Model Reference / Whiten Crusher Modified

Whiten Crusher Modified

Summary

The Modified Whiten Crusher model represents crushing through an internal classification function and a component-specific breakage matrix. The model is based on the Whiten crusher formulation originally proposed by Whiten, Walter and White in “A breakage function suitable for crusher models” Fourth Tewkesbury Symposium, Melbourne, 1979.

The model is suitable for cone crusher simulations where the product size distribution must depend on the feed size distribution, the crusher closed-side gap, and material-specific breakage parameters.

In this DPSIM implementation, the classification function is controlled by CSS, alpha, beta, gamma, and K3 parameters. The breakage function is component-specific and combines tensile and compressive fracture terms.

DPSIM model key: DPSIM.Comminution.ModifiedWhitenCrusher
Category: Comminution
Subcategory: Crushers
Display name: Modified Whiten Crusher

Parameters

#	Parameter	Description
1	Crusher closed-side gap, CSS (mm)	Crusher closed-side gap. It is used as the reference operating setting for the internal classification function.
2	Upper breakage-size offset (mm)	Additive correction used in the calculation of the upper classification size.
3	No-breakage size factor relative to CSS	Multiplier applied to CSS to define the lower classification size. Particles at or below this size are not selected for breakage.
4	Full-breakage size factor relative to CSS	Multiplier applied to CSS to define the upper classification size. Particles at or above this size are fully selected for breakage.
5	Classification transition exponent	Exponent controlling the shape of the transition region in the classification function.
6	[Component] tensile-breakage fraction	Component-specific mixing factor between tensile and compressive breakage terms.
7	[Component] tensile-breakage exponent	Component-specific exponent of the tensile breakage term.
8	[Component] compressive-breakage exponent	Component-specific exponent of the compressive breakage term.

Model Description

The Modified Whiten Crusher model calculates the product size distribution using a matrix formulation based on internal classification and breakage. For each component, the product retained-mass vector is calculated as:

$P_{c} = (I - C)\left( I - B_{c}C \right)^{- 1}F_{c}$

Where:

Symbol	Description
$P_{c}$	Product retained-mass vector for component c.
$F_{c}$	Feed retained-mass vector for component c.
$I$	Identity matrix.
$C$	Diagonal internal classification matrix.
$B_{c}$	Component-specific breakage matrix.

The classification matrix C defines the probability that particles in each active size class are selected for breakage. Two characteristic sizes are calculated:

$K_{1} = \alpha\ CSS$

$K_{2} = \beta\ CSS + \gamma$

Where:

Symbol	Description	Unit
$CSS$	Crusher closed-side setting.	mm
$\alpha$	No-breakage size factor relative to CSS.	dimensionless
$\beta$	Full-breakage size factor relative to CSS.	dimensionless
$\gamma$	Upper breakage-size offset.	mm
$K_{1}$	Size below which particles are not selected for breakage.	mm
$K_{2}$	Size above which particles are fully selected for breakage.	mm

For each active size class i, the diagonal term of the classification matrix is:

$C_{ii} = 0,\ d_{i} \leq K_{1}$

$C_{ii} = 1,\ d_{i} \geq K_{2}$

$C_{ii} = 1 - \left( \frac{d_{i} - K_{2}}{K_{1} - K_{2}} \right)_{3}^{K},\ K_{1} < d_{i} < K_{2}$

Where $d_{i}$ is the size-class opening converted from µm to mm.

For each component c, the cumulative breakage function is calculated as:

$B_{c\left( x;y_{j} \right)} = K_{c}\left( \frac{x}{y_{j}} \right)^{n_{c}} + \left( 1 - K_{c} \right)\left( \frac{x}{y_{j}} \right)^{m_{c}}$

Where:

Symbol	Description	Unit
$B_{c\left( x;y_{j} \right)}$	Cumulative breakage function for component c.	fraction
$x$	Product size boundary at which the cumulative breakage function is evaluated.	µm
$y_{j}$	Lower boundary of the parent size class j.	µm
$K_{c}$	Tensile-breakage fraction for component c.	fraction
$n_{c}$	Tensile-breakage exponent for component c.	dimensionless
$m_{c}$	Compressive-breakage exponent for component c.	dimensionless

The implementation limits $\frac{x}{y_{j}}$ to the interval from 0 to 1 before evaluating the breakage function.

The breakage matrix is then obtained from the cumulative breakage function. In this modified truncated formulation, material selected for breakage is redistributed only to size classes finer than the parent class:

$b_{ij} = 0,\ i \leq j$

$b_{ij} = B_{c\left( D_{i - 1};y_{j} \right)} - B_{c\left( D_{i};y_{j} \right)},\ i > j$

Where:

Symbol	Description	Unit
$b_{ij}$	Fraction of broken material from parent class j reporting to product class i.	fraction
$D_{i - 1}$	Upper boundary of product size class i.	µm
$D_{i}$	Lower boundary of product size class i.	µm
$y_{j}$	Lower boundary of parent size class j.	µm

The model solves the matrix equation independently for each component. The calculated component retained masses are then recombined into the total product size distribution and the product component-by-size matrix.

The product stream preserves the feed solids flowrate and water flowrate. The model recalculates only the product retained size distribution and the component fractions by size class.

After solving the active classes, the pan receives the residual mass required to close the component and total solids balance.