# Crusher 2 Model Theory

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# Model Theory - Fixed Partition Method

The user defines a fixed size distribution (partition) as the product from the Crusher.

The model will produce the user defined size distribution even if this would result in the crusher producing a coarser product than feed. This is best demonstrated using an example:

 Size' Range' (mm) Feed Fraction (%) Required Discharge Fraction (%) +10.00 20 15 +1.70 10 15 +0.85 15 20 +0.50 15 10 +0.10 20 10 +0.01 15 10 -0.01 5 20 (sum) (100) (100)

In the above example, there is enough material in the +10.0mm and +1.7mm feed size ranges to meet the required discharge product size distribution. For the +0.85mm size range only 15% is available where 20% is requested.

# Model Theory - Whiten Crusher Method

This is an empirical model incorporating breakage and classification, based on the key parameter $T_{10}$ for breakage, and parameters $K_1, K_2$ and $K_3$ for classification.

The form of the Whiten classification function is:

$C(x) = 1-\left(\cfrac{K_2-x}{K_2-K_1}\right)^{K_3}, \qquad K_1 \lt x \lt K_2$

Where the parameters K1, K2 and K3 can be entered directly or calculated from equipment variables and operating conditions.

The actual breakage function is determined as a spline fit to test data which is specified as well.

If the key parameters are calculated, the following correlations are used:

1. K1 = A0 + A1*CSS - A2*TPH + A3*F80 + A4*LLEN
2. K2 = B0 + B1*CSS + B2*TPH + B3*F80 - B4*LHR + B5*ET
3. T10 = D0 - D1*CSS + D2*TPH - D3*F80
where
CSS = Closed Side Setting (mm)
TPH = Throughput (dry t/h)
F80 = 80% passing size of Feed (mm)
LLEN = Length of face of mantle liner (mm)
LHR = Liner age (hours)
ET = Eccentric Throw (mm)
A0 to A4 are parameters for K1, obtained from data fitting.
B0 to B5 are parameters for K2, obtained from data fitting.
D0 to D3 are parameters for T10, obtained from data fitting.

Notes:

1. These are linear offsets to base values which depend on the operating conditions and crusher configuration. See MCC for further discussion.
2. The parameters A0, B0 correspond to base values of K1, K2
3. The parameters A1, A2, A3, A4, B1.. B5 are hidden and by default set to zero. Un-hide these fields if you need to adjust the K values based on operating conditions and crusher configuration.
4. So K1=A0 and K2=B0 if other parameters are not set.

## Ore Specific Breakage

The ore properties are given as a table (MCC Table 6.1, General appearance function for crusher model)

This data is determined by various ore tests. The default values are those from MCC.

A second table is provided for ore specific comminution energy, Ecs. This gives the size specific energy needed to achieve a particular degree of breakage.

Reference

Napier-Munn et al "Mineral Comminution Circuits - Their Operation and Optimisation (MCC)", Chapter 6

# Model Theory - Selection/Breakage Method

In the subsequent discussion we use the following notation, loosely following that of Taveres for discrete bin PSD modelling.

The SysCAD PSD model is based on a user supplied sieve series, with sizes $D_0, \,D_1\dots D_N$ which form N bins, with bin k containing particles in the size range $D_{k-1}\leq x \lt D_k$. Each bin has a characteristic size $d_k$, which is typically the geometric mean of the upper and lower sizes, though this can be modified by a user supplied correction.

The new comminution model implements a number of open literature correlations for selection and breakage. The overall breakage function is composed of both a selection function (breakage probability) and a breakage distribution (fragment size distribution).

## Breakage Probability Selection Functions

In the following equations $S_i$ is the probability that a particle in bin i breaks.

### Vogel Selection Function

$S_i = 1-e^{-f_{mat}d_i k(W_{m,kin}-W_{m,min})}$
where:
fMat = Material Parameter that characterises the resistance of particulate material against fracture in impact comminution.
Wm_kin = Mass-specific kinetic impact energy.
Wm_min = Mass-specific threshold energy for particle breakage, i.e. the specific energy which a particle can take up without fracture.

Note that in the SysCAD model, a size independent specific energy is specified, ie $x\times W_{m,min}$

k = Number of impact events.

Reference

Vogel-Peukert, Breakage Behaviour of Different Materials - Construction of a Mastercurve for Breakage Probability, Powder Technology, 2003.

### Austin Selection Function

$S_i = S_1 \left( \cfrac{d_i}{d_1} \right)^\alpha$
where:
S1 = Material Parameter. 0 <= S1 <= 1
$\alpha$ = Parameter. 0 <= $\alpha$ <= 1
d1 = Minimum particle size. 0 <= d1 <= dmax
dmax = Maximum particle size in the feed stream

Reference

Austin L.G. et al, A Rapid Computational Procedure For Unsteady-State Ball Mill Circuit Simulation, SME-AIME, 1984.

## Breakage function for discrete bins

The Breakage function is actually a continuous function.

The Reid and Stewart form (basically a double Schumann equation) is
$B(X) = \Phi\left(\cfrac{X}{X_0}\right)^\gamma+(1-\Phi)\left(\cfrac{X}{X_0}\right)^\beta$
This calculates the fraction of particles of size $X_0$ that will end up smaller than size X after breakage.

In the discrete form, we start with the following cumulative breakage curves.

Here $B_{ji}$ = Fraction of particles from a bin of characteristic size i breaking below size j

### Reid-Stewart form (Perry)

$B_{ji} = \Phi\left(\cfrac{D_j}{d_i}\right)^\gamma+(1-\Phi)\left(\cfrac{D_{j}}{d_i}\right)^\beta$
where:
$\Phi$ = Experimentally determined material parameter.
$\gamma$ = Experimentally determined material parameter.
$\beta$ = Experimentally determined material parameter.

Reference

Perry et al Perry's Chemical Engineers' Handbook 6th or 7th Edition, McGraw-Hill 1984.

### Austin Breakage Function

$B_{ji} = \Phi\left(\cfrac{D_j}{d_i}\right)^\gamma+(1-\Phi)\left(\cfrac{D_{j+1}}{d_i}\right)^\beta$
where the parameters are the same as for the Reid-Steward form above.
The Austin form allows for complete breakage of all selected particles in a bin since
$\frac{D_{j+1}}{d_j}\gt 1$ will force the cumulative function to be greater than 1 (and thus set to 1 for breakage to the same bin).

### Vogel Breakage Function

$B_{ji} = \cfrac12\left(\cfrac {D_j}{d_i}\right)^q\left[1+\tanh\cfrac{D_j-x'}{x'}\right]$

### Tavares Breakage Function

This implementation uses a single constant $t_{10}$ supplied as a parameter
$B_{ji} = 1-(1-t_{10i}) ^{\left[\frac9{d_i/D_j-1}\right]^\alpha}$
where
$\alpha$ = Material Parameter based on experimental data.
t10 = Material Parameter

Reference

Tavares Optimum Routes for Particle Breakage by Impact, Powder Technology 142, 2004.

### Logarithmic form

$B_{ji} = A\ln\cfrac{D_j}{d_i}+1$
where
A = Material Parameter

### User Weibull form

$B_{ji} = 1-\exp \left[-\left(\cfrac{x - x_u}{x^*-x_u}\right)^n \right]$
where
$x = \cfrac{D_j}{d_i}$
n = User Parameter. n>0
xu = User Parameter
x* = User Parameter: x* > xu