Model Reference / Generic Matrix Crusher

Generic Matrix Crusher

Summary

The Generic Matrix Crusher model represents crushing through an internal classification function and a generic breakage matrix. The model follows the matrix approach to breakage introduced by Broadbent and Callcott in “A matrix analysis of processes involving particle assemblies” Philosophical Transactions of the Royal Society of London, 1956, and uses the crusher classification-breakage structure commonly associated with Whiten crusher models.

The model should be used when a simple matrix crusher representation is required, with two user-defined classification sizes and a fixed Callcott-style breakage function. It is suitable for generic jaw crusher or crusher-stage simulations where a calibrated empirical matrix response is sufficient.

DPSIM model key: DPSIM.Comminution.MatrixJawCrusher
Category: Comminution
Subcategory: Crushers
Display name: Generic Matrix Crusher

Parameters

#	Parameter	Description
1	No-breakage size, K1 (mm)	Size below which particles are not selected for breakage. Particles at or below this size pass through the crusher without being selected for breakage.
2	Full-breakage size, K2 (mm)	Size above which particles are fully selected for breakage. Particles at or above this size have classification value equal to 1.
3	Classification transition exponent	Exponent controlling the shape of the transition region between K1 and K2 in the internal classification function.

Model Description

Model Description

The Generic Matrix 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:

$P_{c} = (I - C)(I - B\ C)_{c}^{( - 1)F}$

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	Generic breakage matrix.

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

K_1

K_2

Where:

Symbol	Description	Unit
K_1	No-breakage size.	mm
K_2	Full-breakage size.	mm
n	Classification transition exponent.	dimensionless

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

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

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

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

Where d_i is the size-class opening converted from µm to mm.

If K_2≤K_1, the model uses a step classification function:

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

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

The breakage matrix B is calculated from a cumulative Callcott-style breakage function:

$B(x;y) = 1.58\left( 1 - \exp\left( - \frac{x}{y} \right) \right)$

Where:

Symbol	Description	Unit
B(x;y)	Cumulative fraction of broken material smaller than size x, for a parent size y.	fraction
x	Product size boundary at which the cumulative breakage function is evaluated.	µm
y	Parent size used by the model.	µm

The implementation limits B(x;y) to the interval from 0 to 1 before using it in the breakage matrix.

For each active parent class j, the model uses the parent size:

$y_{j} = D_{j}$

The breakage matrix is obtained from differences between cumulative breakage values at the upper and lower boundaries of each product size class:

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

$b_{ij} = B\left( D_{i};y_{j} \right) - B\left( D_{i + 1};y_{j} \right),\ i \geq j$

Where:

Symbol	Description	Unit
b_ij	Fraction of broken material from parent class j reporting to product class i.	fraction
D_i	Upper boundary of active product size class i.	µm
$D_{i + 1}$	Lower boundary of active product size class i.	µm
y_j	Parent size for active size class j.	µm

The model applies the same classification and breakage matrices to each component independently, using the component retained-mass vector as the feed vector. 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.

The breakage matrix uses a fixed Callcott-style cumulative breakage function. It does not include component-specific breakage parameters.