(422b) A New Hybrid One-Dimensional Particle Attrition Model for Conveying Systems
The attrition of particles during pneumatic conveying has been investigated experimentally in a large number of studies [1–5]. However, all are limited by the fact that the only way to experimentally examine particle attrition for a specific system is to compare the particle size distribution at the inlet to that at the outlet. Since particle attrition is governed by a vast number of parameters, it is extremely difficult to empirically correlate the particle size distribution at the end of the pipeline with that at the inlet, even for a large data set. For this reason, in the design of a new conveying line, there are no definitive 'rules' for obtaining the optimum operating conditions (pressure drop, air and particle mass flow rates, pipe diameter, number of bends, etc.) that will give the desired particle size distribution at the pipe outlet. One way to overcome this problem is to apply computational methods to predict the outcome product, but due to the complexity of the available models and the need for expensive computational resources, only a few researchers have implemented particle breakage in their simulations.
In the discrete element method (DEM) approach, particle breakage can be simulated by one of three different models . Nonetheless, of these, only the ‘fragments spawning’ model of Brosh et al.  has been implemented to examine particle breakage in pneumatic conveying . In this model, computational fluid dynamics (CFD)-DEM simulations are combined with the four following cumulative empirical comminution functions, which define the characteristics of the particles as a function of external loading: 1) the strength distribution to describe the compression force that would cause a particle to break ; 2) the breakage function, which is produced by horizontal impact system, to describe the sizes of the fragments after a breakage event ; 3) the equivalence function to describe the relationship between the impact velocity and the equivalent compression load that would cause the same breakage probability as that if the particles had collided with a particular impact velocity ; and 4) the fatigue function  to take into account the weakening of the particle strength due to repeated loading. By implementing these functions within DEM simulations it becomes possible to determine whether a discrete particle will or will not break in a particular collision. Therefore, in this method, the fatigue and breakage for every discrete particle are calculated within the 3D domain. This tool, while effective, has the major drawback of a high computational cost, since the number of particles increases as the particles break into finer segments.
An alternate approach is to adopt a macroscopic model for two-phase flow and to simultaneously predict particle breakage, albeit with reduced accuracy but with the advantage of far less computation time. Such a degradation model was proposed by Chapelle et al.  and implemented on a large-scale dilute-phase pneumatic conveying line . In this degradation model, a breakage matrix was derived from single impact tests and then implemented on every impact event in the simulation. The zero-dimensional flow model used by Chapelle et al.  was based the assumption that all particles travel at the same velocity (and therefore impact at the same velocity). Furthermore, the collisions were assumed to take place perpendicular to the pipe wall and only inside bends, thereby neglecting particle-particle collisions, collisions in straight pipe sections, and fatigue phenomena. There is thus a need for a model that will address these drawbacks.
In the present study, a new method is presented for calculating particle attrition due to impact events under steady-state flow conditions in conveying pipeline systems. The model consists of two parts: a one-dimensional two-phase flow model for calculating the fluid dynamics , and the one-dimensional breakage algorithm (ODBA) developed in this study for determining particle collisions and breakage with respect to the flow field.
In the ODBA, the methodology of Kalman et al.  is adopted for combining probability comminution functions (strength distribution, breakage, equivalence and fatigue functions) as the particle characteristic tool with the dynamics of the particles inside the system, namely, the ‘system behavior tool.’ The system behavior tool in the previously mentioned fragments spawning model is a 3D CFD-DEM simulation that describes the explicit motion of every discrete particle in a 3D domain. In contrast, the ODBA utilizes ‘machine functions’ as the system behavior tool. The machine functions are the collision frequency function, which describes the distribution of time between each collision, and the collision velocity function, which defines the distribution of the collision impact velocity. These functions must be known or developed prior to applying the ODBA.
The impact velocity function is used as the input for the comminution functions that determine whether a particle will break or weaken, and the collision frequency function is used to locate the positions of the collisions. These functions were developed in the present study for a particular experimental system  for investigating dilute phase pneumatic conveying of potash particles at several superficial gas velocities. The new method facilitates the calculation of the particle size distribution at any location along the axial direction of the pipeline.
In CFD-DEM breakage simulations, every simulated case provides a standalone result for the particle size distribution, which is accurate only for those specific simulation conditions. In principle, this approach facilitates a better classification of the attrition phenomena than the ODBA, since it can characterize every particle's collision event in a 3D domain. Nevertheless, it is impossible to apply this method in practice for long-range conveying systems and for process optimization because of the high computational cost. To overcome this drawback, the ODBA was developed as a fast calculation method for predicting attrition in conveying pipelines. Note that for developing the machine functions required for the ODBA, CFD-DEM simulations were conducted without taking particle breakage into account, which speeds up the simulation time in comparison with full CFD-DEM breakage simulations. Thus, in the present study, a set of CFD-DEM simulations was conducted for different conditions, the collision data was recorded, and correlations for the velocity and collisions frequency distributions were developed. In addition, the newly developed model was validated by comparing the model predictions with experimental results for conveying potash .
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