# Crusher2

Navigation: Main Page -> Models -> Size Distribution Models

## General Description

This model implements predictive comminution with a number of open literature correlations. The model can handle multiple ore streams with different selection and breakage functions and includes optional internal classification to model various closed circuit configurations.

• Selection/Breakage Models
• Whiten Crusher Model
• Ball Mill Model

The specific model is selected with the Method drop down list on the Crusher tab.

### Diagram

The diagram shows the default drawing of the Crusher, with all of the streams that must be connected to the unit.

The physical location of the streams connecting to the Crusher is unimportant. The user may connect the streams to any position on the unit.

## Inputs and Outputs

 Label RequiredOptional InputOutput Number of Connections Description Min Max. Feed 1 Required In 1 20 The feed to the Crusher Product Required Out 1 1 The product from the Crusher

## Model Theory - Selection/Breakage Method

In the subsequent discussion we use the following notation, loosely following that of Taveres[1] 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.
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

$S_i = S_1(\frac{d_i}{d_1})^\alpha$
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
$B(X) = \Phi\left(\frac{X}{X_0}\right)^\gamma+(1-\Phi)\left(\frac{X}{X_0}\right)^\beta$
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)

$B_{ji} = \Phi\left(\frac{D_j}{d_i}\right)^\gamma+(1-\Phi)\left(\frac{D_{j}}{d_i}\right)^\beta$
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

$B_{ji} = \Phi\left(\frac{D_j}{d_i}\right)^\gamma+(1-\Phi)\left(\frac{D_{j+1}}{d_i}\right)^\beta$
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
$\frac{D_{j+1}}{d_j}\gt1$ will force the cumulative function to be greater than 1 (and thus set to 1 for breakage to the same bin).

#### Vogel Breakage Function

$B_{ji} = \frac12\left(\frac {D_j}{d_i}\right)^q\left[1+\tanh\frac{D_j-x'}{x'}\right]$

#### 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

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

#### User Weibull form

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

#### Notes

• For all these distributions, the fractional breakage from bin i into bin j is then just the difference of the cumulative breakages between the upper size and lower size.
• The Reid, Taveres, Log and Weibull distributions are scale invariant and only depend on the ratio of mother to daughter particle size.
• The Vogel breakage distribution depends on the impact velocity

### Multiple Impact Events

In some crushing operations, material is subject to multiple impact events. The parameter NStressEvents is used to model this.

With N multiple impact events, the results are identical to having N individual crusher units with the same parameters and one impact event:

Following the PSD distribution through the multiple events:

#### Fractional Impact Events

The number of stress events need not be an integer. It is possible to have 1.5 stress events, which would correspond to a situation where some of the feed undergoes two impacts.

The result of having two crushers with 2.5 impact events is the same as having five crushers with a single impact event or any other combination adding up to the same total:

### Impact Velocity

The selection (and some of the breakage) models involve a velocity or specific energy which is user supplied.

• For an impact mill, this is just the impact velocity;
• For a hammer mill, it is calculated from rotation speed and diameter.

### Classification

When internal classification is turned on, additional parameters are available for the classification function

With internal classification, particles of larger sized are retained in the mill; the overall internal flow of material can be represented as follows.

The model allows user entry for all the parameters (Selection, Breakage and Classification). If there are multiple ores, data is only displayed for a single ore, and the user can select which ore to display.

Using this representation, we can model other configurations. A simple closed circuit milling system has the following transfer function:

## 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 K3 for classification.

The form of the Whiten classification function is:

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

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% pass size of Feed (mm)
LLEN = Liner Length (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 optimization (MCC)", Chapter 6

## Data Sections

The default sections and variable names are described in detail in the following tables. The default Crusher access window consists of three sections. This number may increase or decrease, based on user configuration.

1. Crusher2 tab - Contains general information relating to the unit.
2. Crusher tab will have different fields depending upon the Method selected by the user:
• Fixed Partition - The fields visible if the user selects the 'Fixed Partition' method.
• Whiten Crusher - The fields visible if the user selects the 'Whiten Crusher' method.
• Selection/Breakage - The fields visible if the user selects the 'Selection/Breakage' method.
3. Selection/Breakage - Visible when method is set to Selection/Breakage.
4. Power - Visible when method is set to Whiten Crusher.
5. QFeed - Optional tab, only visible if ShowQFeed is enabled. This page shows the properties of the mixed stream as the feed to the crusher2 model.
6. QProd - Optional tab, only visible if ShowQProd is selected.
7. Info tab - Contains general settings for the unit and allows the user to include documentation about the unit and create Hyperlinks to external documents.
8. Links tab, contains a summary table for all the input and output streams.
9. Audit tab - Contains summary information required for Mass and Energy balance. See Model Examples for enthalpy calculation Examples.

### Crusher2 Page

 Symbol / Tag Input/ Calc Description Common First Data Section Requirements On Check Box This enables the unit. If this box is not checked, then the material will pass straight through the crusher with no change to size distribution. ShowQFeed Tick Box QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed stream. See Material Flow Section. ShowQProd Tick Box QProd and associated tab pages (eg Sp) will become visible, showing the properties of the products. See Material Flow Section. SizeForPassingFracCalc Input (global) Size fraction for % Passing calculation. The size fraction input here will be shown in the results section. FracForPassingSizeCalc Input (global) Fraction passing for Size calculation. The fraction input here will be shown in the results section. Intervals Ascending Toggle Button (Intervals_Ascending), the size distribution will be displayed with the largest sizes at the top of the list. Descending Toggle Button (Intervals_decending) The size distribution will be displayed with the smallest sizes at the top of the list. Power Calculation; BondWorkIndex / BondWI Input The Bond Work Index of the rock. This number will be used to calculate the power required to produce the final product. Results VectorCount Results The number of solids for which size distribution data has been specified. SizeCount Results The number of size intervals in the size distribution. Power Calc The calculated power to grind the product. Power = 10.0 * Solid Mass Flowrate * BondWI * (1.0/Sqrt(P80) - 1.0/Sqrt(F80)) P80 and F80 are in μm. Stream Result Summary The following fields are shown in a table for the Feed and Product streams. MassFlow / Qm Calc The mass flow rate of the Feed/Product streams. SolidMassFlow / SQm Calc The solid mass flow rate of the Feed/Product streams. LiquidMassFlow / LQm Calc The liquid mass flow rate of the Feed/Product streams. VolFlow / Qv Calc The volumetric flow rate of the Feed/Product streams. Temperature/ T Calc The temperature of the Feed/Product streams. Density / Rho Calc The density of the Feed/Product streams. SolidFrac / Sf Calc The solids fraction of the Feed/Product streams. LiquidFrac / Lf Calc The Liquid fraction of the Feed/Product streams. Passing Calc Returns the % Passing of the user defined size (defined above) of the Feed/Product streams. Passes Calc Returns the size that has the user defined % Passing (defined above) of the Feed/Product streams.

### Crusher Page - Fixed Partition

The fields that will be visible when the Method selected is 'Fixed Partition'.

 Tag / Symbol Input / Calc Description/Calculated Variables / Options IndividualCurves CheckBox If this is ticked, then the user enters discharge partition curves for each individual solid with a partition curve.If the box is NOT ticked, then the user enters a single curve that is used for all of the solids.This is shown in the image below. Curves Input The user enters the required fraction of material in each size interval to match the required discharge partition curve(s). (see image below)

### Crusher Page - Whiten Crusher

The fields that will be visible when the Method selected is 'Whiten Crusher'. See Model Theory - Whiten Crusher Method for more information.

 Tag / Symbol Input / Calc Description/Calculated Variables / Options ShowResults CheckBox If this box is ticked, then the user will see a 'Results' tab with the selection / breakage values displayed. CalcKs CheckBox If this box is ticked, then the user enters all the parameters required to calculate the K1 and K2 values.If the box is NOT ticked, then the user specifies the K values directly.Note that K3 is specified directly by the user in both cases. The fields shown below are only visible if the 'CalcKs' box is ticked. (Except for the K values, which are inputs when the box is NOT ticked) ClosedSideSetting/CSS Input The Closed Side Setting of the crusher. LLEN Input Liner Length LHR Input Liner age ET Input Eccentric Throw A0 to A4 Input Parameters required to calculate K1:K1 = A0 + A1*CSS - A2*TPH + A3*F80 + A4*LLEN B0 to B5 Input Parameters required to calculate K2:K2 = B0 + B1*CSS + B2*TPH + B3*F80 - B4*LHR + B5*ET K1 Input/Calc Size below which all particles escape breakage. K2 Input/Calc Size above which all particles are crushed. K3 Input Exponent. This should be fixed at 2.3. CalcT10 CheckBox If this box is ticked, then the must enter all of the parameters required to calculate T10.If the box is NOT ticked, then the user specifes T10 directly. D0 to D3 Input Parameters required to calculated T10:T10 = D0 - D1*CSS + D2*TPH - D3*F80These fields are only visible if 'CalcT10' is ticked. T10 Input/Calc 10% Passing. Calculated T values Ti Calc The calculated T values will be displayed here. BreakageAtSize Input Ore Specific Data - One set for each ore Ti data Input/Calc The user enters the required data for the General appearance function for the crusher model into this table.

### Crusher Page - Selection/Breakage

The fields that will be visible when the Method selected is 'Selection/Breakage'.

Tag / Symbol Input / Calc Description/Calculated Variables / Options
ShowRecalc CheckBox Tick this box to show when data arrays are recalculated.

#### Impact Methods

Impact. This method uses simple user defined Impact velocity.
ImpactVelocity Input or Result If the 'ImpactType' = Impact, then the user will specify this parameter. It is calculated for the Hammer Mill.
Hammer. The impact velocity is calculated from the rotation speed and diameter.
RotorSpeed Input Rotational Speed for a Hammer Mill.
RotorDiameter Input Rotor Diameter for a HammerMill.

#### Breakage Probability Selection Methods

Vogel-Peukert. Uses the Vogel Selection Function. The user enters the following variables for each ore type.
fMat Input Material Parameter that characterises the resistance of particulate material against fracture in impact comminution.
Wm_min Input Mass-specific threshold energy for particle breakage, i.e. the specific energy which a particle can take up without fracture.
k Input Number of impact events.
Austin. Uses the Austin Selection Function. The user enters the following variables for each ore type.
S1 Input Material Parameter. 0 <= S1 <= 1
x1 Input Minimum particle size. 0 <= x1 <= Maximum particle size in the feed stream.
a Input Parameter. 0 <= a <= 1
User. The user enters their own Selection data on the Selection/Breakage tab.

#### Breakage Functions

Reid-Steward. Uses the Reid-Steward Breakage Function. The user enters the following variables for each ore type.
Phi Input Experimentally determined material parameter.
Gamma Input Experimentally determined material parameter.
Beta Input Experimentally determined material parameter.
Austin. Uses the Austin Breakage Function. The user enters the same variables as for the Reid-Steward method, shown above.
Vogel-Peukert. Uses the Vogel-Peukert Breakage Function. The user enters the following variables for each ore type.
c Input Experimentally determined material parameter.
d Input Experimentally determined material parameter.
MinFragSize Input The minimum size to which the material can break.
Tavares . Uses the Tavares Breakage Function. The user enters the following variables for each ore type.
alpha Input Material Parameter based on experimental data.
t10 Input Material Parameter.
Logarithmic. Uses the Logarithmic Breakage Function. The user enters the following variable for each ore type.
A Input Material Parameter.
Weibull. Uses the Weibull Breakage Function. The user enters the following variables for each ore type.
n Input User Parameter: n>0
xu Input User Parameter.
xStar Input User Parameter: xStar>xu
User. The user enters their own Breakage data on the Selection/Breakage tab.
NStressEvents Input The number of Multiple Impact Events. See Multiple Impact Events. This may be an integer, or it may be a fractional value, such as 1.5.
If the user requires internal classification, then they can tick the 'Classification' box and the following fields all relate to these calculations.
Classification Checkbox If this box is ticked, then the unit will apply internal classification.
ClassificationType Napier Munn Uses the Napier Munn classification.
Whiten Uses the Whiten classification.
User The user enters their own classification data on the Selection/Breakage tab.
PostScreen Input Transfer Model for closed loop classifier.
K1, K2, K3 Input Parameters for Napier Munn classification model.

### Selection/Breakage Page

The selection, breakage fraction, cumulative breakage (and classification if enabled) data is available in tabular form. When User options are selected data may be entered manually.

If User options are selected, the parameters can be entered manually (or by user calculation in a PGM):

 Tag / Symbol Input / Calc Description/Calculated Variables / Options Display Ore Input Which ore to display if multiple ores are present in feed. DisplayCumulative CheckBox Display Cumulative breakage

### Power Page

This tab will be visible when the Method selected is 'Whiten Crusher'. See Model Theory - Whiten Crusher Method for more information.

 Tag / Symbol Input / Calc Description/Calculated Variables / Options Display Ore Input The user may type in the required solid index, or use the 'Prev' and 'Next' buttons to scroll between defined solids. NoLoadPower/Pn Input The power draw of the Whiten Crusher with no feed. Scaling/A Input The dimensionless scaling factor for the actual crusher. TotalPendulmPower/Pp Calc The calculated Pendulum Power. CrusherPower/Pc Calc The crusher power draw with the feed load. Pendulum ECS Data: kWh/ton.These values are per solid ore type (use the scroll buttons to move between ore types) Init Size (mm) Input The initial material size for each of 3 tests. T10=10 Input The T10 value for each of the 3 tests. T10=20 Input The T10 value for each of the 3 tests. T10=30 Input The T10 value for each of the 3 tests.

## Adding this Model to a Project

Insert into Configuration file

Sort either by DLL or Group.

 DLL: Comminution.dll → Units/Links → Size Alteration: Crusher2 OR Group: Size Distribution → Units/Links → Size Alteration: Crusher2