Predicting Fragmentation

©Dr. B. C. Paul 2000

Note – This series of slides portrays the author’s summary of knowledge commonly held by people well studied in the field. As indicated in the slides the original

contributions to the state of knowledge by Kuznetsov, Cunningham, and Calvin Konya are noted and recognized. Some of the formuli contained here in have been adapted for English units by the author from original formuli from the recognized

contributors.

Fragmentation Prediction

• Screening Every Shot to Get Data is very Difficult

• Need to Get a Mathematical Model that Approximately Fits– Like Bell Curve for test distributions

• Formula Used is Rossin Ramler

• Schuman Plot Fits Crusher Distributions well but not Blast Fragments

Rossin Ramler Distribution

• R = 100 * e ^ { ( x/ xc ) ^ n }

– Where• R is the percent retained on a screen of size x openings

• xc is the characteristic size for the distribution (it is a parameter similar to the mean in a normal distribution)

• n is the uniformity - high values indicate a narrow spread is size while low values indicate large spread (it is a parameter like variance in the normal distribution)

• Called a Two Parameter Distribution

Limitations in Fitting Mathematical Distributions

• Remember that blast fragmentation distribution are product of three different families formed by three different mechanisms– Usually design to limit boulder zone

– Crush zone tends to be naturally small

• Unbounded distribution - tell you that there is a certain percentage of blast fragments from your quarry the size of the moon– Take the outer about 5 or maybe 2% with a grain of salt

Developed a Series of Empirical Equations for Predicting xc n

• Use a more common blasters parameter called d50

– d50 means the size where 50% passes

– d50 more popular in Europe

– US traditionally likes d80 (80% passing size)

• Mathematical relationship between d50 and xc

– xc = d50 / {0.693 ^ ( 1/n ) }

Empirical Equations

• Developed from Work of a Russian Scientists Kuznetsov in late 60s– Equation with modifications

– d50 = Rf * [ ( 1.25 * PF ) -0.8 ] * [ (Ch / 2.2) (1/6) ] * [ (115 / E ) (19/30) ] / 2.54

– Equation shown is adapted to U.S. units

• PF is Powder Factor in lbs/ton

• Ch is the Charge per hole in lbs

Kuznetsov’s Equation

• E is the relative weight strength– parameter for explosives from manufacture– Usually developed from a “Bubble Test”

• Fire Underwater and see how big the splash is

– Original work was based on E = 100 for TNT (The Russians had a lot of military surplus they used in their mines)

– Adapted to U.S. Practice with ANFO = 100

Kuznetsov’s Equation Continued

• Rf is the Fudge Factor Rock Factor

– Soft Rocks 6 - 7– Medium Rocks (such as Quarry Limestone) 9– Hard Igneous Rocks 12 - 14

• Most rocks will be 7 to 13 (Kuznetsov worked with some very extreme appetite ores)

Modern Adaptations of Kuznetsov’s Equations

• Late 1980’s Dr. Paul adapted to describe crater shot data– Values in 6 to 7 range typical for medium rocks– 5.5 for soft– harder than 9 or 10 rare

• Crater Shots Tend to Produce finer characteristics for same powder factor– down size is loss of uniformity

Mid 1980s Cunningham developed an equation for n

• Cunningham was a South African

• Named his integrated Message Kuz-Ram

• n = ( 2.2 - 0.168 * B / De ) * [ 1 - W / B ] * [ 1 + (A -1) /2] * (PC - J) / L– B is Burden in feet

– De is hole diameter in inches

– A is the spacing to burden ratio

– PC is the length of the powder column

– J is the subgrade

– L is the bench height

Notes on the Cunningham Equation• W is the drill hole deviation

– bottom of hole deviates from the perfect pattern

– generally know what % deviation for a given drilling distance

• A is the Spacing to Burden Ratio– Reaches optimum value at 2

• As shown with Konya method 2 is optimum only for limited conditions where formula was developed– Suggest [ 1 + (A - 1)/2 ] be set equal to 1.5 if Konyas method

was used

Applying the Equations

• Get d50 from the Kuznetsov Equation

• Get n from Cunningham’s Equation

• Use the Mathematical Relationship to get xc from d50 and n

• Put Parameters in Rossin Ramler Distribution

• Check key sizes with the so called Kuz-Ram method