Application of finite and infinite functions in determining of the metalworking dusts particle size distribution

. Particle size distribution is an important parameter for dust quality and properties assessment as well as a factor to estimate the air dustiness influence on humans and the choice of an appropriate dedusting means. The academic literature analysis shows that, unlike technological dust, there is practically no information available about theoretical models of metal machining dust size distribution. In addition, researchers rarely use finite functions, although an appropriate particle size distribution determination is crucial for dedusting ventilation systems designed for machine tools. The article examines the metal machining dust particle size distribution description using finite and infinite functions, and discusses the features and advantages of their application. As a result, it is concluded that infinite functions are good at describing dust with a high extremity of outliers. On the contrary, finite functions appropriately describe dust with low tailedness, the measure of which is the kurtosis. It should be noted that the obtained results are related to the dust analysis methods, including sieve analysis when selecting a sieve size range.


Introduction
Determining the metal machining dust particle size distribution (PSD) is an important issue in manufacturing processes related to various materials machining [1].The PSD of dust specifies the physical properties of the dust-air mixture, especially aerodynamic ones [2][3][4].Precise PSD determination is crucial for the development of effective methods of dust extraction, aspiration and emission control, and therefore for assurance the workplace safety [5][6][7].At the same time, the PSD description is a complex phenomenon due to dust variety in dependence on the workpiece material and the parameters and type of machining operation [8,9].Dust usually contains particles of various sizes and shapes, which causes difficulties in its description and modeling [10].In addition, dust generation and movement processes in manufacturing systems are often dynamically heterogeneous, which makes the accurate PSD assessment difficult [11][12][13][14][15].
The research performed by Sanchidrián et al. [25], is particularly worth attention as it describes rock fragments using 17 different functions.Also, the paper [25] contains finite distributions, the application of which will be considered in more detail in this article.However, the academic literature lacks enough information about dust produced by metal machining.Only the source [26] describes mathematical methods for the correct averaging of size and mass parameters of dust produced by solid metal cutting.In addition, researchers rarely apply finite functions in this scientific field.
The main objective of this article is to ascertain the appropriateness of finite and infinite functions for the statistical description of metal machining dust PSD.Understanding and developing effective methods of machining dust PSD definition is relevant for manufacturing plants aimed at optimizing their processes and improving working conditions.
The present paper discusses issues associated with the dust PSD description, and also presents approaches based on finite and infinite functions.The features, advantages and limitations of these methods are analyzed and their applicability in various metalworking processes is considered.

Materials and methods
Infinite distributions are a type of probability distributions whose range of values is unlimited and they have no lower and upper bounds.If the distribution function is bounded from below at point 0, then it is called semi-infinite.
One of the most well-known infinite distributions is the normal distribution.Another example is a uniform distribution over the interval (-∞, +∞).In this case, the probability is the same for all values in the infinite range.
To describe continuous distributions, the complementary cumulative distribution function (CCDF) () is used.Here are the semi-finite distributions applied to describe the size  of metal machining fragments (dust, swarf): Rosin-Rammler or Weibull: where   ,  are the scale and shape parameters of the distribution, respectively. Log-normal: where erfc is a complementary error function; ,  are the mean and standard deviation of the particle size natural logarithm.Log-logistic distribution: where   ,  are the scale and shape parameters of the distribution.
There is another method of modeling probability distributions: finite (or re-scaled) functions, which, unlike semi-finite ones, are bounded over a given interval.The finite distribution of a continuous random variable is accomplished from a semi-finite one by substituting  for  in the following way: where   is the maximum particle size.The   parameter is replaced in the same way.Using this method, the finite versions of the previously determined distributions can be obtained: Finite Rosin-Rammler or Weibull distribution: Finite log-normal distribution: Finite log-logistic distribution: The main feature of finite functions is that they allow the limitations and characteristics of real data.In the case of metal machining dust PSD, finite functions are useful in describing it, considering the restrictions of the maximum fragment size resulting from physical or technological limitations.
To carry out the dust analysis, the most common method of sieve analysis with a throwaction vibrating screen was used.Sampling of cast iron dust and swarf, as well as a metalabrasive mixture, was performed at points in the working area of the machinist operating drill presses and grinding machines.It should be noted the correct choice of dust sampling points is crucial not only from a methodological standpoint, but also to determine whether the sample is representative and whether the final result is reliable.Moreover, there are several limitations to the usability of infinite and finite functions in the description of dust PSD.Thus, infinite distributions require a sufficient dataset of dust particle sizes.In case of insufficient data or incorrect sampling, modeling with finite distributions may lead to inaccurate results.
In addition, when using infinite distributions, it must be considered that they assume an unlimited range of values.Real experimental data, may have a physical or logical restriction on particle size, and infinite distributions may not take these restrictions into account.In such cases, it is reasonable to apply finite distributions that are limited to a given range of values including the upper limit value.
In return, the upper limit is determined by the sieve analysis accuracy and the correct sieve size range selection.

Results
Finite functions show a higher quality of data approximation compared to the infinite in the case of dust and chips produced by cast iron drilling: the root mean square error (RMSE) is on average 1.17 times less.This result was achieved by adding the   parameter.At the same time, the machining process generates large dust particles (more than 10%), for which maximum size is not determined.The least squares method is used for approximation.Figure 1 shows CCDFs for both types of Rosin-Rammler distribution in comparison with the initial data of metal-abrasive drilling dust mixture PSD.The same effect is observed with the log-logistic function when researching this drilling dust: the finite version allows us to achieve a higher quality of modeling compared to the infinite one.The RMSE in this case is about 1.4 times lower.Figure 2 shows a graphical quality comparison of both log-logistic distribution models (finite and infinite) describing size distribution of metal-abrasive drilling dust mixture using their CCDFs.The research of cast iron dust produced by metal grinding showed that the infinite lognormal distribution that does not consider the maximum particle size better describes the available data (RMSE is less than 1.78 times) and makes it possible to more appropriately determine the distribution parameters.Figure 3 shows a graph of CCDFs for both types of lognormal distribution in comparison with the initial data of metal-abrasive grinding dust mixture PSD.Infinite functions describing the distribution of such grinding dust make it possible to achieve more precise modeling.In this case, the RMSE is less by about 6%. Figure 4 shows the quality of both types of the Rosin-Rammler distribution describing the particle sizes of a mixture of metallic and abrasive drilling dust.

Discussion
The advantage of infinite distributions in the PSD approximation is their flexibility in the description of various sized dust particles, considering both small and large ones, by adjusting the parameters to fit the data.Finite distributions can describe real data more accurately, especially if that have unusual shapes or extreme values (outliers).The use of infinite distributions requires sufficient data.A lack of data, an incorrect dust sampling method, or inaccurate parameter settings during modeling can cause erroneous results and data distortion.In addition, when using infinite distributions, it should be taken into account that they assume an unlimited range of values.
Real data may have technological limitations on the particle size that are not considered by infinite distributions.In such cases, it is more appropriate to use finite distributions that allow the values range control, which provides to more accurately describe the PSD of dust.At the same time, they are less flexible compared to infinite ones and capable of a precise approximation over a wider range of values.
The research has found that the key factor in determining the type of function (finite or infinite) describing a particular metal dust is the tailedness of the initial data distribution.Numerically, the parameter of dust property is the kurtosis: for distributions with a high positive kurtosis value, infinite functions are the optimal choice.On the contrary, with a small or negative kurtosis, a function from the finite class should be chosen.
Thus, as the kurtosis increases, depending on the metal machining process, there is a transition from finite distributions to infinite ones.A certain transition region of several dust types arises between them, for which the choice of the distribution type remains ambiguous.Presumably, the dust of metal milling machining is exactly like this.The issue defining the boundaries of such a transition region, which determines the distribution type choice, is promising and interesting.
Another potential direction for further research is the selection of a specific function for each metal machining process that most appropriately characterizes the resulting dust.
On the whole, the use of finite distributions to describe the metal machining dust PSD is a useful approach that allows one to consider the limitations and features of the data.However, it is necessary to thoroughly choose the appropriate distribution and take into account its limitations when interpreting the results and making decisions based on modeling.

Conclusions
Academic literature review and analysis show that, unlike well-researched technological dust, there are practically no studies devoted to the development of theoretical models of the PSD for dust resulting from metal machining.
Finite functions are rarely used to describe the PSD of dust, although an appropriate PSD determination is crucial for dedusting ventilation systems designed for machine tools.
Infinite functions describe dust with a high extremity of outliers better while, on the contrary, finite functions appropriately describe dust with low tailedness, the measure of which is the kurtosis.
The kurtosis is a parameter that facilitates the choice of the optimal function type (finite or infinite) to describe the distribution.
The proposed approach to the dust PSD description using finite distributions is promising and can be implemented to study of various technological processes of material machining.

Fig. 2 .
Fig. 2. Comparison of finite and infinite log-logistic functions describing iron drilling dust PSD.

Fig. 3 .
Fig. 3. Comparison of finite and infinite log-normal functions describing cast iron grinding dust PSD.