A skew normal-Laplace model of partition curve based on probability characteristics
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.
Section snippets
Methods
According to the physical meaning of NI model, the partition curve accords with normal cumulative distribution, therefore its differential curve can be regarded as probability density curve which shows the separation density fluctuation and error distribution clearly. In order to obtain the differential curve, a continuous MHT model function with the highest fitting accuracy was used to represent the actual partition curve. Model parameters were got by nonlinear curve fitting method, and then
Theory
In probability theory, the central limit theorem states that the sum of a large number of independent random variables approximates a normal distribution. Assuming that the separation process was solely based on density and the variation of separation density was caused by various independent random factors, each of which had a minimal impact, the total variation of separation density would typically subject to a normal distribution. However, particle size has an obvious influence on gravity
Probability characteristics
A set of partition coefficient data from Appendix A including different types of gravity separators was used in NI model and MHT model fitting. The fitting curves and differential curves are shown in Fig. 1 and RMSE values are listed in Table 2. It can be seen that the MHT model curves are closer to the data points and RMSE values of MHT model are much lower than NI model, therefore the MHT model curve could be used to represent the actual partition curve. The differences between NI model and
Conclusions
The differential partition curve which represents the probability distribution of separation density provides new sights into separation evaluation. Asymmetry and fat tail characteristics of partition curve are summarized according to probability and error distribution theory. The Laplace distribution was used to express the fat tail and a piecewise function was used to describe the asymmetry. A three-parameter skew normal-Laplace model of partition curve was built. Parameters σ1 and σ2 can be
CRediT authorship contribution statement
Qifeng Zhou: Conceptualization, Methodology, Software, Validation, Investigation, Writing - original draft, Writing - review & editing. Lubin Wei: Conceptualization, Methodology, Formal analysis, Writing - review & editing, Supervision, Project administration, Funding acquisition. Yalin Dong: Conceptualization, Software, Methodology, Investigation. Tao Wei: Methodology, Resources.
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
This study was financially supported by the National Natural Science Foundation of China (NSFC) (No. 51574252).
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