Dependence of wall stress ratio on wall friction coefficient during the discharging of a 3D rectangular hopper
Graphical abstract
Introduction
Hoppers are widely used in industry for protection, storage and delivery of solids, and frequently have cylindrical [1], [2] or rectangular shapes [3], [4], [5]. Compared to cylindrical hoppers, rectangular hoppers are more attractive in view of efficient space utilization and low fabrication cost [4]. However, the flow behavior of the solids in rectangular hopper is more complicated because of the geometric asymmetry [4]. When designing hoppers, safe operation is the most important issue to be considered in charging and discharging. Improperly designed hoppers may collapse while discharging, leading to disastrous consequences.
The wall normal stress is a critical variable for safe hopper design. It is related to particle and hopper properties, which is described in the Janssen equation [6], [7]:where Pw is the wall normal stress, k is the wall stress ratio and Dh is the hydraulic diameter of rectangular hopper. As demonstrated by the Janssen equation, wall normal stress is scale dependent. Therefore, the wall normal stress obtained in small hoppers cannot be used for the design of large scale hoppers [3], [8], [9]. Compared to wall normal stress, wall stress ratio, a dimensionless parameter calculated by dividing the wall normal stress by the vertical normal stress, is less scale dependent, especially for hoppers with sufficiently large sizes. From the wall stress ratio, the wall normal stress can be calculated by Janssen's equation. Hence the wall stress ratio is an important parameter for hopper design.
For the prediction of wall stress ratio, Balevičius et al. have proposed a model with assumption of negligible effect of wall friction [7]. However, because the wall friction changes the stress distribution in the granular material near the wall, it changes the wall stress ratio [10]. Walker's model [7], which relates the wall stress ratio to the macroscopic wall friction angle and effective internal friction angle, has theoretically integrated the effect of wall friction, indicating that the wall stress ratio changes monotonically with wall friction coefficient. While the macroscopic wall friction angle is related to the microscopic wall friction coefficients, the effective internal friction angle, which is based on the assumption of critical equilibrium state that bulk solid tends to slip, is independent of the microscopic wall friction coefficients [7], [11]. This makes the prediction inaccurate because the solids are seldom in critical equilibrium state before entering into the converging flow zone of hoppers [12]. Moreover, the stress ratio of one wall may also be influenced by the friction coefficient of its adjacent walls, because the load transmitted to one wall could be influenced by its adjacent walls [13]. However, this problem has not yet been addressed.
Recently, Discrete Element Method (DEM) is frequently employed for an in-depth understanding of solid flow behaviors because the simulation based on DEM is reliable and includes dynamic behaviors. A good fit between simulation with DEM and experiments was observed for the wall normal stress during charging and discharging of a rectangular hopper [7]. The arch structure in the force network shown by DEM simulation [14] was reproduced by experiments [15], [16], and the exponential decay of the normal force distribution indicted by DEM simulation was consistent with the experiments [17]. On the basis of the transient behaviors of the force network, Masson and Martinez related the wall normal stress with the particle properties [18], and Zhu and Yu provided an outline of the flow regions in hopper discharging [1].
The multiscale information from DEM also makes it possible to understand how wall fiction coefficient affects wall stress ratio. Masson and Martinez determined the wall stress ratios in a 2D hopper when the internal friction coefficient was 1 and the wall friction coefficients were 0.5 and 1 [18]. Landry et al. determined the wall stress ratios in a 3D cylinder hopper when the internal friction coefficient was 0.5 and the wall friction coefficients were 0.5 and 2 [19]. Both of these studies show significant changes of the wall stress ratio with the wall friction coefficient. Therefore, for a reliable prediction of wall stress ratio, wall friction coefficient has to be considered. However, the relation between the wall friction coefficient and the wall stress ratio has not been established, and the mechanism how the wall friction coefficient affects the wall stress ratio has not been revealed.
In this work, we focus on the effect of the wall friction coefficient (i.e. side wall friction coefficient and face wall friction coefficient) on the wall stress ratio in discharging a 3D rectangular hopper. A possible mechanism is proposed by analyzing the force networks in the hoppers with different wall friction coefficients. Finally, the effective internal friction angle for the calculation of wall stress ratio is modified to relate to the wall and internal friction coefficients. This avoids the use of the critical equilibrium state assumption and makes the prediction more reliable.
Section snippets
DEM simulation
The rectangular hopper and particle system under investigation are shown in Fig. 1 with the hopper geometries and particle properties [20] listed in Table 1. The hopper heights are 0.54 m and 0.62 m for the hoppers with widths of 0.10 m and 0.15 m, respectively, and the particle number changes between 12,800 and 22,100 accordingly. To understand how the stress ratio of one wall depends on the friction coefficients of its adjacent walls, we have considered the conditions that the friction
DEM validation
To validate the reliability of our DEM code, particle discharging from a batch flat-bottomed hopper, which has been studied by Balevičius et al. both numerically and experimentally [7], is simulated. Almost all the simulation conditions we adopted here (as seen from Table S1 in the Supporting information) are identical to those used by Balevičius et al. [7], except the particle size distribution and the spring and damping coefficients, which have minimal effects on overall stress distribution
Conclusions
The effect of wall friction coefficient on wall stress ratio during discharging of a 3D rectangular hopper has been studied by DEM. Under the investigated conditions, the side wall stress ratio first increases and then slightly decreases, instead of changing monotonically with the increase of side wall friction coefficient. Meanwhile, the face wall stress ratio first increases slightly and then remains almost constant. With the increase of the face wall friction coefficient, the side wall
Acknowledgments
This work is financially supported by the Major State Basic Research Development Program (2012CB720501).
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