Quantitative comparison of experimental and Mohr-Coulomb finite element method simulation flow characteristics from quasi two-dimensional flat-bottomed bins
Graphical abstract
Introduction
Many finite element method (FEM) studies exploring hopper flow of particulate materials are available in the literature [[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]]. These studies differ in terms of the shape and size of the hopper under consideration, complexity of the underlying constitutive model, implemented FEM analysis techniques, and investigated hopper flow characteristics. While some studies are concerned with the cohesive arching phenomenon [3,13,15,22], the majority of these studies focus on the filling or discharging hopper wall stress profiles. The stress profiles are obtained for conical and wedge-shaped symmetric hoppers as well as eccentric hopper systems [1,2,[6], [7], [8], [9],[16], [17], [18], [19],21]. Verification of the FEM wall stress profiles is achieved via comparisons to the approximate analytical theories of Janssen, Walker, and Enstad.
Few FEM studies investigate other hopper flow characteristics such as the velocity profiles [11,20] and mass discharge rates (MDRs) [[10], [11], [12]]. Böhrnsen et al. [20] implemented a hypoplastic model for a wedge-shaped hopper using a plane-strain FEM model and made quantitative comparisons to experimental velocity profiles. Their experimental setup consisted of a large-scale hopper with an outlet velocity controlled by a conveyor belt. The velocity profile was obtained along three horizontal paths at different heights from the outlet and compared with FEM results. The FEM analysis overpredicted the velocity profile at smaller heights, but underpredicted the velocity profiles at larger heights. Zheng and Yu [11] implemented an Eulerian FEM analysis technique using Abaqus [23] and simulated the discharge of particulate material through a conical hopper. The material was represented using the in-built Mohr-Coulomb elasto-plastic constitutive model of Abaqus without additional modifications. The dimensionless average discharge velocity profile obtained for one hopper was qualitatively compared with experiments conducted by Cleaver and Nedderman [24] on a different hopper. The internal friction angle in the model was adjusted within the range of experimentally measured values to give a better match with experimental velocity profiles.
Wang et al. [10] implemented an Arbitrary Lagrangian-Eulerian (ALE) analysis technique within Abaqus and simulated the discharge of a generic particulate material through a wedge-shaped hopper using a linear Drucker-Prager constitutive model. The FEM MDR was shown to match well with the extended Beverloo correlation, i.e., the Beverloo correlation with the modification made by Rose and Tanaka [25]. A similar verification of the hopper MDR was completed by Zheng and Yu [11] in their work. Zheng et al. [12] investigated the effects of different model parameters on the FEM predicted hopper MDR and proposed modifications to the Beverloo correlation.
There are many experimental studies [20,[26], [27], [28], [29], [30], [31]] available in the literature for different hopper geometries, for different discharge conditions, and for different particulate materials; however, a major hurdle in using these experimental data for direct comparisons to FEM predictions is that the experimental studies rarely report the parameters needed for the FEM constitutive model. The present work aims to provide a one-to-one quantitative comparison of FEM and experimental flow characteristics for the same bin geometry. This work is inspired by the experiments of Maiti et al. [30], where particle image velocimetry (PIV) was used to measure the velocity profiles of sand discharging through quasi two-dimensional, flat-bottomed bins with centered (concentric) and off-center (eccentric) exits. A detailed comparison to Maiti et al.'s work was not possible due to the unavailability of some of the model parameters as well as some concern regarding the influence of sand particles sticking to the bin walls. Therefore, PIV experiments on similar laboratory-scale bins were conducted in the present work. After measuring Mohr-Coulomb constitutive law material properties, an Eulerian FEM analysis, implemented in Abaqus/Explicit, was implemented to simulate the complete discharge process. A quantitative comparison of four different hopper flow characteristics: the velocity profiles, the MDR, the duration of steady MDR (TSS), and the free surface profile contours are presented. Comparisons are made for concentric as well as eccentric bins to thoroughly assess the accuracy of the Mohr-Coulomb FEM model.
This paper is organized in the following manner. Section 2 describes the design of the bin geometry and the experimental procedures. Section 3 includes information about the high speed camera and the PIV settings based on the guidelines of Sarno et al. [[32], [33], [34]]. Section 4 outlines the finite element method (FEM) simulation setup and details of the implemented constitutive model. Quantitative comparisons of FEM and experimental flow characteristics are presented in Section 5.
Section snippets
Bin design and experimental setup
The present experimental setup was inspired by the work of Maiti et al. [30], but with slight modifications. The geometry consisted of a quasi two-dimensional, flat-bottomed bin with an exit that could be positioned at different locations at the base of the bin (Fig. 1). In order to reduce the tendency of particles sticking to the bin walls due to electrostatic effects, which has the potential of interfering with PIV measurements, the bin walls were made of tempered glass. Detailed drawings and
High speed camera and PIV settings
The high-speed camera was positioned such that the complete bin length was captured by the camera's square field of view with the maximum possible resolution setting of 1024 × 1024 px. Hence, the spatial resolution of the captured images was 0.168 mm/px or 4.3 px/dparticle. The camera frame rate was chosen such that the PIV random error observed in the extracted velocity profiles was minimized. Table 1 shows the frame rates used to obtain the velocity profiles. The camera shutter speed of
FEM simulation setup
Similar to previous FEM hopper studies [11,12,14], this work implemented the Eulerian analysis technique of Abaqus/6.14–6 [23] to avoid excessive distortion of mesh elements during the discharge process. Fig. 5 shows the simulation setup using a coarse mesh for concentric (Fig. 5a) and eccentric bins (Fig. 5b). An Eulerian material outflow boundary condition at the bin outlet and a void inflow boundary condition at the bin top surface ensured that material discharged freely through the outlet
Quantitative comparison of flow characteristics
Discharge flow characteristics obtained from the experiments are compared with the converged FEM DS and NDNS simulations. Specifically, the steady state velocity profiles, the MDR and TSS, and the free surface profiles during discharge are compared for both bins. At least three experimental trials were performed with approximately the same initial material fill height of 200 mm to demonstrate the reproducibility of the experimental data.
Conclusions
The velocity profiles, mass discharge rates (MDRs), durations of steady MDR (TSS), and free surface profiles predicted by Eulerian FEM simulations using a Mohr-Coulomb material model are compared with experiments performed on laboratory-scale concentric and eccentric bins. Two extreme cases of no material dilation and no softening (FEM NDNS) and maximum dilation with softening (FEM DS) are also compared.
There is a negligible difference in the velocity profiles predicted by the FEM DS and FEM
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The authors would like to acknowledge the financial support provided by Genentech, Inc., United States (Grant ID CLL017900) for a portion of this work.
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