Kuz–Ram Mean Fragment Size

Kuz-Ram is Claude Cunningham's 1983 marriage of two older pieces of work: Kuznetsov's equation for the mean fragment size a given charge produces in a given rock, and the Rosin-Rammler distribution to spread that mean into a full size curve. The name is the two authors' initials. Presented at the first International Symposium on Rock Fragmentation by Blasting at Lulea and refined at the second at Keystone in 1987, it became and remains the standard first estimate of blast fragmentation.

The equation reads $X_m = A (V_0/Q_e)^{0.8} Q_e^{1/6} (115/E)^{19/20}$, and it is not dimensionally homogeneous — the constants absorb the units. $V_0$ is the rock volume broken per hole in cubic metres, $Q_e$ the explosive per hole in kilograms, $E$ the relative weight strength against ANFO at 100, and the answer comes out in centimetres. The 115 is the relative weight strength of TNT, the reference Kuznetsov's original work was written against. The dominant term is $(V_0/Q_e)^{0.8}$, which is the inverse powder factor raised to the 0.8 power: more explosive per unit volume gives finer muck, roughly as the 0.8 power, and that sensitivity is the most trustworthy thing in the model.

Now the honesty, because it is the whole point of the page. THE ROCK FACTOR $A$ IS A JUDGEMENT AND THE PREDICTION RIDES ENTIRELY ON IT. $A$ runs from about 1 for very weak, heavily jointed material to 13 or 14 for hard, massive, tightly jointed rock, and it is assembled from ratings for rock mass description, joint spacing, joint orientation relative to the face, density and hardness — read off descriptive tables by a person standing at a face. Two competent engineers will differ by something like ±30 %. That is not carelessness; it is the resolution of the method. And because $A$ multiplies the answer LINEARLY and one for one, the fragmentation prediction moves with them by the same ±30 %. A Kuz-Ram result quoted to three significant figures is quoting the arithmetic, not the fragmentation.

What follows is a rule about how to use the model. Kuz-Ram predicts a trend well and an absolute size poorly. Hold the rock factor fixed and compare two patterns on the same rock, and the ratio between the answers is worth trusting — that is the model doing what it is good at. Take the absolute number to a crusher vendor as a specification and it is not. The far more valuable use is to run the model BACKWARDS: measure the fragmentation of a shot you actually fired, by image analysis or screening, and back out the rock factor your own ground really has. A calibrated $A$ absorbs everything the model gets wrong about your particular rock along with everything it gets right, and a few of them from different areas of the pit give you a site-specific fragmentation model instead of a borrowed one.

Four further limitations are worth knowing. The model UNDER-PREDICTS FINES badly, because the crushed zone immediately around each hole follows quite different physics from the fracture network further out — the modified and extended Kuz-Ram models exist mostly to patch exactly this. It has no term for DELAY TIMING, which real experience says matters a great deal. Its treatment of jointing is a rating inside $A$ rather than a mechanism, and in a rock mass whose natural block size is smaller than the predicted fragment size the joints govern completely, the blast merely loosening what was already there. And $E$ is a chemical energy figure: it says nothing about how much of that energy actually couples into the rock, which depends on the borehole pressure and on how well the charge fits its hole. Notice how little $E$ moves the answer in any case — going from ANFO at 100 to a strong emulsion at 130 improves the mean size by only about 22 %. Within the range of commercial products, WHERE you put the explosive matters more than WHICH explosive it is.