HPGR (Torres)

Description

This article describes the Torres and Casali (2009) model for comminution by High Pressure Grinding Rolls (HPGR).^[1]

Model theory

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Excel

The Torres HPGR model may be invoked from the Excel formula bar with the following function call:

=mdUnit_HPGR_Torres(Parameters as Range, Size as Range, HPGRFeed as Range, OreSG as Range, Selection as Range, Breakage as Range)

Invoking the function with no arguments will print Help text associated with the model, including a link to this page.

Inputs

The required inputs are defined below in matrix notation with elements corresponding to cells in Excel row ( $i$ ) x column ( $j$ ) format:

Parameters={\begin{bmatrix}{\text{ Selection function method}}\\{\text{ Breakage function method}}\\D{\text{ (m)}}\\L{\text{ (m)}}\\s_{0}{\text{ (m)}}\\R_{\rm {p}}{\text{  (bar)}}\\\rho _{\rm {a}}{\text{ (t/m}}^{3}{\text{)}}\\C_{\rho }{\text{ (frac)}}\\a{\text{ (frac)}}\\C_{\rm {IP}}{\text{ (frac)}}\\N_{\rm {B}}\end{bmatrix}},\;\;\;\;\;\;Size={\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\\\end{bmatrix}},\;\;\;\;\;\;{\mathit {HPGRFeed}}={\begin{bmatrix}(Q_{\rm {M,F}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,F}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}},\;\;\;\;\;\;OreSG={\begin{bmatrix}(\rho _{\rm {S}})_{1}{\text{ (t/m}}^{\text{3}}{\text{)}}&\dots &(\rho _{\rm {S}})_{m}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\end{bmatrix}}

Selection={\begin{cases}{\begin{bmatrix}{\begin{bmatrix}S_{1}^{\rm {E}}\\\vdots \\S_{n}^{\rm {E}}\\\end{bmatrix}}_{1}\dots {\begin{bmatrix}S_{1}^{\rm {E}}\\\vdots \\S_{n}^{\rm {E}}\\\end{bmatrix}}_{m}\end{bmatrix}}&{\text{ Selection function method}}=0{\text{ (User defined)}}\\&&\\{\begin{bmatrix}{\begin{bmatrix}S_{1}^{\rm {E}}\\\zeta _{1}\\\zeta _{2}\\d_{{\rm {p}}1}{\text{ (mm)}}\\\end{bmatrix}}_{1}\dots {\begin{bmatrix}S_{1}^{\rm {E}}\\\zeta _{1}\\\zeta _{2}\\d_{{\rm {p}}1}{\text{ (mm)}}\\\end{bmatrix}}_{m}\end{bmatrix}}&{\text{ Selection function method}}=1{\text{ (Herbst-Fuerstenau)}}\\&&\\{\begin{bmatrix}{\begin{bmatrix}\alpha _{0}\\\alpha _{1}\\\alpha _{2}\\d_{\rm {crit}}{\text{ (mm)}}\\\alpha _{02}\\\alpha _{12}\\\end{bmatrix}}_{1}\dots {\begin{bmatrix}\alpha _{0}\\\alpha _{1}\\\alpha _{2}\\d_{\rm {crit}}{\text{ (mm)}}\\\alpha _{02}\\\alpha _{12}\\\end{bmatrix}}_{m}\end{bmatrix}}&{\text{ Selection function method}}=2{\text{ (Austin)}}\\\end{cases}}

Breakage={\begin{cases}{\begin{bmatrix}{\begin{bmatrix}b_{11}&&0\\\vdots &\ddots &\\b_{n1}&\dots &b_{nn}\\\end{bmatrix}}_{1}\\\vdots \\{\begin{bmatrix}b_{11}&&0\\\vdots &\ddots &\\b_{n1}&\dots &b_{nn}\\\end{bmatrix}}_{m}\\\end{bmatrix}}&{\text{ Breakage function method}}=0{\text{ (User defined)}}\\&&\\{\begin{bmatrix}{\begin{bmatrix}\beta _{0}\\\beta _{1}\\\beta _{2}\\\beta _{02}\\\end{bmatrix}}_{1}\dots {\begin{bmatrix}\beta _{0}\\\beta _{1}\\\beta _{2}\\\beta _{02}\\\end{bmatrix}}_{m}\end{bmatrix}}&{\text{ Breakage function method}}=1{\text{ (Austin and Luckie)}}\\&&\\{\begin{bmatrix}{\begin{bmatrix}K\\n_{1}\\n_{2}\\n_{3}\\y_{0}{\text{ (mm)}}\\\end{bmatrix}}_{1}\dots {\begin{bmatrix}K\\n_{1}\\n_{2}\\n_{3}\\y_{0}{\text{ (mm)}}\\\end{bmatrix}}_{m}\end{bmatrix}}&{\text{ Breakage function method}}=2{\text{ (King)}}\\&&\\{\bigg [}\;\;\;{\bigg ]}&{\text{ Breakage function method}}=3{\text{ (Natural selection)}}\\\end{cases}}

where:

${\text{Selection method}}$ is the selection function method, 0 = User, 1 = Herbst-Fuerstenau, 2 = Austin
${\text{Breakage method}}$ is the breakage function method, 0 = User, 1 = Austin & Luckie, 2 = King, 3 = Natural selection
$n$ is the number of size intervals
$m$ is the number of ore types.

Results

The results are displayed in Excel as an array corresponding to the matrix notation below:

{\mathit {mdUnit\_HPGR\_Torres}}={\begin{bmatrix}{\begin{bmatrix}\delta {\text{ (t/m}}^{3}{\text{)}}\\x_{\rm {c}}{\text{ (mm)}}\\\alpha _{\rm {IP}}{\text{ (deg.)}}\\G_{\rm {s}}{\text{ (t/h)}}\\U{\text{ (m/s)}}\\{\text{Rolls rotational speed (rpm)}}\\{\dot {m}}{\text{ (t.s/m}}^{3}{\text{.h)}}\\F{\text{ (kN)}}\\W{\text{ (kWh/t)}}\\P{\text{ (kW)}}\\\end{bmatrix}}&{\begin{array}{c}{\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,P}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,P}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,P}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,P}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,E}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,E}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,E}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,E}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,C}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,C}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,C}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,C}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{n}{\text{ (mm)}}\\\end{bmatrix}}&{\begin{bmatrix}{\begin{bmatrix}S_{1}^{\rm {E}}\\\vdots \\S_{n}^{\rm {E}}\\\end{bmatrix}}_{1}\dots {\begin{bmatrix}S_{1}^{\rm {E}}\\\vdots \\S_{n}^{\rm {E}}\\\end{bmatrix}}_{m}\end{bmatrix}}&{\begin{bmatrix}{\begin{bmatrix}B_{1,2}\\\vdots \\B_{n,2}\\\end{bmatrix}}_{1}\dots {\begin{bmatrix}B_{1,2}\\\vdots \\B_{n,2}\\\end{bmatrix}}_{m}\end{bmatrix}}&{\begin{bmatrix}y_{1}{\text{ (m)}}\\\vdots \\y_{N_{\rm {B}}}{\text{ (m)}}\\\end{bmatrix}}&{\begin{bmatrix}P_{1}{\text{ (kW)}}\\\vdots \\P_{N_{\rm {B}}}{\text{ (kW)}}\\\end{bmatrix}}\\\\\\\\\\\\\\\\\end{array}}\end{bmatrix}}

where:

${\text{Rolls rotational speed}}$ is the rotational speed of the rolls (rpm)
$Q_{\rm {M,P}}$ is the mass flow rate of particles in the overall, combined product (t/h), equivalent to $p_{i}^{\rm {HPGR}}$ in Torres and Casali's nomenclature
$Q_{\rm {M,E}}$ is the mass flow rate of particles in the edge product (t/h), equivalent to $p_{i}^{\rm {E}}$ in Torres and Casali's nomenclature
$Q_{\rm {M,C}}$ is the mass flow rate of particles in the centre product (t/h), equivalent to $p_{i}^{\rm {C}}$ in Torres and Casali's nomenclature
$B_{ij}$ is the cumulative breakage function, i.e. $b_{ij}=B_{ij}-B_{i+1,j}$ .

Example

The images below show the selection of input arrays and output results in the Excel interface.

Figure 5. Example showing the selection of the Parameters (blue frame) array in Excel.	Figure 6. Example showing the selection of the Size (red frame), HPGRFeed (purple frame) and OreSG (green frame) arrays in Excel.	Figure 7. Example showing the selection of the Selection (pink frame) and Breakage (brown frame) arrays in Excel.
Figure 8. Example showing the outline of the Results (light blue frame) array in Excel.

SysCAD

The sections and variable names used in the SysCAD interface are described in detail in the following tables.

MD_HPGR page

The first tab page in the access window will have this name.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Tag	Display	This name tag may be modified with the change tag option.
Condition	Display	OK if no errors/warnings, otherwise lists errors/warnings.
ConditionCount	Display	The current number of errors/warnings. If condition is OK, returns 0.
GeneralDescription / GenDesc	Display	This is an automatically generated description for the unit. If the user has entered text in the 'EqpDesc' field on the Info tab (see below), this will be displayed here. If this field is blank, then SysCAD will display the unit class ID.
Requirements
On	CheckBox	This enables the unit. If this box is not checked, then the material will pass straight through the crusher with no change to the size distribution.
NumParallelUnits	Input	The number of parallel, identical units to simulate: Feed is divided by the number of parallel units before being sent to the unit model. Unit model products are multiplied back by the same value and returned to the SysCAD product streams. All unit model result values are shown per parallel unit.
Method	Fixed Discharge	The discharge particle size distribution is user defined. Different distributions can be used for different solids.
	Morrell-Shi-Tondo	The throughput, power draw and product size distribution are determined by the Morrell-Tondo-Shi model. Different parameters can be used for different solids.
	Torres	The throughput, power draw and product size distribution are determined by the Torres model. Different parameters can be used for different solids.
	Campos	The throughput, power draw and product size distribution are determined by the Campos model. Different parameters can be used for different solids.
Options
ShowQFeed	CheckBox	QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed stream.
ShowQProd	CheckBox	QProd and associated tab pages (eg Sp) will become visible, showing the properties of the products.
SizeForPassingFracCalc	Input	Size fraction for % Passing calculation. The size fraction input here will be shown in the Stream Summary section.
FracForPassingSizeCalc	Input	Fraction passing for Size calculation. The fraction input here will be shown in the Stream Summary section.
Stream Summary
MassFlow / Qm	Display	The total mass flow in each stream.
SolidMassFlow / SQm	Display	The Solids mass flow in each stream.
LiquidMassFlow / LQm	Display	The Liquid mass flow in each stream.
VolFlow / Qv	Display	The total Volume flow in each stream.
Temperature / T	Display	The Temperature of each stream.
Density / Rho	Display	The Density of each stream.
SolidFrac / Sf	Display	The Solid Fraction in each stream.
LiquidFrac / Lf	Display	The Liquid Fraction in each stream.
Passing	Display	The mass fraction passing the user-specified size (in the field SizeForPassingFracCalc) in each stream.
Passes	Display	The user-specified (in the field FracForPassesSizeCalc) fraction of material in each stream will pass this size fraction.

HPGR page

The HPGR page is used to specify the input parameters for the HPGR model.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Torres
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
Parameters
RollDiameter / D	Input	Diameter of the rolls.
RollLength / L	Input	Length of the rolls.
Gap / s0	Input	Gap between the rolls.
OperatingPressure / Rp	Input	Operating pressure of the rolls.
FeedBulkDensity / Rhoa	Input	Bulk density of solids in the feed.
CakeDensityFactor	Input	Cake density factor.
EdgeFraction / a	Input	Fraction of the product collected from the edge of the rolls length, appearing in the edge product stream.
ForceAngleFactor	Input	Factor applied to alphaIP when computing power draw.
NumBlocks / Nb	Input	Number of model blocks, i.e. instances of the Herbst-Fuerstenau mill model along the length of the rolls.
Selection
Method	User	The user specifies the selection function.
	Herbst and Fuerstenau	The Herbst and Fuerstenau selection function is used.
	Austin	The Austin selection function is used.
OreSpecific	CheckBox	Ore-specific parameters, allows the selection function to be separately input for all species. Default is all species have the same set of single input properties. This option is only available if there is more than one species in the project with the size distribution property.
The fields below are only visible if Herbst-Fuerstenau is selected.
S1E	Input	Input parameter of the Herbst-Fuerstenau selection function.
Zeta1	Input	Input parameter of the Herbst-Fuerstenau selection function.
Zeta2	Input	Input parameter of the Herbst-Fuerstenau selection function.
dp1	Input	Input parameter of the Herbst-Fuerstenau selection function.

The fields below are only visible if Austin is selected.
Alpha0	Input	Input parameter of the Austin selection function.
Alpha1	Input	Input parameter of the Austin selection function.
Alpha2	Input	Input parameter of the Austin selection function.
dCrit	Input	Input parameter of the Austin selection function.
Alpha02	Input	Input parameter of the Austin selection function.
Alpha12	Input	Input parameter of the Austin selection function.

Breakage
Method	User	The user specifies the breakage function.
	Austin and Luckie	The Austin and Luckie breakage function is used.
	King	The King selection function is used.
	Natural Selection	The Natural selection function is used.
OreSpecific	CheckBox	Ore-specific parameters, allows the breakage function to be separately input for all species. Default is all species have the same set of single input properties. This option is only available if there is more than one species in the project with the size distribution property.
The fields below are only visible if Austin and Luckie is selected.
Beta0	Input	Input parameter of the Austin and Luckie breakage function.
Beta1	Input	Input parameter of the Austin and Luckie breakage function.
Beta2	Input	Input parameter of the Austin and Luckie breakage function.
Beta01	Input	Input parameter of the Austin and Luckie breakage function.

The fields below are only visible if King is selected.
K	Input	Input parameter of the King breakage function.
n1	Input	Input parameter of the King breakage function.
n2	Input	Input parameter of the King breakage function.
n3	Input	Input parameter of the King breakage function.
y0	Input	Input parameter of the King breakage function.

Results
CakeDensity / delta	Display	Solids density of the cake.
CriticalSize / xc	Display	Critical size.
InterparComprAngle / alphaIP	Display	Interparticle compression angle.
Throughput / Gs	Display	Solids mass throughput flow rate of the HPGR.
RollsVelocity / U	Display	Circumferential velocity of the rolls.
RollsSpeed	Display	Rotational speed of the rolls.
SpecificThroughput / mDot	Display	Specific throughput parameter of the HPGR unit.
CompressionForce / F	Display	Compression force of the rolls.
GrossPower / P	Display	Gross power draw of the HPGR unit
SpecificEnergy / W	Display	Specific energy of grinding.

Selection page

The Selection page is used to specify or display the selection function values.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Distribution
Name	Display	Shows the name of the SysCAD Size Distribution (PSD) quality associated with the feed stream.
IntervalCount	Display	Shows the number of size intervals in the SysCAD Size Distribution (PSD) quality associated with the feed stream.
SpWithPSDCount	Display	Shows the number of species in the feed stream assigned with the SysCAD Size Distribution (PSD) quality.
Selection
Size	Display	Size of each interval in mesh series.
MeanSize	Display	Geometric mean size of each interval in mesh series.
Selection	Input/Display	Value of the selection function for each size interval, for each ore species.

Breakage page

The Breakage page is used to specify or display the breakage function values.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Distribution
Name	Display	Shows the name of the SysCAD Size Distribution (PSD) quality associated with the feed stream.
IntervalCount	Display	Shows the number of size intervals in the SysCAD Size Distribution (PSD) quality associated with the feed stream.
SpWithPSDCount	Display	Shows the number of species in the feed stream assigned with the SysCAD Size Distribution (PSD) quality.
Breakage
Size	Display	Size of each interval in internal mesh series.
Breakage	Input/Display	Value of the breakage function for each parent size interval, progeny size interval and ore species.

Power page

The Power page is used to display power draw of model blocks along the length of the rolls.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Power
Position / y	Display	Position along the rolls length.
Power / Pk	Display	Power draw of model block at position along the rolls length.

About page

This page is provides product and licensing information about the Met Dynamics Models SysCAD Add-On.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
About
HelpLink		Opens a link to the Installation and Licensing page using the system default web browser. Note: Internet access is required.
Information		Copies Product and License information to the Windows clipboard.
Product
Name	Display	Met Dynamics software product name
Version	Display	Met Dynamics software product version number.
BuildDate	Display	Build date and time of the Met Dynamics Models SysCAD Add-On.
License
File		This is used to locate a Met Dynamics software license file.
Location	Display	Type of Met Dynamics software license or file name and path of license file.
SiteCode	Display	Unique machine identifier for license authorisation.
ReqdAuth	Display	Authorisation level required, MD-SysCAD Full or MD-SysCAD Runtime.
Status	Display	License status, LICENSE_OK indicates a valid license, other messages report licensing errors.
IssuedTo	Display	Only visible if Met Dynamics license file is used. Name of organisation/seat the license is authorised to.
ExpiryDate	Display	Only visible if Met Dynamics license file is used. License expiry date.
DaysLeft	Display	Only visible if Met Dynamics license file is used. Days left before the license expires.

Additional notes

When the edge stream is connected, the following applies to mass splits:

Solid species without a particle size distribution property are split to the edge stream according to the overall mass split of the default particle size distribution species selected in the SysCAD Project Configuration
If the default particle size distribution species is not present in the unit feed, the overall split of all other species with particle size distributions combined is used, as determined by the model.
Liquids species are split between the edge and product streams in the same proportion as solid species.
Gas phase species report directly to the product stream without split.

References

↑ Torres, M. and Casali, A., 2009. A novel approach for the modelling of high-pressure grinding rolls. Minerals Engineering, 22(13), pp.1137-1146.

[Torres_and_Casali_(2009)-1] Torres, M. and Casali, A., 2009. A novel approach for the modelling of high-pressure grinding rolls. Minerals Engineering, 22(13), pp.1137-1146.

[1]

HPGR (Torres)

Contents

Description

Model theory