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Incorporation of a Fractal

Breakage Mode into the BrokenRock ModelDosti Dihalu and Bas Geelhoed
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Purpose of this presentation

To investigate how the variance changes withparticle size reduction (comminution) Broken Rock Model (Minnit 2007)

the breakage pattern of a particle is regular, thefragments will all have the same size and weight

Here proposed: Fractal Broken Rock Model

the breakage pattern of a particle is a fractal, thefragments will have unequal sizes and weights


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Outline

1. Introduction1a. Sampling

1b. Theory of Sampling (TOS)

1c. Developments in TOS

2. Methodology: FBRM

3. Worked-out example4. Future Work

5. Conclusions


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1. Introduction

1a. Sampling1b. Theory of Sampling (TOS)

1c. Developments in TOS


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1. IntroductionVariance Estimator

2 2

2 21

3

1( ) (A )

f (B )

(C )

ba tc h N

i i b a tch

ib a tch b a tch

b

qV m c c

q c M

g cD

M

K D

M

V = the relative varianceq = the inclusion probability of each particlecbatch = the concentration of property of interest in the population

Mbatch = the (total) mass (or weight) of the populationm i = the mass of the i-th particle in the populationc i = the concentration of property of interest in the i-th particle ofthe populationf = shape factor (unitless quantity)g = size range factor (unitless quantity)

= the liberation factor (unitless quantity)c = the mineralogical factor (specified in the same units asdensity, e.g. g/cm 3 or kg/m 3)

D = nominal size dimension of the particle (specified in units oflength)M = mass of the sample (specified in units of mass)K = fg cD 3-bb = an exponent that can be selected freely, but in practice willbe set to a value that describes how the variance predictionchanges with D


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2. Methodology: FBRM (A)

(B)

(C)

(D)

Model yields 2 N fragmentsof varying particle mass m i

A. particle undergoes N breaking stagesB. particle is broken infragments by generating arandom number (r 0) between 0and M 0C. 2 new fragments (mass M 0,1= r 0 ,mass M 0,2 = M0-M0,1

D. each of the 2 fragments isbroken (breakage point offragment M 0,i = r 0,i ) in fourfragments of mass M 0,1,1 , M0,1,2 ,M0,2,1 , and M 0,2,2


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2. Methodology: FBRM Program

INPUTParticle massParticle concentrationPurity of inclusionNumber of breakinggenerationsMode (BRM/FBRM)

OUTPUT

Fragment number Fragment massFragment concentration


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3. Worked-Out Example

(step 1) population A: 15 particles (c 1, m 1) and 15 others(c 2, m 2)(step 2) population B by breaking population A according to FBRM(step 3) Calculate sampling variance of population B ( Gys formula).(step 4) Calculate for each particle i of population B D i and D.(step 5) Repeat steps 2, 3, and 4 for varying breaking generations N(step 6) Plot the Var(FSE) vs D and deduce the K and b value graphically .


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3. Worked-Out Example-Concentration


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3. Worked-out example

Particlenumber

Particle mass Particleconcentration

Purity ofinclusion

1-15 50 20 0.99

16-30 50 5 0.99

Starting population:

D indicates the 95 th percentile value ofthe total collection of obtained D is (Gy, 1979).

In this experiment, the maximum number of breaking generations (N max) was set to N max=10.


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LOG(VAR(FSE)) = 0.88 * LOG(D) - 3.53

-4

-3,8

-3,6

-3,4

-3,2

-3

-2,8

-0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7

L O G ( V a r (

F S E ) )

LOG(D)

3. Worked-out example-Variance vs Diameter


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3. Worked-out example-Interpretation of Results

It can be concluded that - at least in thisexperiment- either the parameter b, or theparameter K, or both b and K at the sametime, depend on the particle size D.

Based on this graph, the best choice ofparameter b (that provides the best fit) is thevalue of b=0.88 2.5 (see also Franois-Bongaron & Gy, 2002)


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4. Future Work Using the FBRM to investigate how the parameters f,

g, , and c from Gys theory change with D (Equation1B).

Upscaling the one-dimensional fractal breakage

model to a three-dimensional fractal breakagemodel. Implementing more realistic particle concentration

profiles in addition to the step function that is usedhere.

Implementing a more realistic way of assigning adiameter (D i) to each output fragment.


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