Transcript Predicting Fragmentation

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 KuzRam method