Whiten Crusher

Summary

This model represents the crushing through the breaking and classification functions as proposed by Whiten, W. J.; Walter, G. W.; White, M. E. (1979). A breakage function suitable for crusher models. IV Tewkesbury Symposium, Melbourne, pp. 19.1–19.3.

The model represents crushing through an internal classification function and a component-specific breakage matrix. The model follows the Whiten crusher structure, where particles are first classified according to their probability of being selected for breakage and then redistributed among finer size classes by a breakage function.

The model should be used for cone crusher or similar compression crusher simulations when the product size distribution must depend on the feed size distribution, closed side setting, feed coarseness, and material-specific breakage parameters.

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

Parameters

#	Parameter	Description
1	CSS (mm)	Closed side setting of the crusher. It is used to calculate the internal classification function.
2	[Component] Coarse Breakage - U exponent	Component-specific exponent used in the coarse breakage term ( $B_{1}$ ). Higher values change the steepness of the coarse breakage distribution.
3	[Component] Fine Breakage reference size (mm)	Component-specific reference size used in the fine breakage term ( $B_{2}$ ).
4	[Component] Fine Breakage - V exponent	Component-specific exponent used in the fine breakage term ( $B_{2}$ ). Higher values change the steepness of the fine breakage distribution.

Derived parameters

#	Derived parameter	Description
1	T(t) - capacity adjustment	Capacity-related correction calculated from feed solids flowrate using the Whiten spline points.
2	Fraction above 1 inch	Fraction of feed retained above 25.4 mm. This value is used in the calculation of the upper classification size.

Model Description

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

${P_{c} = \ (I - C)\left( I - b_{cC} \right)_{c}^{\left\{ - 1 \right\} F}}$

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 matrix (C) defines the probability that particles in each size class are selected for breakage. The classification function is calculated from two characteristic sizes:

$K_{1} = \ 0.67CSS$
$K_{2}\ = \ 1.12CSS\ + \ 58.674Q\ + \ 25.4T(t)$

Where:

Symbol	Description	Unit
$CSS$	Closed side setting.	mm
$Q$	Fraction of feed above 1 inch, calculated as the retained fraction above 25.4 mm.	fraction
$T(t)$	Capacity adjustment calculated from feed solids flowrate.	dimensionless
$K_{1}$	Size below which particles are not selected for breakage.	mm
$K_{2}$	Size above which particles are fully selected for breakage.	mm

The value of T(t) is calculated from a cubic spline through the points:

(0,0), (100,-0.0486), (250,-0.085), (400,-0.259)

where the x-coordinate is the feed solids flowrate in tph. For feed rates equal to or greater than 400 tph, the model uses T(t)=-0.259.

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

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

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

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

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

For each component, the breakage matrix is built from a cumulative breakage function composed of two terms:

$B\left( x;d_{p,j} \right) = \alpha\ B_{1\left( x;d_{p,j} \right)} + (1 - \alpha)B_{2(x)}$

with:

$B_{1\left( x;d_{p,j} \right)} = \frac{1 - \exp\left( - \left( \frac{x}{d_{p,j}} \right)^{u} \right)}{1 - \exp( - 1)}$

$B_{2(x)} = 1 - \exp\left( - \left( \frac{x}{1000s} \right)^{v} \right)$

and:

$\alpha = \min\left( 1,\max(0,0.872 + 0.0045\ CSS) \right)$

Where:

Symbol	Description	Unit
$x$	Size boundary at which the cumulative breakage function is evaluated.	µm
$d_{p,j}$	Representative size of parent class j. The implementation uses the DPSIM geometric mean size when available.	µm
$u$	Component-specific coarse breakage exponent.	dimensionless
$s$	Component-specific fine breakage reference size.	mm
$v$	Component-specific fine breakage exponent.	dimensionless
$\alpha$	Mixing coefficient between the coarse and fine breakage terms.	dimensionless

The breakage matrix is obtained from the cumulative breakage function by difference between size boundaries. In descending size order:

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

$b_{jj} = 1 - B\left( D_{j};d_{p,j} \right)$

$b_{ij} = B\left( D_{i - 1};d_{p,j} \right) - B\left( D_{i};d_{p,j} \right),\ i > j$

Where $D_{i}$ is the lower boundary of size class i, and $D_{i - 1}$ is the upper boundary of the same product size interval.

The model solves the matrix equation independently for each component, using the component retained-mass vector as the feed vector. The product component masses are then recombined to form the total product size distribution and the component-by-size matrix.

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