A skew normal-Laplace model of partition curve based on probability characteristics

Highlights

• Probability characteristics of the partition curve were analyzed.

• The differential partition curve is asymmetric and fat-tailed.

• A skew normal-Laplace model with three parameters was developed.

• Accuracy and ability of this model were validated by comparison with other models.

• The parameters transformation law when the separation density changed was provided.

Abstract

The mathematical model of partition curve is fundamental to evaluate and predict the gravity separation performance in coal cleaning industry. Existing models have achieved good fitting accuracy but lack explicit physical meanings. From the perspective of probability and error distribution, the common characteristics of actual partition curve and its differential curve were analyzed according to the generalized central limit theorem. A skew normal-Laplace (SNL) model with three parameters was established based on asymmetry and fat tail characteristics. Parameters in the SNL model represent location, scale and skewness of partition curve, which reflect separation density and error distribution, therefore it is convenient to evaluate the efficiency of the whole process. Compared with the existing models, the SNL model achieved smaller deviation with fewer parameters and conformed better to physical reality. For practical application, the parameters transformation law when separation density changed was provided according to their properties.

Introduction

Gravity separation is one of the main methods of mineral processing and relies on different densities of minerals. The performance of gravity separation is depicted by partition curve, which shows the relationship between average density and partition coefficient. Partition coefficient is the percentage of each density fraction distributed in reject, and the density where partition coefficient equals 50% is often regarded as separation density, which represents the cut point where a separation has taken place. Partition curve and its mathematical model are fundamental to evaluate and predict the gravity separation performance. They are also the bases of coal cleaning plant design. Mathematical models such as Whiten equation (WE) (Wood, 1990), Normal Integral (NI) model (Terra, 1954), Quasi Normal Integral (QNI) model, Erasmus model (EM) (Erasmus, 1976), Hyperbolic Tangent (HT) model, Modified Hyperbolic Tangent (MHT) model, Modified Normal Integral (MNI) model and others (Mohanta et al., 2011) have been developed and the formulas are shown in Table 1. They are the foundation of computer simulation of partition curve. MHT model has higher fitting accuracy than the others in most cases (Mohanta and Mishra, 2009, Reid et al., 1985). NI model and WE model are widely used in practical applications (Dou et al., 2018, Dou et al., 2015, Galvin et al., 2018, Iveson et al., 2015). But these models are obtained by translating, scaling and rotating S-shape functions according to geometric features of the partition curve. Parameters do not have explicit physical meanings except for NI model.

NI model could explain the separation process from the perspective of probability and error distribution theory (Zhang, 1980). It was assumed that the separation density of each moment was influenced by various random errors and followed normal distribution. The partition coefficient of each density fraction was determined by the distribution of separation density, therefore the partition curve of the whole process was in accordance with the normal cumulative distribution. Parameters in the NI model are mean value and standard deviation, which reflect separation density and separation efficiency. Jowett (1986) used the binomial expansion to simulate the probable distribution of components of different densities, it was shown that the partition curve was in a normal cumulative distribution form. However, there are obvious deviations between actual distribution curve and normal cumulative distribution curve at two ends, which limits the accuracy of NI model. In order to precisely describe the shape characteristics of partition curve, NI model has been improved in many ways. Fan and Zhang (1998) put forward location, spread, skewness and kurtosis as characteristic parameters, and combined normal distribution with a polynomial function. Dong et al (2013) built a four-parameter mathematical model including skew and kurtosis coefficients based on the generalized normal distribution. Liu et al (2014) employed the power function’s rational expression in the Edgeworth-Kapteyn generalized normal distribution and got a normalized mathematical model with 4 or 5 parameters.

Although modified models achieve higher fitting accuracy, they are still on the basis of normal distribution and adding extra characteristic parameters increases model complexity. The two-parameter models with relatively low accuracy are yet predominant in practical application and the evaluation index Ep that represents the distribution between partition coefficients 25% and 75% cannot reflect the entire process performance. An accurate model with physical meaning has not been applied. Accuracy is crucial for a mathematical model and parameters with clear physical meanings could be directly used to evaluate the whole process. When separation density changed, parameters could be transformed accordingly, therefore making it convenient in practical application. In this study, the authors employed a new probability distribution in the model building. It was investigated by analyzing the probability characteristics of partition curve and its differential curve. The proposed model achieved higher fitting accuracy with fewer parameters, thus making it superior in utilization.