A New Hyperbolic Function Approach of Rock Fragmentation Size Distribution Prediction Models

Abstract

It is well known that the first stage of mine-to-mill optimization is rock fragmentation by blasting. The degree of rock fragmentation can be expressed in terms of average grain (X₅₀) size and size distribution. There are approaches in which exponential functions are used to estimate the size distribution of the pile that will be formed before blasting. The most common of these exponential functions used to estimate the average grain size is the Kuz–Ram and KCO functions. The exponential functions provide a curve from 0% to 100% using the mean grain size (X₅₀), characteristic size (X_C), and uniformity index (n) parameters. This distribution curve can make predictions in the range of fine grains and coarse grains outside the acceptable error limits in some cases. In this article, the usability of the hyperbolic tangent function, which is symmetrical at origin, in the estimation of the size distribution as an alternative to the exponential distribution functions used in almost all estimation models is investigated. As with exponential functions, the hyperbolic tangent function can express the aggregated size distribution as a percentage with reference to the variables X₅₀ and X_C. It has been shown that the hyperbolic tangent function provides 99% accuracy to the distribution of fine grains and coarse grains of the pile formed as a result of blasting data for the characteristic size (X_C) parameter and the uniformity index (n).

1. Introduction

Since their discovery by Alfred Nobel in the 18th century, explosives are still used in open-pit mining as the most economical and effective method of rock excavation. Since its discovery, the technology of explosives has constantly developed, and numerous studies have been carried out. In this process, one of the issues that researchers focus on is the determination of the size distribution of the pile formed after blasting, which is a criterion for the evaluation of blasting efficiency. While the stack size distribution can contain numerical information about blasting efficiency on its own, the blasting problem remains important in terms of the efficiency of operations.

With a well-designed blasting, it is possible to obtain material that is uniformly distributed, can be loaded efficiently with loaders, has a low swelling factor, and can be transported by transport vehicles in a maximum volume, and the crusher can provide crushing under optimum conditions. Under ideal conditions, in a rock environment where there are no discontinuities, and rock properties are the same, the hole geometry created and the size distribution that can occur with the amount of explosive used can be modelled with very little error. However, due to the fact that there are many controllable and uncontrollable variables (physical and mechanical properties of the rock, discontinuity properties of the slope, etc.) in bench blasting, a realistic model in which the size distribution can be estimated outside of empirical approaches has not been developed. In the developed empirical approaches, the relationship between the main variables of bench blasting (specific charge, slice thickness, bench geometry, rock mass properties, etc.) and the average size constitute the starting point.

Factual part size distribution can only be determined by sieve analysis of all heaps. Subjecting a production-scale blasting batch to sieve analysis in this way is a difficult method to implement in practice. For this reason, empirical approaches are used to determine the size distribution of the heap. These approaches are based on the principle of defining the distribution of a part or all of the bulk with different methods. One of these methods is the visual-observational method, which is a subjective evaluation. In this method, which is carried out by experts, no numerical data are produced, and the blasting efficiency is verbally graded. Another method is to count the size of the particles that cannot be loaded by the loader machines and measure their size. Boulder shots, which can give an approach in terms of blasting efficiency, are one of the indirect solution methods produced due to the difficulties encountered in determining the explosive consumption and size distribution in practice. With the same approach, the loading performance of loader machines or the crushing performance of primary crushers are also indirect methods used to determine blasting efficiency.

Except for sieve analysis of the whole heap on the subject, the data closest to the truth have been obtained by image processing methods within the limits of acceptable margin of error [1,2]. In the first applications of this method, which went through various stages with the development of technology, it was made by counting the grains by hand with the grid method on analogue images [2,3]. Later, in parallel with the development of image acquisition technology, software was developed for analyzing images and evaluating numerical data, and these processes, which were previously semi-automatic, have become an accepted process in the literature today, and commercial software has been developed and offered to the service of practitioners [2,4,5].

The size distribution of the stack formed as a result of bench blasting is very important for the efficiency of the ongoing size reduction processes. The effect of fragmented rock mass on ongoing size reduction processes is discussed with mine-to-mill approaches within the framework of energy use. The physical size of the pile formed by the blast was defined by Kuznetsov in 1973 [6]. Using the mean diameter of the fragments formed by blasting rock, Soviet Mining Science in [6] expresses the average grain size (X₅₀). Ongoing research has shown that the average grain size (X₅₀) as a single value is insufficient to determine its effect on post-blasting operations. The mass size distribution resulting from blasting is not the standard normal distribution characteristic, largely due to the rock mass properties. The blasting pattern and specific charge with rock mass properties can occur as a non-standard normal distribution function of the bulk size distribution resulting from blasting. In non-standard normal distribution characteristics, mean (X₅₀) values can lead to misleading results. On the other hand, heaps of different grain sizes are classified according to their size and expressed in % passing. The determination of the average grain size (X₅₀) in % passing graphs is determined by analytical approaches on the graph. When the percentage passing graph of the pile formed as a result of blasting is examined, it is seen that it has a curved function. The curved structure of the pile formed as a result of blasting is then approached with exponential logarithmic functions. Exponential logarithmic functions provide approximation in the region of average grain size with a very low error rate. However, exponential logarithmic functions are outside the acceptable error limits in the very fine grain region, fine grain region, and coarse grain region. In this article, the hyperbolic tangent function approach is discussed in order to reduce the error rates in the convolution effect in the fine and coarse grain region of exponential logarithmic functions. To estimate the size distribution of fragmented rocks obtained by blasting, three mathematical relationships include the hyperbolic tangent function suggested. In [7], the author approached the closest desired results with precision by experimenting on the powers of the functions and uniformity indices. The power hyperbolic tangent function can predict the size distribution of hard rock fragmentation more accurately and with higher precision with more uniformity in fine and coarse grain sizes [7]. Fragmentation is determined by specific charge, rock mass properties, and blasting pattern. The exponential distribution functions used are limited in relation to specific charge, rock mass property, and blasting pattern. The excess in these variables makes it difficult to estimate the bulk size distribution resulting from blasting. In particular, the % passing function in the curved structure increases this difficulty even more. The hyperbolic tangent function represents the convoluted structure with fewer errors than exponential functions.

In this paper, the hyperbolic tangent function is proposed as a novel approach for the estimation of the heap size distribution, and the durability of the tangent hyperbolic function to realize the curvature of the dimensional distribution curve is investigated. The aim of the study is to produce the tangent hyperbolic function approximation model of dimensional analysis by pre-calculating the blasting data and using the average grain size (X₅₀), characteristic grain size (X_C), and especially the uniformity index. It has been understood that the exponential number in the function is effective in size distribution analysis, as it can be developed further in the future. The accuracy of the function was verified by the heap size distribution determined by the image analysis method in Ref. [8], the Kuz–Ram and KCO models, and curve fitting by least squares method.

2. Estimation of Size Distribution

Pioneering studies on the investigation of the effect of the size distribution resulting from drilling–blasting operations are about the determination of drilling–blasting costs and the efficiency of the excavation, loading, and transportation operations of the size distribution. Mackenzie concluded in his study that drilling–blasting costs remain constant or increase per ton depending on the degree of size distribution [9]. According to Currie, material size should be decisive in the sizing of primary crushers in efficiency and energy consumption optimization [10]. Currie estimated that the maximum size of the feed material for primary crushers was 1520 mm [10]. Tunstall and Bearman investigated the effects of size distribution on crushing–comminution operations and suggested that the maximum size of the feed good should be 75–80% of the primary jaw crusher inlet opening and 80% of the inlet opening for impact crushers [11].

Nielsen and Kristiansen investigated the effect of size distribution on crushing–comminution by examining many field blasts and laboratory-scale blasts [12]. In their study, they found that size distribution has a significant effect on crushing–comminution and grinding. Nielsen subjected four different rock types to laboratory-scale blast tests and investigated ball mill-grinding efficiency [13]. In these tests, which he carried out by increasing the amount of specific charge, he revealed that the micro cracks predicted to form during blasting, especially along the mineral grain boundaries, have a direct relationship with grindability [14].

Workman and Eloranta examined the effect of size distribution on crushing–comminution and grinding efficiency in terms of energy consumption [15]. They stated that mining operations consume large amounts of energy, and the Bond Work Index reveals the relationship between blasting cost and energy cost. In their study, Workman and Eloranta calculated that if they increased the specific charge amount from 0.33 kg/ton to 0.45 kg/ton in order to reduce the average size distribution resulting from blasting from 40 cm to 30 cm, their total cost (blasting, crushing–comminution, and grinding) would decrease by USD 0.39 per ton [15].

The most popular model for estimating the size distribution is the Kuz–Ram model developed by Cunnigham in [16,17]. The Kuz–Ram model was obtained by combining the empirical equation proposed by Kuznetsovin [6] for the estimate of mean size (X₅₀) and the size distribution function proposed by Rosin and Rammler [18] in [16]. Afterwards, in the studies on the estimation of the size distribution, either the development of an alternative model was attempted by claiming that the Kuz–Ram model was insufficient, or new coefficients were developed to eliminate the inadequacy of this model. In this sense, the Kuz–Ram model has been the starting point where it is discussed whether or not the predictability of the size distribution of the pile formed as a result of bench explosions may or may not be possible.

On one hand, research has been reported suggesting that it is not possible to predict the size distribution in any way due to the uncontrollable rock mass properties in blasting and that the size distribution formed as a result of blasting is very different from the Kuz–Ram estimation model, whereas on the other hand, research has been carried out that argues that there should be a general approach with the least error for the estimation of the size distribution, trying to confirm this with studies.

Ouchterlony and Sanchidrián considered the historical development of estimation models of blasting-induced heap size distribution [19]. Accordingly, the efforts to find better formulas by researchers from the USBM, Queens University (Katsabanis and colleagues), and the JKMRC were discussed. Model improvement partly focused on a better fines prediction (JKMRC), also incorporating a largest fragment size (Swebrec function and KCO model).

2.1. Kuz–Ram Model

Kuznetsov proposed the formula given in Equation (1) between the amount of explosives per unit volume (specific charge) and the average size as a function of the rock mass in [6].

$$X_{50}=A{({\displaystyle \frac{V_o}{Q_e}})}^{0.8}{Q}_e^{1/6}$$

(1)

where the X₅₀ is average size (cm), A is rock factor, V_o is volume to be blasted per hole (slice thickness x distance between holes x bench height, m³), $Q_e$ is nitroglycerin-based explosive (kg) used per hole. The average size distribution given in Equation (1) is proposed for the nitroglycerine-based detonation rate of highly explosive materials. The power of these detonators is higher than the commonly used ANFO. Therefore, according to the situation where Equation (1) is used as ANFO, the correction coefficient is given as in Equation (2). In the case where ANFO is used, S_ANFO =100 [6].

$${X_{50}=A{\left({\displaystyle \frac{V_o}{Q_e}}\right)}^{0.8}Q}_e^{{\displaystyle \frac{1}{6}}}{\left({\displaystyle \frac{S_{\mathit{ANFO}}}{115}}\right)}^{-{\displaystyle \frac{19}{30}}}$$

(2)

One of the important parameters used in bench blasting is the amount of specific charge (q, kg/m³), which is defined as the amount of explosive used per unit volume. The specific amount of charge is given as follows:

$${\displaystyle \frac{1}{\mathrm{q}}}={\displaystyle \frac{{\mathrm{V}}_0}{{\mathrm{Q}}_{\mathrm{e}}}}$$

(3)

In this case, when the intended average size distribution is determined, the required specific amount of charge can be calculated using Equations (2) and (3), as in the following equation.

$$q={\left[{\displaystyle \frac{A}{X_{50}}}{Q}_e^{1/6}{\left({\displaystyle \frac{115}{S_{Anfo}}}\right)}^{19/30}\right]}^{1.25}(\mathrm{kg}/{\mathrm{m}}^3).$$

(4)

The amount of explosive used and the amount of specific charge can be used as the magnitude in the equations given for estimating the size distribution, while the effect of rock mass has been proposed as a factor between 7 and 13. Cunnigham tried to eliminate the lack of rock factor since it does not reflect the characteristic features of the rock mass [16]. The rock mass blasting index proposed by Lilly regarding rock mass blastability is used as the basis for determining the rock factor used in the equation of mean size estimation [20].

$$A=0.06(RMD+RDI+HF)$$

(5)

where RMD = rock mass description = 10 (powdery/friable), JF (if vertical joints), or 50 (massive):

JF = Joint Factor = JPS + JPA = Joint Plane Spacing + Joint Plane Angle.

JPS = 10 (average joint spacing SJ < 0.1 m), 20 (0.1 m oversize < X), or 50 (>oversize).

JPA = 20 (dip out of face), 30 (strike ┴ face), or 40 (dip into face).

RDI = Rock density influence = 0.025·ρ (kg/m³)—50.

HF = Hardness factor, uses compressive strength σc (MPa) and Young’s modulus E (GPa).

HF = E/3 if E < 50, and σc/5 if E > 50.

In practice, due to the difficulties in determining the rock mass number (RMD), the A value has been made practical as 7 for medium hard rock masses, 10 for hard multi-fissure rock masses, and 13 for hard low-fissure rock masses.

Rosin and Rammler described the dimension distribution function as in Equation (6) in Ref. [18].

$$R_m=1-e^{{-({\displaystyle \frac{X}{X_C}})}^n}$$

(6)

where $R_m$ is the ratio of material passing in the specified dimension (%), X is the specified size (sieve opening, mm), n is the uniformity index, and X_C is the scale factor described as the characteristic dimension (mm). In order to draw a distribution curve according to this equation, it is sufficient to know the uniformity index (n) and characteristic dimension (X_C). When relationship 6 is rearranged to determine the characteristic dimension (X_C), it can be expressed as in Equation (7).

$$X_C={\displaystyle \frac{X}{\sqrt[n]{-\mathit{ln}(1-R_m)}}}$$

(7)

Cunningham determined the distribution by assuming the X₅₀ value as the mean size value proposed by Kuznetsov (X = X₅₀) and =0.5 (50%) [6,16]. Accordingly, the characteristic dimension formula of relationship 7 can be written as in Equation (8).

$$X_C={\displaystyle \frac{X_{50}}{\sqrt[n]{0.693}}}$$

(8)

In order to determine the Rosin and Rammler distribution function given in relationship 6, the uniformity index must be known [18]. Cunningham provided the factors related to explosive and rock mass in the estimation of the size distribution with the Kuznetsov equation and proposed the uniformity index (n) as given in Equation (9) for the determination of variables related to hole pattern and blast geometry [16,17,21].

$$n=\left(2.2-14{\displaystyle \frac{B}{D}}\right){\left[{\displaystyle \frac{1+\frac{S}{B}}{2}}\right]}^{-0.5} \left(1-{\displaystyle \frac{W}{B}}\right){\left[{\displaystyle \frac{\left|BCL-CCL\right|}{L}}+0.1\right]}^{0.1}\left({\displaystyle \frac{L}{H}}\right)$$

(9)

where B is burden (m); S is space (m); D is the hole diameter (mm); W is drilling accuracy, where (m) W = 0.1 + (0.03 H); H is the bench height (m); BCL is the bottom charge length (m); CCL is the column charge length (m). The uniformity index (n) is the main coefficient that determines the slope of the Rosin and Rammler distribution curve in [18]. A high uniformity value indicates a steep and uniform distribution of the size distribution, while a low uniformity coefficient indicates that the size distribution of the heap is not uniform. As seen in correlation 9, the burden/hole diameter ratio decreases the uniformity coefficient, and the distance between the holes/burden increases the uniformity coefficient. Normally, the value of n varies between 0.75 and 1.5. Cunningham stated that there are some neglected parameters in the application of the Kuz–Ram size distribution estimation model [16,17,21]. The first of these is that the firing sequence and delay interval are not evaluated in the model, and the second is that the explosive energy is relatively included in the model. In addition, he emphasized that the main variable affecting the fragmentation size distribution is rock mass properties, especially in very discontinuous structures [22].

In addition, forecast models give realistic values in homogeneous, discontinuous solid rocks. Rock mass properties are used in forecasting models as coefficients or as numerical values associated with certain discontinuity properties. Doucent investigated the relationship between the RMR classification system, Q classification system, and RQD rock quality indicator and size distribution, which are among the rock mass classification systems [23]. Accordingly, he reported the result that low rock mass properties cause non-uniform size distribution in classification systems and suggested that rock mass classification systems should be used in order to take realistic approaches in estimation models of size distribution.

Different analysis methods were used to eliminate the obvious errors of the size distribution in fine grains and coarse grains. Cho et al. suggested that the distribution of fine grains in the bulk could be determined using numerical modeling methods [24]. Ouchterlony and Sanchidrián pointed out that size estimation models particularly neglected discontinuity and water content, leading to errors [19]. Li et. al. adopted support vector regression (SVR) techniques as basic estimation tools for determining the size distribution, and they used five types of optimization algorithms [25]. Kemeny introduced “Practical Techniques for Determining the Size Distribution of Blasted Benches, Waste Dump and Heap Leach Sites” [26].

2.2. Kuznetsov–Cunningham–Ouchterlony (KCO) Model

The KCO distribution model was proposed by Ouchterlony to eliminate the shortcomings of the Kuz–Ram model [27]. Ouchterlony proposed a function different from the Rosin–Rammer distribution function, considering that the Kuz–Ram distribution model is inaccurate in estimating the size distribution in the ratio of fine and coarse dimensions. Ouchterlony published the size distribution model using this function, known as the Swebrec function, in 2005 and called it the KCO (Kuznetsow, Cunningham, and Ouchterlony) distribution model [27]. The three parameters used in the Swebrec distribution function are used in the dimension distribution model. X₅₀ is the average size at which 50% of the material passes through the sieve, X_max is the largest particle size, and b is the curve undulation parameter that determines the convolution of the distribution function. This coefficient is similar to the uniformity index n proposed to the Rosin–Rammer distribution function. The equations used in the KCO model are given in Equations (10) and (11).

$$R_m={\displaystyle \frac{1}{\left\{1+{\left[\frac{ln\left(\frac{X_{max}}{X}\right)}{ln\left(\frac{X_{max}}{X_{50}}\right)}\right]}^b\right\}}}$$

(10)

and

$$b=\left[2·ln2·ln\left({\displaystyle \frac{X_{max}}{X_{50}}}\right)\right]n$$

(11)

where $R_m$ is the proportion of additive material in the specified size (%), b is the curve undulation parameter of the distribution function, X is the determined sieve opening (cm), lnX is the mean size (same as the Kuz–Ram model, cm), n is the uniformity index (same as the Kuz–Ram model), and X_max is the largest part size (cm) [27].

3. Tangent Hyperbolic Function Model

The Rosin and Rammler distribution function and the Kuznetsov–Cunningham–Ouchterlony (KCO) distribution function estimate the additive curve using the variables mean grain size (X₅₀) and characteristic grain size (X_C). It has been reported that the estimation curves give results outside the acceptable error limits in the fine grain and coarse grain ranges. The suitability of the tangent hyperbolic function was investigated in order to realize the curvature of the curve of the dimensional distribution. As in other estimation models, mean grain size (X₅₀) and characteristic grain size (X_C) were used in the tangent hyperbolic function approach. The tangent hyperbolic function is defined mathematically by $tanhx={\displaystyle \frac{sinhx}{coshx}}={\displaystyle \frac{e^x-e^{-x}}{e^x+e^{-x}}}$ for all real x. In this function, if ${\left({\displaystyle \frac{X}{X_C}}\right)}^n$ is written instead of x, $y=tanh {\left({\displaystyle \frac{X}{X_C}}\right)}^n={\displaystyle \frac{e^{{\left(\frac{X}{X_c}\right)}^n}-e^{-{\left(\frac{X}{X_c}\right)}^n}}{e^{{\left(\frac{X}{X_c}\right)}^n}+e^{-{\left(\frac{X}{X_c}\right)}^n}}}$ is obtained for all real number, and this function is a symmetrical function at origin.

In this study, a new approach model of rock fragmentation size distribution prediction is recommended as follows:

$$R_m=Ctanh{\left({\displaystyle \frac{X}{X_c}}\right)}^n=C{\displaystyle \frac{e^{{\left(\frac{X}{X_c}\right)}^n}-e^{-{\left(\frac{X}{X_c}\right)}^n}}{e^{{\left(\frac{X}{X_c}\right)}^n}+e^{-{\left(\frac{X}{X_c}\right)}^n}}}$$

(12)

where n is the uniformity index, and using the definition of the tangent hyperbolic function, the new approach model is considered as below:

$$R_m=CtanhB{X}^n$$

(13)

where the tangent hyperbolic function is a symmetrical function at origin. In this study, using the symmetry property of the function, the approximation model is considered for positive numbers.

The function is defined by finding the coefficients C and B by the curve fitting and least squares method using the dimension distribution data.

In Equation (12), we work using the analytical and basic properties of the defined function, and using Equation (13) we find the function by curve fitting. When the image analysis data of the dimension distribution models are used in [8], it is seen that the C coefficient of Equations (12) and (13) is the same.

Since it is known that the grain size for the function described in Equation (12) is 100 percent below the distribution when it is magnified considerably, the following is considered:

$$\underset{X\to \infty }{\lim }{R}_m=C\underset{X\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}tanh{\left({\displaystyle \frac{X}{X_c}}\right)}^n=C\underset{X\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{\displaystyle \frac{e^{{\left(\frac{X}{X_c}\right)}^n}-e^{-{\left(\frac{X}{X_c}\right)}^n}}{e^{{\left(\frac{X}{X_c}\right)}^n}+e^{-{\left(\frac{X}{X_c}\right)}^n}}}=C.1=100.$$

The new model function of the rock fragmentation size distribution is defined as given below:

$$R_m=100tanh{\left({\displaystyle \frac{X}{X_c}}\right)}^n$$

(14)

and the inverse function of $R_m$ that is grain size function is obtained as follows:

$$X=X_c\left(\sqrt[n]{Argtanh{\displaystyle \frac{R_m}{100}}}\right)=X_c\left(\sqrt[n]{ln\sqrt{{\displaystyle \frac{100+R_m}{100-R_m}}}}\right)$$

(15)

When C = 100 is written in the function defined in Equation (12), $R_m$ is obtained:

$$R_m=100tanhB{X}^n$$

(16)

In the next section, we estimate the size distribution for the calculated X_C and n numbers of the pre-calculated blasting operations as an example application of the tangent hyperbolic function model defined in Equation (14). At the same time, we find the most suitable curve by applying Equation (16) with the least squares method using the size distribution obtained from the WipFrag v2.6 software program.

4. Comparison of Size Distribution Estimation Models

Dimension distribution estimation models were compared with the size distribution determined by image analysis of the stack obtained as a result of a bench blast. In the image analysis method, a scatter graph was created by taking images from four different regions, representing the pile formed as a result of blasting [8]. The WipFrag program was used in the analysis. Application blasting was planned as a single row in an actively operating limestone quarry. The technical data of the blasting are given in

Table 1. Application Blast Technical Data.

Hole Diameter	89.0 mm
Bench Height	13.0 m
Hole Length	14.3 m
Burden	2.0 m
Space	2.5 m
Charging	56.0 kg
Column Height	10.0 m
Stemming	4.3 m
The Rock Factor	10

, and the image analysis of the formed pile and the size distribution graph is given in Figure 1.

For the calculation of the average grain size (X₅₀) and the number of uniformity index (n) in order to determine the Kuz–Ram estimation distribution of the application blast, the technical details given in Table 1 are given in Equations (17) and (18).

$$X_{50}=10{\left({\displaystyle \frac{(2.5\times 2\times 13)}{56}}\right)}^{0.8}{56}^{{\displaystyle \frac{1}{6}}}{\left({\displaystyle \frac{100}{115}}\right)}^{{\displaystyle \frac{19}{30}}}=24.1 \mathrm{cm}$$

(17)

and

$$n=\left(2.2-14{\displaystyle \frac{2}{89}}\right){\left[{\displaystyle \frac{1+\frac{2.5}{2}}{2}}\right]}^{-0.5}\left(1-{\displaystyle \frac{0.35}{2}}\right){\left[{\displaystyle \frac{\left|1.3-10\right|}{10}}+0.1\right]}^{0.1}\left({\displaystyle \frac{10}{13}}\right)=1.27 .$$

(18)

Based on these data, it is necessary to calculate the characteristic grain size in order to determine the distribution curve of the Kuz–Ram estimation model. Accordingly, the characteristic grain size (X_C) is computed as follows:

$$X_C={\displaystyle \frac{24.1}{\sqrt[1.27]{ln\left(2\right)}}}=32.15 \mathrm{cm}.$$

(19)

The calculated size distribution values of the application blast using the Rosin and Rammler distribution function given in Equation (6) at the sieve openings determined using these values are given in

Table 2. Predicted Distribution Values using Estimation models in specified sizes: WipFrag image processing program, Kuz–Ram, KCO, and tangent hyperbolic approximation models.

Size (cm)	WipFrag Program Distribution Values (%)	Kuz–Ram Model Distribution Values (%)	KCO Model Distribution Values (%)	Tangent Hyperbolic Approximation Model Distribution Values (%)
162.56	100.00	99.96	100.00	99.99
81.28	93.57	96.07	99.07	99.69
40.64	75.47	73.96	76.79	87.32
20.32	48.43	42.84	42.61	50.67
10.16	23.27	20.74	22.03	22.74
7.62	13.60	14.91	17.03	15.93
6.35	9.27	12.03	14.58	12.67
5.08	5.40	9.21	12.14	9.57
3.81	2.30	6.49	9.72	6.65
2.54	0.37	3.94	7.26	3.97
1.91	0.10	2.75	6.00	2.77
1.27	0.00	1.66	4.66	1.65
0.95	0.00	1.15	3.94	1.14
0.64	0.00	0.69	3.16	0.68

.

Similarly, for the KCO estimation model, it is necessary to calculate the three-element Swebrec distribution function variables given in Equation (10). Two of these variables, the mean size distribution (X₅₀) and the uniformity index (n), are the values calculated above for the Kuz–Ram distribution function. The undulation parameter, which is the third element of the Swebrec distribution function, is calculated with the help of the function given in Equation (11). Accordingly, the coefficient $b$ given in Equation (11) is calculated:

$$b=\left(2ln2 ln\left({\displaystyle \frac{107.4}{24.1}}\right)\right)1.27 = 2.77.$$

(20)

The size distribution values used in the KCO dispersion estimation model and calculated at the sieve openings determined according to the Swebrec function given in Equation (10) are given in Table 2.

Using the numbers $X_C=32.15$ in Equation (14) and n = 1.27, which we obtained before blasting, the approach model of the rock fragmentation size distribution function is obtained as follows:

$$R_m=100 tanh0.01219X^{1.27}$$

(21)

where ${\left({\displaystyle \frac{1}{32.15}}\right)}^{1.27}=0.01219$. Using the data in Table 2, the prediction values obtained from this function are given in Table 2.

The function given in Equation (16) as $R_m=f\left(X\right)=100tanhB{X}^n$ is considered to approximate a definition range of the function that is gridded by $\left(X_i,R_{m_i}\right)$ on ${\mathcal{R}}^2$ and $R_{m_i}=f\left(X_i\right)$ data given for the function of single variable at the distinct points where X is the size, and $R_m$ is the distribution values obtained by the WipFrag program.

The single variable function given by Equation (16) approximates the definition range, in which $R_{m_i}=f\left(X_i\right)$ passes through each point in the range. The problem of the function approximation satisfying the data with minimum error is called finding the most optimal function.

Let us suppose that there exists a nonlinear function with independent variable X, and the dependent variable $R_m$ can be formulated. Finally, the least squares method can be used for determining the coefficients B and n, which are obtained by various methods using the generalized inverses, especially the least squares method, which is applied to the best approximate solution for the inconsistent system of the linear equations [28,29]. This curve defined by Equation (16) can be transformed to the linear form as follows:

$$\mathrm{U}=m+nlnX=g\left(X\right)$$

(22)

where $\mathrm{U}=ln\left(\mathrm{A}\mathrm{r}\mathrm{g}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}{\displaystyle \frac{R_m}{100}}\right),\mathrm{a}\mathrm{n}\mathrm{d} m=lnB$.

We now consider Equation (22) to approximate over the range. Assuming that $U_i=g\left(X_i\right)$ data are given for the function of the single variable at the distinct points in the range, there is a best function approximation that is the optimal curve on ${\mathcal{R}}^2$. In the experimental data obtained from the WipFrag program, there are about 14 measurements involving size values. The least squares method known curve-fitting method is applied to the curve in (22). B = 0.01233 and n = 1.216 numbers are computed, and the best approximation tangent hyperbolic function with the curve fitting using the tangent hyperbolic function is obtained as follows:

$$R_m=100tanh 0.01233X^{1.216}.$$

(23)

Using the data in Table 2, the prediction values obtained from this function are given in Table 2. When the hyperbolic model proposed using the X_C value and uniformity index before blasting is compared with the best curve found when the least squares method is used on the data obtained from WipFrag, it can be seen that the coefficients and powers of the functions are quite close:

$$R_m=100 tanh0.0122X^{1.27} \mathrm{and} R_m\approx 100 tanh0.0123X^{1.22}.$$

According to this result, the model created in this study is suitable for dimensional analysis. B can be computed in two ways as $B={\left({\displaystyle \frac{1}{X_C}}\right)}^n$ or $B=e^m$ using curve fitting.

This result shows that the approximation formula defined by Equation (10) can be used from the beginning by using the numbers X_C and n.

Comparing the data calculated from the tangent hyperbolic approximation model with data obtained from other models, the graphs below show the close relationships between the prediction data. The graphs of the data in Table 2 are given as follows, where x_w shows the data obtained from the WipFrag program, x_KR is the predicted distribution value using Kuz–Ram, x_KCO is the value obtained from KCO model, x_cf is the predicted value using the curve fitting-formula, and y is the distribution value using the tangent hyperbolic approximation model.

As can be seen in Figure 2, there are differences in the distribution of both the prediction models used and the application blast stack image analyses in thin and coarse sizes. On the other hand, the average size and the distribution percentages of approximately 50% of this dimension are very close to each other in both the prediction models and the distribution values of image analysis. Considering that the KCO model was developed to eliminate the coarse and thin size deficiencies of the Kuz–Ram model and that the curl coefficient (n) was added to minimize the errors with the real data, it can be said that large and thin image analysis programs are insufficient. This is the natural result of the fact that when an image of a particular area is taken, the image resolution cannot distinguish between fine grains. In addition, errors that occur in large size are the errors caused by the slope of the stack and its third dimension. In this sense, considering the errors of image analysis acceptances, the average size of the pile to be formed by the proposed distribution models before making a blast can be estimated within acceptable limits with the Kuz–Ram model developed by Cunnigham [16]. However, although it is reported that the size distribution of the pile has a low error rate in the application of the KCO model developed by Ouchterlony [27], a realistic estimate of the bulk and thin size remains theoretical in the pile size distribution estimation due to the lack of the largest grain size determination, especially in cases where the rock mass discontinuity properties are decisive. In addition, the JKMRC models, which are envisaged to eliminate the coarse and thin size deficiencies of the examined models, are not evaluated here because they require input parameters that are difficult to determine and measure in practice, such as subjecting almost the entire batch to sieve analysis.

In this study, a model was developed using the tangent hyperbolic functions of size distribution. In the approach model defined in Equation (21), it was found by using the average size and characterization values obtained from the WipFrag program. It was observed that the results obtained from the approximation model have an accuracy of R² = 0.99, and by using the inverse function, the percentage of the distribution can be given, and the dimensions can be found with this precision. Since this model was created by using blasting parameters, it allows us to find the distribution percentage of function values.

The feature of the curve approach defined in Equation (23) is that it defines the model using size distribution data. Naturally, creating the experimental data and finding the coefficients and power of the function will give us the best function. By finding the best approach, this provides the opportunity to easily find the desired size and distribution percentage from the distributions from the inverse function, with an accuracy of R² = 0.99. Therefore, when Equations (21) and (23) are compared, it is seen that the proposed tangent hyperbolic function approach can be used effectively in size distribution estimations with the use of blasting data.

5. Conclusions

The size distribution of the pile formed as a result of bench blasting directly affects both the efficiency of blasting and the ongoing operations. In this sense, it remains important to determine the size distribution by considering the design parameters and rock mass properties before blasting. In this study, a new function is proposed to eliminate the deficits of the Kuz–Ram and KCO models, which are widely used in estimating bulk size distribution. The estimation of size distribution was proposed through a new function approach using the tangent hyperbolic function, which is a hyperbolic function. When this approximation function was compared with the data obtained from other classical methods, R² = 0.99, and it was seen that the results were quite close. The resulting function and its inverse function enabled us to estimate the data effectively and faster. The desired values were easily obtained with the newly defined hyperbolic tangent function and the estimation formula for the size that characterizes the average size according to the approximation formula. When using the experimental data, average characterized dimensions can be calculated in advance. When the formulas in other models were compared with the calculations in this new approach model, substantially accurate results were found.