HPGR (Torres)
Description
This article describes the Torres and Casali (2009) model for comminution by High Pressure Grinding Rolls (HPGR).^{[1]}
Model theory
Torres and Casali (2009) proposed a phenomenological HPGR model that combines equipment specifications, operating variables and ore characteristics to predict throughput, power draw, and size reduction performance.^{[1]}
Figure 1 presents the principal dimensions of an HPGR machine, which are relevant to throughput and power draw behaviour.
Torres and Casali suggest that particles traversing the space between the two counterrotating rolls of an HPPR are subject to two modes of particle breakage:
 Single particle compression, where individual particles are nipped and crushed between the rolls in isolation from neighbouring particles.
 Particle bed compression, where particles are packed into close contact with their neighbours and break as the ensemble is drawn into the rolls and stressed.
Figure 2 illustrates the arrangement of the single particle and particle bed compression zones in relation to HPGR geometry.
The throughput, power draw, and size reduction submodels are described in detail below.
Throughput
The solids mass throughput flow rate, (t/h), of an HPGR may be estimated as:
where:
 is the bulk density of discharge cake solids (t/m^{3})
 is the operating gap (m)
 is the length of the rolls (m)
 is the peripheral rolls velocity (m/s).
By rearrangement, the peripheral rolls velocity () for a given solids mass throughput flow rate () is:
The bulk density of the discharge cake solids () can be specified in terms of the solid material density, (t/m^{3}), by:
where is the cake density factor (frac). A cake density factor () of 0.80  0.85 may be typical for operating HPGR machines.^{[2]}
Power
The compression force, (kN), applied to the material in the bed compression zone due to the operating pressure of the rolls, (bar), is:
The start of the bed compression zone is defined by the size of the interparticle compression angle, (rad), which is:
where is the diameter of the rolls (m) and is the bulk density of the feed (t/m^{3}).
The compression force exerts a torque on each roll, which results in the following expression for power draw:
where is the ratio of the angle at which the compression force acts to the theoretical interparticle compression angle. Torres and Casali suggest a value of . This implementation allows a userspecified value of , so that the power draw of a simulated machine may be matched to the measured power draw of an operating machine.
Finally, the specific energy consumption, (kWh/t), is:
Single particle compression
The flow chart in Figure 3 illustrates the progression of feed material through the single particle compression zone, the interparticle compression zone, and into the final products.
Particles larger than the the critical size, (mm), are broken in the single particle compression regime. The critical size is defined from the roll geometry as:
The mass flow rate of solids in size interval which are selected for single particle compression, (t/h), is:
where:
 is feed solids mass flow rate by size and ore type (t/h), equivalent to in Torres and Casali's (2009) nomenclature
 is the size of the square mesh interval that mass is retained on (mm)
 , i.e. descending size order from top size () to sub mesh ( mm).
All particles subject to single particle compression are assumed to break, and the product size distribution is determined from:
where:
 is the mass flow rate of particles in size interval appearing in the product (t/h)
 is the breakage function, the mass fraction of a broken particle of original size appearing in size (w/w).
Interparticle compression
Power distribution profile
The interparticle compression zone extends across the full length of the rolls. The operating pressure applied to the rolls is assumed to adopt a parabolic distribution, with the lowest pressure being experienced at the edges and highest pressure at the centre.
The roll length is discretised into blocks, each of which experiences a particular pressure and hence power consumption during grinding. Figure 4 illustrates the block discretisation and parabolic shape of the power distribution across the rolls.
The position (m) of a block is:
Then, the power consumed by each block during grinding, (kW) is:
Size reduction
A plug flow grinding regime is assumed for the grinding process within each block. The plug flow (i.e. batch) population balance equation for grinding is:
where
 is the rate of breakage of particles of size in each block (h^{1})
 is the velocity of the plug flow material, which is assumed constant and equal to the rolls velocity (m/s).
The height of the particle compression zone, (m), is:
The appearance function, is defined as:
where is the mass flow rate of feed to the interparticle compression zone, i.e. .
The rate of breakage () is:
where is the invariant specific rate of breakage for each size interval (alternatively referred to as the specific energy selection function), and the mass holdup in each block, (t), given by:
Selection and breakage functions
 Main article: Mill (HerbstFuerstenau)
The single particle and interparticle compression models require representations of the selection () and breakage () functions.
The specific energy, plug flow population balance grinding formulation presented by Torres and Casali (2009) is effectively a subset of the HerbstFuerstenau mill model. As such, this implementation of the Torres and Casali HPGR model embeds a full instance of an HerbstFuerstenau mill model into each discretised block .
The complete range of selection and breakage functions available to the HerbstFuerstenau mill model are therefore also available for the Torres and Casali HPGR model.
Torres and Casali's original HPGR model adopted the HerbstFuerstenau selection function (King, 2012) and the Austin and Luckie (1972) breakage function.^{[3]}^{[4]} However, slight differences in otherwiseequivalent formulations mean that the parameters published in Torres and Casali (2009) or other sources are invalid for this implementation and will require refitting.
HPGR products
The total HPGR product, (t/h), is the sum of products from each discretised block :
The product specifically from the edge region of an HPGR, (t/h), may be determined from:
where , the number of edge blocks is:
and is the edge fraction (frac).
The product from the centre region of an HPGR, (t/h), is found by linear interpolation on the difference, i.e.:
Specific throughput
The specific throughput of an HPGR, (t.s/h.m^{3}), is defined as:^{[5]}
Specific throughput is a common HPGR performance metric used for design and operational assessment.
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 () x column () format:
where:
 is the selection function method, 0 = User, 1 = HerbstFuerstenau, 2 = Austin
 is the breakage function method, 0 = User, 1 = Austin & Luckie, 2 = King, 3 = Natural selection
 is the number of size intervals
 is the number of ore types.
Results
The results are displayed in Excel as an array corresponding to the matrix notation below:
where:
 is the rotational speed of the rolls (rpm)
 is the mass flow rate of particles in the overall, combined product (t/h), equivalent to in Torres and Casali's nomenclature
 is the mass flow rate of particles in the edge product (t/h), equivalent to in Torres and Casali's nomenclature
 is the mass flow rate of particles in the centre product (t/h), equivalent to in Torres and Casali's nomenclature
 is the cumulative breakage function, i.e. .
Example
The images below show the selection of input arrays and output results in the Excel interface.
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. 
Method  Fixed Discharge  The discharge particle size distribution is user defined. Different distributions can be used for different solids. 
MorrellShiTondo  The throughput, power draw and product size distribution are determined by the MorrellTondoShi 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 userspecified size (in the field SizeForPassingFracCalc) in each stream. 
Passes  Display  The userspecified (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 

HerbstFuerstenau  
HelpLink  Opens a link to this page using the system default web browser. Note: Internet access is required.  
Requirements  
NumParallelUnits  Input  The number of parallel, identical units to simulate:

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 HerbstFuerstenau 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  Orespecific 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 HerbstFuerstenau is selected.  
S1E  Input  Input parameter of the HerbstFuerstenau selection function. 
Zeta1  Input  Input parameter of the HerbstFuerstenau selection function. 
Zeta2  Input  Input parameter of the HerbstFuerstenau selection function. 
dp1  Input  Input parameter of the HerbstFuerstenau 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  Orespecific 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 AddOn.
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 AddOn. 
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, MDSysCAD Full or MDSysCAD 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.
See also
References
 ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} ^{1.4} ^{1.5} Torres, M. and Casali, A., 2009. A novel approach for the modelling of highpressure grinding rolls. Minerals Engineering, 22(13), pp.11371146.
 ↑ Dunne, R.C., Kawatra, S.K., and Young, C.A. (eds), 2019. SME mineral processing and extractive metallurgy handbook. Society for Mining, Metallurgy & Exploration.
 ↑ King, R.P., 2012. Modeling and Simulation of Mineral Processing Systems. Elsevier.
 ↑ Austin, L.G. and Luckie, P.T., 1972. The estimation of nonnormalized breakage distribution parameters from batch grinding tests. Powder Technology, 5(5), pp.267271.
 ↑ Kawatra, S.K. ed., 2006. Advances in comminution. Society for Mining, Metallurgy & Exploration.