Rosin–Rammler Passing Fraction

Paul Rosin and Erich Rammler published this distribution in the Journal of the Institute of Fuel in 1933, fitting the size distribution of ground coal. It is mathematically a Weibull distribution, and it describes broken and ground brittle material remarkably well across an enormous range of scales — from a muck pile of metre boulders down to a ball mill product at fifty micrometres. Two parameters do all the work.

The first is the CHARACTERISTIC SIZE $x_c$, and it is neither the mean nor the median. Put $x = x_c$ into the equation and the exponent becomes exactly 1, so the passing fraction is $1 - e^{-1} = 0.632$ — for EVERY uniformity index at once. That is the definition: $x_c$ is the size that 63.2 % of the material passes, and that specific number falling out of $e$ is what makes it characteristic rather than arbitrary.

The second is the UNIFORMITY INDEX $n$, and it is the parameter worth spending money on. It sets how steep the curve is. A low $n$ means a wide spread — plenty of fines and plenty of oversize around the same average. A high $n$ means everything close to the same size. Blasted rock usually lands between 0.7 and 2.2, and Cunningham's own correlation puts it near 1.1 to 1.5 for ordinary bench patterns, rising with drilling accuracy and with a spacing-to-burden ratio near the recommended value. What the plant downstream cares about is not the average size but the CONSISTENCY: a low $n$ means the same tonnage arrives as a mixture of boulders the crusher chokes on and fines that were ground for nothing.

Two traps, and both cost real money. The first is which fraction you are looking at. Rosin and Rammler wrote their distribution as the RETAINED fraction, $R(x) = \exp[-(x/x_c)^{n}]$, and much of the blasting literature still quotes it that way; this page gives the PASSING fraction, $1 - R$, because that is how a screen analysis and every P80 in mineral processing are reported. The two are complements, and confusing them turns an 80 % passing size into a 20 % passing size. The second is that THE MEAN IS NOT THE P80. A blast quoted at a 30 cm mean has an 80 % passing size well above it — the ratio is $(\ln 5/\ln 2)^{1/n}$, about 1.9 at $n = 1.2$ — so a crusher sized on the mean will meet rock nearly twice that size for a fifth of its feed. Specify crushers and grinding circuits on P80, never on the mean, and say which one you mean whenever you quote a number.

Fitting the two parameters is done on one particular plot and for a good reason. Take logarithms twice: $\ln[-\ln(1-P)] = n \ln x - n \ln x_c$. That is a straight line in $\ln x$ whose slope is $n$ and whose intercept gives $x_c$, which is why the Rosin-Rammler plot exists and why fitting from a single screen point — which this page will happily do — is much weaker than fitting from the whole analysis. If you must use one point, use one near the middle of the distribution, between about 30 and 70 % passing: a fit anchored out in either tail is fitting exactly the part of the curve the distribution describes worst.

Which is the model's honest limitation. It has NO TOP SIZE. As $P$ approaches 100 % the predicted size runs away to infinity, so the distribution predicts an infinitesimal fraction of infinitely large boulders. It fits the middle of a size distribution well and both tails badly. Design against P80 or P95 and use judgement — not this equation — for what the largest block on the muck pile will be.