of Engineering, Edinburgh, UK. – Correspondence: Dr. Y. Yang (jamesyang12@gmail.com), Chinese Academy of Sciences, Institute of Rock and Soil Mechanics, State Key Laboratory of Geomechanics and Geotechnical Engineering, Xiaohong- shan, Wuchang, Wuhan 430071, China. Slip plane 1295