Elsevier

Powder Technology

Volume 301, November 2016, Pages 186-196
Powder Technology

Acceleration of kinetic Monte Carlo simulation of particle breakage process during grinding with controlled accuracy

https://doi.org/10.1016/j.powtec.2016.05.059Get rights and content

Highlights

  • Breakage process simulation is accelerated by reducing the system size dynamically.

  • A feedback approach is used to control simulation accuracy under a given threshold.

  • Simulation accuracy is analytically expected on the simulation run-time.

Abstract

An acceleration algorithm for the kinetic Monte Carlo simulation of breakage process during grinding is introduced. We show that a feedback approach can be used to accelerate simulation as much as possible while control the simulation accuracy under a threshold. This is implemented by introducing a quantitative measurement of simulation accuracy and an estimation method during simulation run-time. The analysis is supported by numerical results showing significant acceleration and accuracy being well controlled below the given threshold.

Introduction

In mineral processing industry, ore grinding is the fine phase in the process of comminution after coarse phase of size reduction such as crushing. In this stage, the sizes of ore particles are further reduced in grinding mills. Grinding operation is expensive as the energy and grinding media (such as steel balls, rods or the ore rock itself) consumptions are extremely high [1]. Optimizing the comminution design and operation requires intensive knowledge of the comminution process, which is traditionally obtained by carrying out experiments on a prototype or real process. Yet, the cost and time of building prototype process and experiments on field turn out to become the most serious obstacle to the development and application of new optimization techniques [2]. This is because conducting experiments on operating processes may cause off-quality product and improperly set experiments may even damage process equipment.

Simulation can provide a cheaper and safer alternative to conducting experiments on field. Quantitative simulation with verified model is useful for many purposes [3]. It can be used to evaluate the design of comminution process and validate operational optimization strategy [4], [5]. For grinding process, particle size distribution (PSD) is most of interest. The purpose of the grinding process is to control PSD within an economic optimum particle size range [6], [7], [8]. Simulation can be used to evaluate whether the comminution design can reduce ore to the desired PSD range [3] and the process control strategy can stabilize the product PSD [9], [10], [4]. Since PSD is difficult to measure online, simulation of the time evolution of PSD can be used as a predictive model for the control of PSD [11], [8], [12], [13].

Accurate simulation of the PSD dynamics during grinding is challenging due to the highly complex nature of particulate processes. Grinding process involves a large population of particles in a solid phase or dispersed in a liquid phase. Also, the system dynamics is often described by stochastic equations because it is not practical to observe and trace the exact state of each particle. The time evolution of PSD is well modeled by the population balance equation (PBE) [14], [15], which describes the growth, birth and death processes of the particle population in the mill. Unfortunately, the analytical solutions to PBE of strict form are available only for some of the simplest cases [16], [17], [18]. In many cases, it needs to resort to numerical techniques, e.g. the method of moments [19], the sectional methods [20], [21] and the kinetic Monte Carlo (KMC) method [22], [23], [17], [24], [25], etc.

The KMC method refers to a kind of Monte Carlo algorithm for evolving systems dynamically from state to state [26]. For the analysis of particulate system PBE, the KMC method is probably the most widely used because of its flexibility and the inherently stochastic nature [27], [28], [24], [25], [22]. The high computational cost is the primary issue in applying KMC method to grinding simulation. The computational cost of KMC is proportional to the population size of the grinding system [29]. Therefore, with the raw KMC method, e.g. the constant volume KMC method (CV-KMC) [22], the simulation will slow down quickly with a growing number of particles in a unit volume of particle population due to particle breakage. A solution to this problem is to constrain the growth of population size during simulation. Such an approach can accelerate the KMC computation but it will sacrifice accuracy which is in reverse proportional to the population size. One representable example of such approach is the constant number KMC (CN-KMC) [30], [31], [32], where the population size of the simulated system is intentionally kept constant. However, it can be shown that the accuracy of the CN-KMC approach is constantly decreasing during the simulation.

Since the establishment of KMC method as a powerful numerical simulation tool in physics, chemistry, biology, materials science, etc., the heavy computational demand of KMC method has also become one of the major challenges [33]. Therefore, improving the computational efficiency has always been at the heart of KMC simulation research [34]. One key approach to simulation acceleration is based on the idea of time scale separation [35]. In most physical and chemical systems, there coexist in the KMC simulation both slow processes with relatively very small transition rates and fast processes with much larger transition rates than the slow processes. The KMC simulation of such system exhibits stiffness because too much computational resources will be consumed by the fast processes. The scale separation approach attempts to overcome the problem by partitioning the system into “fast” and “slow” sections, and approximating out the effects of the fast processes, e.g. in [36], [37], [38]. The problem of stiffness also exits in the KMC simulation of breakage process, where the particles of smaller size classes have much larger transition probabilities because of their large number. However, scale separation cannot be readily applied to breakage process simulation because the particle sizes are nearly continuously distributed over the entire size range and it is difficult to draw a clear line to distinguish the “fast” and “slow” populations. Another important acceleration method is the τ-leap procedure [39], [40], [41]. It is a coarse grained approach to the acceleration of KMC simulation by executing many events in each time step τ. The value of τ must satisfy the leap condition, i.e. the propensity function should not change by a significant amount. The τ-leap method provides a user-specified parameter ε for accuracy/speed control. Application of the τ-leap method to the KMC simulation of particle breakage process was reported in [42], where the simulation was accelerated by more than 200 times compared to the raw KMC method. However, this was achieved under a relatively large value of ε which indicates a significant decrease in simulation accuracy.

Successful application of KMC method to particulate system simulation should ensure that the computational cost is reasonably low for practical use while keeping accuracy under a certain threshold. In this paper, we develop a feedback approach to control both the computational cost and the simulation accuracy so that the most acceleration can be achieved under a given accuracy threshold. The basic idea is to adjust the system size so as to accelerate and let the accuracy below a threshold. There are two keys to achieve this goal: 1) a dynamic estimator of the accuracy during simulation running; 2) means to manipulating the accuracy.

This paper begins with a brief background introduction to the KMC simulation methods. Next, the main problems of the related approaches are outlined. Then, the main results are presented. Those establish a new solution to the problem of acceleration with accuracy control. Finally, numerical results are presented to support the analysis, and the main contributions are outlined.

Section snippets

Assumptions

This work focuses on the simulation of PSD transitions in mill system. We assume batch grinding in a perfect mixing mill, which means that the mill is charged with material and operated without continuous discharge and feed, and particles of different sizes are mixed perfectly in the mill. The batch grinding model is useful in that it is the basis for PSD prediction in control and optimization.

For the scheduling mechanism of simulation, we assume an event-driven approach, which means that in

Description of problems

It is computational demanding to simulate particle breakage process using the KMC approach when the system scale is large [22]. The main challenge is to speed up the simulation while maintaining the simulation accuracy. Difficulties arise because these two factors often contradict. This paper addresses this problem.

KMC with accuracy control

We consider the task of speeding up the simulation of breakage process during grinding, and in the meantime control the accuracy to a desired level. Our goal is to accelerate the simulation as much as allowed by the desired accuracy. The key idea is to estimate the simulation accuracy and use it to dynamically adjust the system size so as to keep the accuracy within the desired threshold. This idea is illustrated in Fig. 8, in which the ED-KMC simulation accuracy is controlled by a closed loop

Numerical results

Simulation experiments are carried out with the configuration listed in Appendix. The CV-KMC, CN-KMC and the proposed ED-KMC-AC algorithms are compared in terms of the speed and accuracy. Note that each simulation is repeated 10 times and then the presented results are the average over the repeated outputs.

Discussions and conclusions

In this paper, we have shown that a feedback approach, using an accuracy measure in terms of coefficient of variation, can control the accuracy of event-driven Kinetic Monte Carlo simulation of particle breakage process during grinding. The key idea of the proposed ED-KMC-AC algorithm is to accelerate the simulation by reducing the system size, i.e. the number of particles, with a reference to a given accuracy threshold and to adjust the number of removal particles in order to keep the accuracy

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    This work is supported by the Natural Science Foundation of China under Grant 61473071 and the Fundamental Research Project of Key Laboratory of Liaoning Provincial Education Department under Grant LZ2015033.

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