The Case for the Median Fragment Size as a Better Fragment Size Descriptor than the Mean

Abstract

Cunningham’s use of x ₅₀, the median fragment size, instead of the mean $\left\langle x \right\rangle$ in the main prediction equation of the Kuz–Ram model has several times been pointed out as a mistake. This paper analyses if this mistake is important using dimensional analysis and by reanalyzing the historical Soviet data behind Kuznetsov’s original equation for the mean. The main findings in this paper are that: (1) Cunningham’s mistake has no proven effect in practice and would only be relevant as long as he used Kuznetsov’s equation for the rock factor A, i.e. till 1987. (2) Kuznetsov’s equation has its roots in the characteristic size of the Rosin–Rammler (RR) functions fit to the sieving data as a way to determine the mean, not only in the mean itself. (3) The key data set behind Kuznetsov’s equation just as easily provides a prediction equation for x ₅₀ with the same goodness of fit as the equation for the mean. (4) Use of x ₅₀ instead of the mean $\left\langle x \right\rangle$ in a dimensional analysis of fragmentation leads to considerable mathematical simplifications because the normalized mass passing at x ₅₀ is a constant number. Non-dimensional ratios like x ₅₀/x _max based on two percentile sizes also lead to such simplifications. The median x ₅₀ as a fragment size descriptor thus has a sounder theoretical background than the mean $\left\langle x \right\rangle$. It is normally less prone to measurement errors and it is not rejected by the original Soviet data. Thus, Cunningham’s mistake has led the rock fragmentation community in the right direction.

Appendices

Appendix 1: The Data of Baron and Sirotyuk (1967)

In their Table 1, Baron and Sirotyuk (1967) present numerical sieving data for 12 different types of rock obtained from large size blasts with Ø75 mm blast holes. They are given as masses retained for given fractions (PDF form) and have been converted to mass passing (CDF form) in Table 2.

Table 2 Summed percentages retained on mesh from Table 1 in Baron and Sirotyuk (1967)

The table gives data for five different mesh sizes; 50, 100, 200, 300 and 600 mm. For all but type 2, a gabbro, the mass passing the 50 mm mesh, P(50 mm) is larger than 30 %, for types 7 and 8 it is larger than 50 %. The data range 50–600 mm is large enough to establish that the coarse range behavior follows the RR function well, like generally in Sanchidrián et al. (2009, 2012, 2014).

Baron and Sirotyuk (1967) have ‘linearized’ the sieving data by the standard transformation Y _RR = log{log[1/(1 − P)]} = constant + n × X _RR where X _RR = log(x) and the constant contains the term –n × log(x ₀). These equations are valid for both base 10 and natural logarithms and would work equally well with x ₅₀ instead of x ₀ since the latter are related trough ln(x ₀) = ln(x ₅₀) − ln(ln(2))/n. Baron and Sirotyuk have used the base 10 logarithms. These regressions, which are linear in n and n × log(x ₀) assume though that the data follow the RR function well and they will not work with the Swebrec function.

Curve fits in P vs. x space with both the RR and the Swebrec function have been performed here, using the mathematical forms P(x) given by Eqs. 1 and 8, respectively. For the RR, n and x ₅₀ were used as fitting parameters, for the Swebrec function, b, x _max/x ₅₀ and x _max. This makes the fitting procedure non-linear in the parameters and gives a different result than the linear regression of Baron and Sirotyuk (1967). The program TableCurve2D (Systat Software 2014) and the basic weighting function w(x) = 1 were used.

To judge the goodness of fit we use first r ², the coefficient of determination, as it is directly related to the sum of residuals squared that the least squares regression minimizes. We also use the smaller, DOF adjusted value, DOF r ², to account for the fact that a function with two parameters (RR) will probably do a poorer job in absolute terms than one with three parameters (Swebrec).

The results of these non-linear fits are given in Table 3.

Table 3 Parameters for RR and Swebrec functions fitted to data of Table A1

The statistics (mean ± Std dev) for the data in Table 3 are r ² = 0.984 ± 0.020 for the RR fits and r ² = 0.994 ± 0.005 for the Swebrec fits. For 11 of the 12 rock types in Table 3 the Swebrec fit has a higher r ² value than the RR fit. These values are all quite high and support Kuznetsov’s assertion that the RR function is a good representation of blast fragmentation over the range 50–600 mm, i.e. a bit more than one order of magnitude in size.

The diagrams in Fig. 2 show at the bottom of each the fitted function with the data as filled squares. At the top of each diagram residuals bars may be seen. The upper pair of diagrams compare the RR and Swebrec fits for rock type 4 for which the r ² difference 0.0527 is the largest. The bottom pair does the same for rock type 12 for which this difference −0.0002 is the smallest and the RR fit thus is marginally better than the Swebrec fit.

Fig. 2

Comparison of RR (left) and Swebrec (right) curve fits for rock types 4 (top) and 12 (bottom)

The data for most of these rocks already at x = 50 mm show a slight upward trend in ln(P) vs. ln(x) space. This trend is inherent both in most sieving data and in the Swebrec function. The concave upwards behavior for smaller fragment sizes is clearly visible in the top right diagram for rock type 4 in Fig. 2. It would probably make itself more and more felt if the data range were extended downwards (Ouchterlony 2005, 2009a). The combination of higher r ² values and the concave trend make the Swebrec function a better choice than the RR.

In relative terms, DOF r ² values in Table 3, we find that the Swebrec and RR functions each give a better fit for half of the rock types. So relatively speaking the RR function does as good a job of representing the sieving data, but not as good a job in absolute terms.

Appendix 2: The Data of Marchenko (1965)

According to Kuznetsov (1973), Marchenko’s (1965) fragmentation data refers to 26 full-scale blasts from four quarries and mines with rock of different hardness. Kuznetsov’s running text makes no mention of either (1) the term mean, (2) how the fragmentation measurements were made or (3) how the means were calculated. The only mention of a related term is the words “Av. fragment length” in Kuznetsov’s (1973) Tables 1–4. Strictly, this term could be taken to mean either the mean $\left\langle x \right\rangle$ or the median x ₅₀ but the latter interpretation is highly improbable in this case.

Kuznetsov (1973) does not mention if the muck piles behind the data in his Tables 1–4 are well-represented by the RR function or not and he gives no n values either. This probably indicates that he has arrived at his “Av. fragment length” or $\left\langle x \right\rangle$ values via the experimental mean $\left\langle {x_{Z} } \right\rangle$ rather than via the RR characteristic size x ₀ from curve fitting.

The data ranges of Kuznetsov’s (1973) Tables 1–4 are given in Table 4. This table shows (1) that there is a wide range of q values, (2) that the blast holes have been relatively large and (3) that the fragments are relatively large too. When reanalysing Kuznetsov’s data it became apparent that data for at least 4 of the 26 rounds are corrupt and that reading the text does not always lead to the correct values to use in the equation for $\left\langle x \right\rangle$, so some reasonable choices based on good faith have been made in our analysis. We may further wonder about the accuracy of the fragmentation measurements, given the scale of the blasts. See e.g. the comment “…the very rough nature of the measurements.” on p. 148 (Kuznetsov (1973). Any sieving done of the large blasts must have been limited and calculating $\left\langle {x_{Z} } \right\rangle$ is not trivial, see Appendix 4.4.

Table 4 Range of checking $\left\langle {x_{Z} } \right\rangle$ prediction in Kuznetsov’s (1973) Tables 1–4 data

In relation to the data in Tables 1–4, Kuznetsov (1973) mentions for Table 2 on p. 147 that: “The mean deviation of the experimental data from the theoretical data is ±15 %”. For his Tables 2 and 3 the numbers ±12 and ±15 % are mentioned, respectively. If we calculate the same deviations the numbers in Table 4 result. There is general agreement with Kuznetsov’s figures whose predicted numbers are not far from the measured ones. To achieve this degree of fit he has adjusted the f hardness values somewhat to get the A values he uses in his Eq. (14), see Eqs. 5 and 7.

We should also note the large range of the deviations in Table 4, −42 to 55 %. This corresponds to a variation in A values in the range 0.6 ≤ A′/A ≤ 1.6. This is not far from the range 0.5–2.0 mentioned by Cunningham (2005).

Appendix 3: The Data of Kuznetsov (1973)

The only meaningful fragmentation data that (Kuznetsov 1973) himself provides and that can be used to test the RR fits is given by his Table 6, which gives 8 points on the fragment size distribution for a blast with 20 tons of TNT, see Table 5. Table 6 gives the curve fitting parameters for the RR and Swebrec functions, together with the coefficients of determination, r ² and DOF r ². The Swebrec fitting was made using the usual weight function w(x) = 1/√x. The main justification is that the number of individual fragments in the largest bins becomes too small to represent a continuous function accurately and it improves the fit in the fines range.

Table 5 Fragment size distribution values from Kuznetsov (1973), Table 6

Table 6 Curve fitting parameters based on Kuznetsov (1973), Table 6 data

Figure 3 shows a plot of the fitted functions. The data is clearly best represented by the Swebrec function. The r ² values are so far apart that it makes no difference if r ² or DOF r ² is used as a measure, unlike for the data in Appendix 1.

Fig. 3

Plots of fitted functions against data from Kuznetsov’s (1973) Table 6

The data in Kuznetsov’s (1973) Table 5 come from an American nuclear blast. There are 4 points in the range 25–1500 mm. The RR fit gives r ² = 0.9998 with n = 0.78 and the Swebrec fit r ² = 0.9999. The data in his Table 5 are simply too few and too approximate to make a distinction the RR and Swebrec fits meaningful. So, in summary, the data in Kuznetsov (1973) and references supports the use of the RR function for coarse range fragmentation. The RR is not the best fitting alternative though.

Appendix 4: The Digitized Data of Koshelev et al. (1971)

4.1 4.1 Introduction

As mentioned in Sect. 2.2, the paper of Koshelev et al. (1971) contains the first known derivation of Kuznetsov’s equation for $\left\langle x \right\rangle$. To support the use the RR function there are several linearization diagrams of the type ln{ln[1/(1 − P)]} vs. ln(x) but there are no individual sieving data. To support the derivation of the x ₀ equation there is a diagram with axes ln(10 × Q/V ₀) vs. ln(x ₀/Q ^1/6) where Q/V ₀ = q equals the specific charge plus tabulated data for x ₀, Q etc., but there are no data for V ₀.

There are two main problems here. The first one is related to the terminology that Koshelev et al. (1971) use. Originally, like Kuznetsov (1973), they use x ₀ to denote the characteristic size of the fitted RR function, use $\left\langle x \right\rangle$ to denote the mean fragment size and $\left\langle {x_{Z} } \right\rangle$ to denote the approximate mean calculated from the sieving data as in Eq. 12. Yet Koshelev et al. (1971) on p. 254 express their predecessor Eq. (5.1) to Kuznetsov’s Eqs. (12)–(14) for $\left\langle x \right\rangle$ as

$$ x_{0} \approx \, 10 \times Q^{1/6} \times \left( {V_{0} /Q} \right)^{0.8} , $$

(23)

and state that “…x ₀ is the mean size of a piece, cm”.

x ₀ is explicitly and implicitly referred to as the mean several times in Koshelev et al. (1971) and this seems to be a shift of nomenclature. This shift should probably be understood in the light that Koshelev et al. (1971) nearly always refer to experimental data that follow the RR function well and that in these cases $\left\langle x \right\rangle$ ≈ x ₀ when n ≫ 1. The text on p. 254, preceding their Eq. (5.1) specifically mentions that: “The parameters of the explosions and the analytical results are given in Tables 2 and 3, as well as in Fig. 3a and b, respectively…”. This most probably refers to the x ₀- and n values from RR fits to the experimental data that are given in the tables. These x ₀- and n values lie behind the ln(10 × Q/V ₀) vs ln(x ₀/Q ^1/6) plot in Koshelev’s et al. Fig 4 from which the linear regression yields the constants prefactor 10 and exponent 0.8 in Eq. 23.

On p. 254 Koshelev et al. (1971) also mention that: “In addition, the mean value of $\left\langle {x_{Z} } \right\rangle$ was determined, calculated directly from experiments, using formula (2.16)”. This formula is the same as Eq. 12. Then follows: “The maximal divergence between the values of $\left\langle x \right\rangle$ and $\left\langle {x_{Z} } \right\rangle$, calculated using formula (2.13) was not more than 15 % and, in a majority of cases, not more than 4–6 %”. The formula reads $\left\langle x \right\rangle$ = x ₀ × Γ(1 + 1/n) and implies obtaining $\left\langle x \right\rangle$ from the RR characteristic size x ₀. For the data in their Table 2, n lies in the range 1.25–2.05 and the statistics for Γ(1 + 1/n) become 0.90 ± 0.02.

We do not know if Koshelev et al. (1971) have used this conversion factor when they compute $\left\langle x \right\rangle$. This is anyway an indication of the accuracy they assign to the relation $\left\langle x \right\rangle$ ≈ x ₀, which they say is valid for n ≫ 1. See also Appendix 4.6. In the end it is the approximate equivalence $\left\langle {x_{Z} } \right\rangle$ ≈ $\left\langle x \right\rangle$ ≈ x ₀ that allows Kuznetsov (1973) to write his “semi empirical formula” in terms of $\left\langle x \right\rangle$ instead of in terms of x ₀. The result is his Eq. (12)

$$ \left\langle x \right\rangle = \, 7 \times Q^{1/6} \times \left( {V_{0} /Q} \right)^{0.8} \times \theta^{ - 2/3} . $$

(24)

Here θ is the TNT equivalent of the charge. Considering this background it is more correct to say that Kuznetsov’s formula, Eq. 24, originally refers to the characteristic size x ₀ of the RR function fitted to the sieving data rather than to the mean size $\left\langle x \right\rangle$ of the sieving data. This puts Spathis (2004) statement about Cunningham’s mistake in a somewhat different light.

To gain confidence in how Koshelev et al. (1971) have reached their conclusions, their key diagrams, not only Figures 3a and 4 in the original Russian version of their paper were digitalized and the data analyzed, but also Figures 3b and 5. The scanning of all figures was made with a Canon iR-Adv C7260 copying machine. The digitization was made with PhotoStudio6 from .bmp files with roughly 2700 × 1700 pixels that were rotation corrected to the nearest degree. Figure 3b has been digitized to obtain more data for judging how error prone the determination of $\left\langle {x_{Z} } \right\rangle$ is. Figure 5 is the only figure based on externally published numerical sieving data and has been used as a check of the quality of the digitization and of the plotting made by Koshelev et al. (1971).

Figure 5 in Koshelev et al. (1971) shows ln[ln(1/R)] vs. ln(x) for three sieving curves for “sandstone tests made in England”. Here R = 1 − P is the normalized mass retained. There is a reference to an article by Baron (1962), who confirms that the data come from Grimshaw (1958) who in turn tested one limestone (specimen A) and 12 sandstone specimens (B–M) of cylindrical shape. Koshelev et al. (1971) do not however, tell which three sieving curves are depicted in their Figure 5, nor does Baron (1962).

The digitization of the data in Figure 5 has revealed plotting errors. The results are that:

• 1. The digitization used 200 pixels per axis unit and the readings were obtained in steps of 2 pixels so the reading accuracy is about ±1 pixel or ±0.5 %. This is much smaller than the errors made when drafting the diagram.

• 2. The systematic difference Δln(x) between read-off (digitized) and nominal diagram values is an insignificant 1 % for the horizontal axis or mesh size values. This indicates a digitization which is on average accurate. Most horizontal drafting errors seem to be stochastic with a scatter of ±7 % but individual errors of 10–20 % or more were found.

• 3. Correlation analysis and linear regression was made to find the best matched columns of ln[ln(1/R)] data, i.e. to assign each of the three curves of Koshelev et al. (1971) to the proper specimens of Grimshaw (1958). The agreement between the matched columns data is however, much worse than the normal goodness of fit for Swebrec and RR functions to sieving data,

• 4. The systematic difference Δln[ln(1/R)] between read-off and nominal diagram values of the best matched columns data is a significant −10 % for the vertical axis values. The reason for the latter could unfortunately not be found and this leaves an uncertainty about the quality of the plotting work in Koshelev et al. (1971). The scatter for these values is ±23 % but there are some individual errors that are considerably larger.

• 5. A relatively close scrutiny of the data in and behind Figures 3a, b and 4 in Koshelev et al. (1971) is thus necessary.

4.2 4.2 Figure 3a

Figures 3a and 3b from Koshelev et al. (1971) are shown in Fig. 4. Figure 3a on the left shows 4 sets of sieving data from a series of 11 shots in limestone stones with 20–500 g Hexogen charges. The fragments were sieved and weighed, then plotted as ln[10 × ln(1/R)] vs. ln(x) with x in cm. Straight lines were fitted to these data using linear regression to get values of x ₀ and n. Such data are given in their Table 2.

Fig. 4

Figures 3a + 3b from Koshelev et al. (1971)

The data is analyzed here both to (1) check the suitability of the RR function, (2) to estimate the errors involved in the calculation of $\left\langle {x_{Z} } \right\rangle$ and to check the data behind Figure 4 in Koshelev et al. (1971). The digitized values of the four data series in Figure 3a are expressed as x and P(x) and given in Table 7 where we assume or estimate that Koshelev et al. (1971) have used a series of sieves with decimal mesh sizes, x _est in the final column. We have let this sieve series be a guide for the sorting of the data columns.

Table 7 Digitized data from Fig. 4, left half, converted to x (mm) and P(x) %

This sorting has left empty spaces in all four series of sieve sizes and mass passing values. It is as if the same set of sieves was not used in all tests. This is puzzling but Koshelev et al. (1971) give no indication as to why this might be the case and we do not have any exact numbers to go on. Accepting that the series of sieve sizes x _est in Table 7 are correct we obtain the statistics for the relative ratio x/x _est = 0.975 ± 0.036. This would imply a relatively small error of −2.5 % in the drafting and digitizing of the mesh size values.

The P(x) data in Table 7 were used in RR curve fits in P vs. x space to determine x ₀ and n values. An example from the curve fits is given in Fig. 5. The choice is motivated below.

Fig. 5

The fitting of a RR function in P(x) space to the test no. 2 data in Table 7

The numerical results of this analysis are given in Table 8 in the columns to the right, the 3rd set of data. In the table M is the stone mass, with Koshelev’s et al. (1971) symbol Q ⁰, given within parenthesis. The 1st set of x ₀ and n values refers to Koshelev’s straight line fits and the data from his Table 2. The italicized test numbers 2, 4, 8 and 9 in the table are the test numbers in Fig. 4. To obtain the 2nd set of x ₀- and n values in Table 8, an interpolation of the data in Table 7 to obtain x ₅₀ was made. The straight lines in Figure 3a of Fig. 4 were digitized separately to obtain n. From these values of x ₀ = x ₅₀/(ln2^1/n) were calculated.

Table 8 Stone test data from Table 2 in Koshelev et al. (1971) compared with curve fit analysis of Fig. 4 (their Fig. 3a) data

The quality of the fits is quite good since r ² is 0.9946 or larger. The agreement between the 2nd and 3rd data sets of x ₀ and n values is also reasonable except for the n value for test 2. Where the digitization of the lines in Fig. 4 gave n = 1.803, the P(x) curve fitting gave n = 1.477, which is nearly 20 % lower. The curve fitting result in Fig. 5, which gave the values in the right-hand columns in Table 8, clearly favor the large P(x) values.

The digitized straight line for test 2 in set 2 of Table 8 represents Koshelev’s et al. (1971) linear regression in ln{10 × ln[(1/(1 − P))]} vs. ln(x) space. A linear regression to the digitized ln{10 × ln[(1/(1 − P))]} vs. ln(x) data for test 2 in Fig. 4 was made with the results that x ₀ = 22.8 cm, n = 1.82 and r ² = 0.980. These values agree reasonably well with the digitized line data and it clear that a transformation of the P(x) data plus the subsequent regression in ln{10 × ln[(1/(1 − P))]} vs. ln(x) space returns a best fit that emphasizes the smaller P(x) values. For the fit to the data in Fig. 5 we then obtain a much steeper slope, i.e. a much higher n value than n = 1.477. Thus, we conclude that the two regression methods in cases like this may yield quite different results even if the curve fit results are both in different meanings ‘best’.

If we include the 1st set of x ₀- and n values in Table 8 in the comparison, the agreement is still reasonable for tests 4, 8 and 9 but not for test 2. The 2nd and 3rd sets for test 2 look more like the 1st sets for tests 3, 5 and 6. c.f. italicized entries in columns 4 and 5 in Table 8. Thus, we suspect that Koshelev et al. (1971) have made a mistake in the labeling of the data sets in the left half of Fig. 4.

The data of Table 7 together with RR functions defined by the parameters in Koshelev’s (1971) Table 2 and in Table 8 are shown in Fig. 6, where the dashed line shows the RR curve for test 5. This choice was made by plotting in turn the RR curves for tests 3, 5 and 6 and comparing with the position of the straight line for test 2 in Fig. 4. Test 5 gave the best visual agreement.

Fig. 6

Comparison of P(x) data from digitization with RR functions based on Koshelev’s (1971) Table 2

Our digitized data in Fig. 6 are consistent with Koshelev’s et al. (1971) tabulated RR parameters, which means that our digitization errors must be relatively insignificant and that their data are consistent when we accept a probable mix up on their side of the test 2 and 5 data in Fig 3a of Fig. 4.

4.3 4.3 Figure 3b

The right half of Fig. 4, Koshelev’s et al. (1971) Figure 3b, show blasting tests that were made “in an outcropping of a continuous mass”, which may mean either a crater blast or a natural bench. In Koshelev’s et al. (1971) Table 3, the tests are described by the charge weight Q = 20–500 g, a (broken) rock mass Q ⁰ = 29–1711 (kg?) and the unexplained parameter H (cm?). The question marks indicate that the units kg and cm are guesses. A single hole crater blast seems to be the most probable blast geometry but this is not important here. These data were not used for any prediction equations for x ₀ “since the range of change in x ₀ was too small”, see their p. 254.

The digitized values of the four data series in Figure 3b are expressed as x and P(x) and given in Table 9 where we assume the same series of sieves with decimal mesh sizes, x _est in final column, as in Table 7. The statistics for the relative ratio in Table 9 become x/x _est = 1.015 ± 0.018 instead of 0.975 ± 0.036. The agreement points to a reasonable consistency in our analysis.

Table 9 Data from Fig. 4, right half, converted to x (mm) and P(x)  %

The P(x) data in Table 9 were used in RR curve fits to determine x ₀ and n values. The numerical results of this analysis are given as data set 3 in Table 10. The other data in Table 10 were obtained in the same manner as the corresponding data in Table 8. In further analogy with the data in Table 8 the data in Table 10 shows that the 2nd and 3rd sets for test 1 look more like the 1st set values for test 2, and those for test 3 look more like the 1st set values for test 7. Thus, we suspect that Koshelev et al. (1971) also made a mistake in the labeling of the data sets in Figure 3b of Fig. 4.

Table 10 Stone test data from Table 3 in Koshelev et al. (1971) compared with analysis of Fig. 4 (their Fig. 3b) data

The data of Table 9 together with RR functions defined by the parameters in Koshelev’s (1971) Table 3 and in Table 10 are shown in Fig. 7. If the parameters for their test 2 (dashed line) is used instead of the parameters for their test 1 (solid line), and the parameters for test 7 instead of those for test 3, then visually the RR functions appear to fit the data pretty well.

Fig. 7

Comparison of P(x) data from digitization with RR functions based on Koshelev’s (1971) Table 3

Thus, our digitized data are again consistent with Koshelev’s et al. (1971) tabulated RR parameters, which means that our digitization errors must be relatively insignificant and that their data are consistent when we accept a probable mix up on their side of the data from test 1 with 2 and data from test 3 with 7 or typographical errors to the same effect.

4.4 4.4 Calculation of $\left\langle {x_{Z} } \right\rangle$ from sieving data

The P(x) and x _est data in Tables 7 and 9 make it possible to calculate the experimental mean values $\left\langle {x_{Z} } \right\rangle$ for Koshelev’s et al. (1971) sieving data and to compare them with the x ₀ values from their Tables 2 and 3, which are given in Tables 8 and 10. Note that even if Koshelev et al. use the term mean fragment size for x₀, most probably x₀ refers to the characteristic sizes of fitted RR functions.

The $\left\langle {x_{Z} } \right\rangle$ calculations have been made with the estimated sieve series x _est in Tables 7 and 9. Since Eq. 12 reads $\left\langle {x_{Z} } \right\rangle$ = Σx _iz  × ΔP _i , summed over all size fractions i = 1,…,n, we still need to know (1) what x _iz value to use for a given mesh interval and (2) what is the upper mesh size limit x _nu through which the largest fragment passes. Neither Kuznetsov (1973) nor Koshelev et al. (1971) give any data for x _nu .

Koshelev’s et al. (1971) clearly state that in their Eq. (2.16) on p. 249, Eq. 12, x _iz should be chosen as “the mean size of the ith group”, i.e. the midpoint x _im  = (x _il  + x _iu )/2 of the size interval with lower and upper size limits, x _il and x _iu , respectively. Kuznetsov (1973) presents a slightly different equation for the experimental mean $\left\langle x \right\rangle$ = Σx _i  × ΔP _i , his Eq. (6) on p. 144, and states that “…the probability that a fragment has a dimension lying in the range from x _i to x _i  + Δx _i is simply the relative content of that fraction…”. The two equivalences x _il  = x _i and x _iu  = x _i  + Δx _i follow literally and from his Eq. (6) then follows that Kuznetsov uses the lower interval end point x _i  = x _il as x _iz in the summation to obtain $\left\langle {x_{Z} } \right\rangle$, rather than using the midpoint.

The natural x _iz choice in the $\left\langle {x_{Z} } \right\rangle$ calculation would be the midpoint value as the mean is the point that neutralizes the sum of the moments of all fractions whose gravity centers should then decide the leverage arm. Using x _il is unrealistic as the smallest size fraction does not contribute since x _1l  = 0. Nor does the upper size limit of the largest fraction x _nu then contribute. This bias is worse but it is maybe practical as measuring x _nu accurately in a large muck pile may be difficult.

To estimate the effect of the x _iz choice three alternatives were tried; x _il , x _im and x _iu . This was done while choosing 100 % passing x _nu values from the next mesh size in the estimated sieve series x _est in Tables 7 and 9. For test no. 8 in Fig. 3a we assume that the mesh size following 400 mm would be 500 mm. Then $\left\langle {x_{Z} } \right\rangle$ was calculated for all seven series of sieving data using in each case the three different values of x _iz. The results are given in Table 11.

Table 11 Calculated values for the experimental mean $\left\langle {x_{Z} } \right\rangle$ of the data in Koshelev’s (1971) Fig. 3a, b

We take the $\left\langle {x_{Z} } \right\rangle$ values based on the interval midpoints as references and also as step one in the analysis below. Table 11 shows that the values based on the lower end points are on average 12 % lower and those based on the high end points on average 8 % higher. Thus, a correct choice of x _iz is important.

Of the 21 $\left\langle {x_{Z} } \right\rangle$ values in Table 11 all values except the two italicized ones 86.0 and 93.7 to the lower right are lower than the corresponding x ₀ values. The $\left\langle {x_{Z} } \right\rangle$ values for tests 8 and 9 of Fig 3a are substantially lower, about 35 %. See italicized numbers in columns headed by 8 and 9. For these two tests Table 7 gives P(x) as less than 45 % and the top part of the sieving curve is consequently ill defined.

Step two in the sequel was to extrapolate the P(x) data in Tables 7 and 9 to find the 100 % size x _nu  = x ₁₀₀ for which P(x ₁₀₀) = 100 %. This extrapolation was made in log–log space and could not have been made with the RR fit since the RR function has no top size. The $\left\langle {x_{Z} } \right\rangle$ calculations used the midpoint x _iz values. The results are given in Table 12. The agreement between $\left\langle {x_{Z} } \right\rangle$ and x ₀ is reasonable, about ±15 %, for the 6 of the 7 tests, especially for those where the top passing values P _max lie in the range 70–90 %. The $\left\langle {x_{Z} } \right\rangle$ value 318 mm for test 9 is still 20 % below the x ₀ value 396 mm.

Table 12 Calculated values for the experimental mean $\left\langle {x_{Z} } \right\rangle$ of the data in Koshelev’s (1971) Fig. 3a, b

Step three was to find the rounded x _nu values that give the ‘best’ agreement between the $\left\langle {x_{Z} } \right\rangle$- and the x ₀ values, using the midpoint x _iz values. The results are given in Table 12. The best $\left\langle {x_{Z} } \right\rangle$ values now lie within ±4 % of the x ₀ values. To get this agreement x _nu had to be changed by about −20 to +130 % from the first estimate. From the perhaps more natural extrapolated 100 % value of step 2, x _nu had to be changed by −30 to +110 % to obtain this agreement. Seeing how sensitive the calculations of the experimental mean values $\left\langle {x_{Z} } \right\rangle$ are, the use of a fitted RR curve to determine x ₀ and then to use that the characteristic value x ₀ ≈ $\left\langle x \right\rangle$ to arrive at an estimate of the mean becomes understandable.

Estimating the mean through a RR curve fit to obtain the characteristic size x ₀ is also prone to errors when the data deviate from the RR function. To calculate an estimate of x ₅₀ is much more accurate and needs no assumption about the fragment size distribution. In the worst cases, tests 8 and 9 of Fig 3a, an extrapolation of the data from the 45 % to the 50 % point in linear space or in log–log space suffices. The x ₅₀ values may then be converted to x ₀, compare the different entries in Tables 8 and 10.

Since the mean is especially sensitive to the quality of the data at the top end of the sieving curve, the mean is also more sensitive to the sometimes occurring dust and boulders effect, i.e. when the top end of the curve disintegrates into a relatively small number of discrete blocks rather than being well-represented by a continuous distribution.

4.5 4.5 Figure 4

Having gained relative confidence in the digitization of the previous figures in Koshelev’s et al. (1971), if not always for the quality of drafting and handling of other data in their article, the reanalysis of their important Figure 4 was made. Their figure is shown in Fig. 8 and the digitized symbol positions are given in Table 13.

Fig. 8

Figure 4 from Koshelev et al. (1971)

Table 13 Comparison of digitized data from Fig. 8 with tabulated data from Table 8

Figure 8 contains a log–log plot of ln(10Q/V ₀) vs. ln(x ₀/Q ^1/6) based on Koshelev’s (1971) Table 2, see our Table 8. The order of the entries of digitized data in the left half of Table 13 is not the same though as the order of the entries in Table 8. First, Koshelev et al. have not equipped the data points in Fig. 8 with the corresponding test numbers from Table 8 and this table contains data for 11 tests but Fig. 8 has only 10 data points. Second, Fig. 8 gives values of Q/V ₀, i.e. specific charge linked to breakage volume V ₀, where Table 8 gives the masses Q ⁰ (or M) of the test blocks and Koshelev et al. (1971) do not provide the limestone density ρ that would connect the two values through V ₀ = M/ρ.

Koshelev’s tabulated data were correlated with the digitized data as follows: first, a reasonable density value was chosen so that a series of values for Q/V ₀ based on the tabulated Q- and M values could be calculated. This series was then compared with the corresponding series of digitized data. The series of tabulated data was then sorted to give the best agreement with the series of digitized data. Finally, the density value was fine tuned to 2540 kg/m³ because this makes the mean of the ratios of the tabulated to digitized Q/V ₀ values become very close to one. The reordered tabulated values from Table 8 are given in the right half of Table 13.

These Q/V ₀ ratios lie in the range 0.92–1.09 with the statistics 1.001 ± 0.047. With the exception of test 9, the values of corresponding x ₀/Q ^1/6 ratios lie in the range 0.96–1.01 with the statistics 0.988 ± 0.021. The tabulated data for test 6 have been positioned directly beneath those of test 3 in Table 13. It is difficult to tell if the tabulated test 3 or test 6 data are better matched to the digitized data on the same row as the tabulated test 3 data. The Q/V ₀ ratio 0.997 for test 6 is closer to 1 than the ratio 1.035 for test 3 but the x ₀/Q ^1/6 ratio 0.952 for test 6 lies outside the range 0.96–1.01 for the other tests so the data for test 6 were excluded. There is a small chance that two data points in Fig. 8 have been plotted on top of each other but Koshelev et al. (1971) do not mention this possibility.

The data set for test 9 in Table 13 is strange. The digitized x ₀/Q ^1/6 value of 148.4 in the last row of the third column is almost twice as high as the corresponding tabulated value 76.0. The most probable reason for this discrepancy is a drafting error. Apart from this point, our digitized data are quite consistent with Koshelev’s tabulated data.

What remains is thus a check of the linear regression that Koshelev et al. (1971) have made in Fig. 8, which is based on the characteristic fragment sizes x ₀ from the RR fits and which is the basis for their key Eq. (5.1), Eq. 23. Judging by Fig. 8 Koshelev et al. (1971) have made a linear regression based on the equation

$$ \ln (10Q/V_{0} ) \, = \, \ln 10 + 1/\alpha \times \ln \left( A \right) - 1/\alpha \times \ln \left( {x_{0} /Q^{1/6} } \right),{\text{ case }}1 $$

(25)

They do not provide an explanation for the choice of the exponent value 1/6, however. In our analysis we also use a more general expression with an undetermined β value, ln(x ₀) = ln{A × Q ^β/(Q/V ₀)^α} with the dependent parameter x ₀ in the left member

$$ \ln \left( {x_{0} } \right) \, = \, \ln \left( A \right) + \beta \times \ln \left( Q \right) - \alpha \times \ln \left( {Q/V_{0} } \right),{\text{ case }}2 $$

(26)

Table 14 contains the values of the linear regression parameters for these two cases. As a consistency check (case 0) the slope and intercept values of the line in Fig. 8 were determined by digitization and the intercept converted to an A value though the expression ln(A) = α × [intercept − ln(10)]. Case 2 was run through to check up on the choice of the exponent β = 1/6.

Table 14 Parameters from linear regression to fragmentation data, italicized rows data provide the lines plotted in Fig. 1

When extracting the individual x ₀-, Q/V ₀- and Q values from the digitized values of ln(10Q/V ₀) and ln(x ₀/Q ^1/6), the factor 10 was first removed in the first ln-term. Then with the help of V ₀ = M/2540, where M is given in Table 8, Q was obtained and with Q finally x ₀ from the second term. The digitized data from test 9 were not included. This means that the number of points in the regression was 9 for the digitized data and 10 for the tabulated data.

To find out which is the better fragmentation descriptor, the median or the mean, the same regressions were also made where x ₅₀-data replace the x ₀-data through x ₅₀ = x ₀ × (ln2)^1/n. Note that we only have access to interpolated x ₅₀ values for 4 of the 11 tests in Table 13 and thus cannot base the comparative regression on the interpolated x ₅₀ values. The values of ln(x ₅₀/Q^1/6) are given in Table 13. The regression results are given in the lower half of Table 14.

The results in Table 14, show in general relatively small variations. First, the consistency check of case 0 gives the parameter value A = 9.73 and α = 0.847. Both lie quite close to the reference values A = 10 and α = 0.8 that are given in Eq. 23. For the regressions based on x ₀, A lies in the range 9.08–10.59 with the statistics A = 9.72 ± 0.68 and α in the range 0.77–0.90 with the statistics α = 0.84 ± 0.05. For the case 2, regression with free β values, β = 0.174 on average or slightly larger than the value 1/6 chosen by Koshelev et al. (1971).

For the regressions based on x ₅₀, the individual A values are as expected always smaller than the corresponding A values for the x ₀ regressions. The statistics are A = 7.99 ± 0.77. The same may be said for α, with the statistics 0.83 ± 0.05. The individual β values for the x ₅₀ regressions are on the other hand larger than the corresponding β values for the x ₀ regressions. The average β = 0.189 is in fact closer to 1/5 than the 1/6 proposed by Koshelev et al. (1971) and Kuznetsov (1973).

The transformation x ₅₀ = x ₀ × (ln2)^1/n involves the n values in the x ₅₀ regression. As the values for the coefficient of determination r ² for each x ₅₀ regression is higher than the r ² value for the corresponding x ₀ regression, on average 0.938 vs. 0.931, the introduction of the varying n values has actually improved the fit, if only marginally.

The used n values in Table 8, all except those of test 5, lie in the range 1.28–2.05 and the statistics are n = 1.66 ± 0.30. Taking the mean to calculate the conversion factor in x ₅₀ = x ₀ × (ln2)^1/n we obtain (ln2)^1/n = 0.80. This is close to the A ratio 7.99/9.72 = 0.82. Thus, the tabulated and digitized data are consistent as are the two data groups based on x ₀ and x ₅₀.

The results in Table 14 are summarized in Table 1. The case 1 regression with the tabulated x ₀ data comes closest to Koshelev’s et al. (1971) analysis. Plots of this case and the corresponding regression with the x₅₀ data are given in Fig. 1. The parameter values come from the italicized lines in Table 14.

4.6 4.6 A note on the equivalence $\left\langle x \right\rangle$ ≈ x ₀

During the review process, the matter came up of how accurate the equivalence $\left\langle x \right\rangle$ ≈ x ₀ used by Kuznetsov (1973) and Koshelev et al. (1971) is. This equivalence means that Γ(1 + 1/n) ≈ 1. Their articles are not very specific on this matter. Koshelev et al. in eqns (2.13) to (2.15) give exact formulas for the mean $\left\langle x \right\rangle$ and what they call dispersion D, actually the variance of the RR function plus the approximate equivalences $\left\langle x \right\rangle$ ≈ x ₀ and D ≈ $\left\langle x \right\rangle$ ²/n when n ≫ 1. They blast aluminum rings and find n values in the range 1.9–3.3. For their blasting of rock the n values in their Tables 2 and 3, Tables 8 and 10, are considerably lower and lie in the ranges 1.28–2.05 and 0.91–1.70, respectively. Calling this n ≫ 1 is a bit farfetched.

Kuznetsov (1973) in eqns. (8) to (10) gives the mean and the dispersion (variance) plus the approximate expressions $\left\langle x \right\rangle$ ≈ x ₀ and D ≈ x ²₀ /n when n > 1. Eleven of the twelve fits in Table 3 that Kuznetsov gives as support for the RR function have n values less than 1.0 however. For the smallest value, n = 0.47 and Γ(1 + 1/n) = 2.26 and there is definitely no longer an equivalence $\left\langle x \right\rangle$ ≈ x ₀. For the production blasts in Appendix 2 no n values are mentioned. For the two fits in Appendix 3, n = 0.81 and 0.78, respectively. Thus, vast the majority of examples given by Kuznetsov do not respect his requirement n > 1 for the equivalence.

Γ(1 + 1/n) is larger than 1 when n < 1, equals 1 when n = 1, has a minimum of about 0.886 at n ≈ 2.17 and then approaches 1 when n ≫ 1. To have a symmetric error interval we would set Γ(1 + 1/n) < 1 + (1 – 0.886) ≈ 1.114 which occurs when n > 0.82. We would then have a maximum error of ±12 % in the approximate equivalence $\left\langle x \right\rangle$ ≈ x ₀. Still, five of Kuznetsov’s twelve examples in Appendix 1 have lower n values than 0.8, see Table 3, and if the equivalence $\left\langle x \right\rangle$ ≈ x ₀ were used we would get an underestimate of $\left\langle x \right\rangle$ by more than 12 %, sometimes much more. For Koshelev’s data for which n = 1.28–2.05 we would have Γ(1 + 1/n) ≈ 0.90 on average and get a systematic overestimate of $\left\langle x \right\rangle$ by +10 %.

This inspection of n values in the papers of Koshelev et al. (1971) and Kuznetsov (1973) thus shows that the errors involved in using $\left\langle x \right\rangle$ ≈ x ₀ are (1) of the same order as the differences between the predicted and measured fragmentation values in Appendix 2 and (2) a substantial part of the difference ±30 % between the A values of Kuznetsov’s three classes of A values in Eq. 5.

A last note; the exact expressions for the dispersion D, Eq. (2.14) in Koshelev et al. (1971) and Eq. (9) in Kuznetsov (1973) are not identical and a check with e.g. Weibull (1939) shows that they are both incorrect since D = x ²₀  × [Γ(1 + 2/n) − (Γ(1 + 1/n))²] for the RR function. The approximate expressions for D are incorrect too.