Primary Crusher

Summary

The King Primary Crusher model estimates the product particle size distribution of primary jaw or gyratory crushers using the empirical formulation presented by R. P. King. The model is based on the crusher open-side setting and on a product type index that represents the material breakage characteristic.

The model is suitable for primary crushing applications where the feed contains only a small amount of material finer than the open-side setting and where the product size distribution can be estimated mainly from crusher setting and material type.

DPSIM model key: DPSIM.Comminution.KingPrimaryCrusher
Category: Comminution
Subcategory: Crushers
Display name: King Primary Crusher

Parameters

#	Parameter	Description
1	Product Type Index	Material characteristic
2	OSS (mm)	Crusher open-side setting. It is used as the characteristic size for the normalized product size distribution.
3	Apply feed fines bypass correction (0-disabled/1-enabled)	Optional DPSIM correction. When disabled, the model follows the original King formulation and does not use the feed PSD to calculate the product PSD. When enabled, the model applies an empirical correction to preserve feed fines in the product.
4	[Component] Product Type Index	Component-specific product type index, expressed as percent passing the open-side setting. Typical values are 90 for soft or average material, 85 for spongy material, 82 for hard tough material, and 75 for hard slabby material.

Model Description

The model calculates one product size distribution for each component and then recombines the component products into the final product stream.

For each size class, the model calculates the normalized size ratio:

$r_{i} = \frac{d_{i}}{OSS}$

Where,

Symbol	Description	Unit
$\mathbf{r}_{\mathbf{i}}$	Size relative to the open side setting.
$\mathbf{d}_{\mathbf{i}}$	Aperture of the ith mesh.	mm
$\mathbf{OSS}$	Open-side setting	mm

For each component, the product type index ( $P_{T}$ ), given by the following table.

Crusher work index (kWh/t)	Material characteristic	Product type, P_T
5-10	Soft	90
	Soft spongy	85
10-13	Average	90
	Average spongy	85
>13	Hard brittle	90
	Hard tough	82
	Hard slabby	75

The product size distribution is given by the equation

$P(D) = 1 - \exp\left( - \left( \frac{r}{K_{u}} \right)^{1.5} \right)$ , for $r > 0.5$

And,

$P(D) = 1 - \exp\left( - \left( \frac{r}{K_{L}} \right)^{0.85} \right)$ , for $r \leq 0.5$

With,
$K_{u} = \left( \ln\left( \frac{1}{1 - P_{T}} \right) \right)^{- 0.67}$

And,
$K_{L} = \left( \ln\left( \frac{1}{1 - P_{T}} \right) \right)^{- 0.67}$

And
$P_{b} = 1 - \exp\left( - \left( \frac{0.5}{K_{u}} \right)^{1.5} \right)$

The model preserves the feed solids flowrate and water flowrate. It recalculates only the product size distribution and the component-by-size matrix.

The component-specific Product Type Index used in DPSIM is an extension of the original single-material formulation. It allows different components to use different crushing response parameters.

IMPORTANT: These equations can lead to inconsistent results since they take no account of the distribution of particle sizes in the feed to the crusher and should be applied only when the open-side setting of the crusher is smaller than the $d_{50}$ size in the feed.