## Abstract

Cunningham’s use of x50, the median fragment

size, instead of the mean hxi 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 x50 with the

same goodness of fit as the equation for the mean. (4) Use

of x50 instead of the mean hxi in a dimensional analysis of

fragmentation leads to considerable mathematical simplifications

because the normalized mass passing at x50 is a

constant number. Non-dimensional ratios like x50/xmax

based on two percentile sizes also lead to such simplifications.

The median x50 as a fragment size descriptor thus

has a sounder theoretical background than the mean hxi. 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.

size, instead of the mean hxi 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 x50 with the

same goodness of fit as the equation for the mean. (4) Use

of x50 instead of the mean hxi in a dimensional analysis of

fragmentation leads to considerable mathematical simplifications

because the normalized mass passing at x50 is a

constant number. Non-dimensional ratios like x50/xmax

based on two percentile sizes also lead to such simplifications.

The median x50 as a fragment size descriptor thus

has a sounder theoretical background than the mean hxi. 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.

Original language | English |
---|---|

Pages (from-to) | 143-164 |

Number of pages | 22 |

Journal | Rock mechanics and rock engineering |

Volume | 49.2016 |

Issue number | 1 |

DOIs | |

Publication status | E-pub ahead of print - 15 Mar 2015 |

## Keywords

- Blast fragmentation Prediction equation