HPGR (Morrell-Tondo-Shi)

Description

This article describes the Morrell, Shi and Tondo (1997) model for comminution by High Pressure Grinding Rolls (HPGR), verified for simulation and scale-up by Daniel and Morrell (2004).

Model theory

Figure 1. Schematic diagram showing the Whiten crusher sub-model arrangement of the Morrell-Shi-Tondo approach (after Morrell et al., 1997).

This section is currently under construction. Please check back later for updates and revisions.

The Morrell, Tondo and Shi (1997) model simulates HPGR performance by linking three separate instances of the Whiten crusher model. The crusher instances represent comminution by the sub-processes of pre-crushing, edge crushing, and compressed bed crushing, as illustrated in Figure 1.

Please see Crusher (Whiten) for a full description of the Whiten crusher model.

Excel

The Morrell-Tondo-Shi HPGR model may be invoked from the Excel formula bar with the following function call:

=mdUnit_HPGR_MorrellTondoShi(Parameters as Range, Size as Range, HPGRFeed as Range, SP_Appearance as Range, SP_Ecst10Size-t10-size as Range, Bed_Appearance as Range, Bed_Ecst10Size 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:

$Parameters={\begin{bmatrix}D{\text{ (m)}}\\L{\text{ (m)}}\\U{\text{ (m/s)}}\\x_{\rm {g}}{\text{ (m)}}\\\rho _{\rm {c}}{\text{ (t/m}}^{3}{\text{)}}\\\rho _{\rm {g}}{\text{ (t/m}}^{3}{\text{)}}\\\gamma {\text{ (-)}}\\CpA{\text{ (-)}}\\CpB{\text{ (-)}}\\P_{\rm {NoLoad}}{\text{ (kW)}}\\(k_{\rm {P}})_{\rm {Edge}}{\text{ (kW/kW)}}\\(k_{\rm {P}})_{\rm {Bed}}{\text{ (kW/kW)}}\\(c_{K_{1}})_{\rm {Pre}}{\text{ (mm/mm)}}\\(K_{3})_{\rm {Pre}}{\text{ (-)}}\\(t_{10})_{\rm {Pre}}{\text{ (}}\%{\text{)}}\\(K_{3})_{\rm {Bed}}{\text{ (-)}}\\(c_{K_{1}})_{\rm {Edge}}{\text{ (mm/mm)}}\\(K_{3})_{\rm {Edge}}{\text{ (-)}}\\(t_{10})_{\rm {Edge}}{\text{ (}}\%{\text{)}}\\\end{bmatrix}},\;\;\;\;\;\;Size={\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\\\end{bmatrix}},\;\;\;\;\;\;HPGRFeed={\begin{bmatrix}(Q_{\rm {M,F}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,F}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}$

$SP\_Appearance={\begin{bmatrix}{\begin{bmatrix}(t_{75})_{10}{\text{ (}}\%{\text{)}}&(t_{50})_{10}{\text{ (}}\%{\text{)}}&(t_{25})_{10}{\text{ (}}\%{\text{)}}&(t_{4})_{10}{\text{ (}}\%{\text{)}}&(t_{2})_{10}{\text{ (}}\%{\text{)}}\\(t_{75})_{20}{\text{ (}}\%{\text{)}}&(t_{50})_{20}{\text{ (}}\%{\text{)}}&(t_{25})_{20}{\text{ (}}\%{\text{)}}&(t_{4})_{20}{\text{ (}}\%{\text{)}}&(t_{2})_{20}{\text{ (}}\%{\text{)}}\\(t_{75})_{30}{\text{ (}}\%{\text{)}}&(t_{50})_{30}{\text{ (}}\%{\text{)}}&(t_{25})_{30}{\text{ (}}\%{\text{)}}&(t_{4})_{30}{\text{ (}}\%{\text{)}}&(t_{2})_{30}{\text{ (}}\%{\text{)}}\\\end{bmatrix}}_{\rm {SP,1}}\\\vdots \\{\begin{bmatrix}(t_{75})_{10}{\text{ (}}\%{\text{)}}&(t_{50})_{10}{\text{ (}}\%{\text{)}}&(t_{25})_{10}{\text{ (}}\%{\text{)}}&(t_{4})_{10}{\text{ (}}\%{\text{)}}&(t_{2})_{10}{\text{ (}}\%{\text{)}}\\(t_{75})_{20}{\text{ (}}\%{\text{)}}&(t_{50})_{20}{\text{ (}}\%{\text{)}}&(t_{25})_{20}{\text{ (}}\%{\text{)}}&(t_{4})_{20}{\text{ (}}\%{\text{)}}&(t_{2})_{20}{\text{ (}}\%{\text{)}}\\(t_{75})_{30}{\text{ (}}\%{\text{)}}&(t_{50})_{30}{\text{ (}}\%{\text{)}}&(t_{25})_{30}{\text{ (}}\%{\text{)}}&(t_{4})_{30}{\text{ (}}\%{\text{)}}&(t_{2})_{30}{\text{ (}}\%{\text{)}}\\\end{bmatrix}}_{\rm {SP,m}}\\\end{bmatrix}},\;\;\;\;\;\;SP\_Ecst10Size={\begin{bmatrix}{\begin{bmatrix}(d_{\rm {p}})_{1}{\text{ (mm)}}&\dots &(d_{\rm {p}})_{k}{\text{ (mm)}}\\\end{bmatrix}}\\{\begin{bmatrix}(Ecs_{1})_{10}{\text{ (kWh/t)}}&\dots &(Ecs_{k})_{10}{\text{ (kWh/t)}}\\(Ecs_{1})_{20}{\text{ (kWh/t)}}&\dots &(Ecs_{k})_{20}{\text{ (kWh/t)}}\\(Ecs_{1})_{30}{\text{ (kWh/t)}}&\dots &(Ecs_{k})_{30}{\text{ (kWh/t)}}\\\end{bmatrix}}_{\rm {SP,1}}\\\vdots \\{\begin{bmatrix}(Ecs_{1})_{10}{\text{ (kWh/t)}}&\dots &(Ecs_{k})_{10}{\text{ (kWh/t)}}\\(Ecs_{1})_{20}{\text{ (kWh/t)}}&\dots &(Ecs_{k})_{20}{\text{ (kWh/t)}}\\(Ecs_{1})_{30}{\text{ (kWh/t)}}&\dots &(Ecs_{k})_{30}{\text{ (kWh/t)}}\\\end{bmatrix}}_{\rm {SP,m}}\\\end{bmatrix}}$

$Bed\_Appearance={\begin{bmatrix}{\begin{bmatrix}(t_{75})_{10}{\text{ (}}\%{\text{)}}&(t_{50})_{10}{\text{ (}}\%{\text{)}}&(t_{25})_{10}{\text{ (}}\%{\text{)}}&(t_{4})_{10}{\text{ (}}\%{\text{)}}&(t_{2})_{10}{\text{ (}}\%{\text{)}}\\(t_{75})_{20}{\text{ (}}\%{\text{)}}&(t_{50})_{20}{\text{ (}}\%{\text{)}}&(t_{25})_{20}{\text{ (}}\%{\text{)}}&(t_{4})_{20}{\text{ (}}\%{\text{)}}&(t_{2})_{20}{\text{ (}}\%{\text{)}}\\(t_{75})_{30}{\text{ (}}\%{\text{)}}&(t_{50})_{30}{\text{ (}}\%{\text{)}}&(t_{25})_{30}{\text{ (}}\%{\text{)}}&(t_{4})_{30}{\text{ (}}\%{\text{)}}&(t_{2})_{30}{\text{ (}}\%{\text{)}}\\\end{bmatrix}}_{\rm {Bed,1}}\\\vdots \\{\begin{bmatrix}(t_{75})_{10}{\text{ (}}\%{\text{)}}&(t_{50})_{10}{\text{ (}}\%{\text{)}}&(t_{25})_{10}{\text{ (}}\%{\text{)}}&(t_{4})_{10}{\text{ (}}\%{\text{)}}&(t_{2})_{10}{\text{ (}}\%{\text{)}}\\(t_{75})_{20}{\text{ (}}\%{\text{)}}&(t_{50})_{20}{\text{ (}}\%{\text{)}}&(t_{25})_{20}{\text{ (}}\%{\text{)}}&(t_{4})_{20}{\text{ (}}\%{\text{)}}&(t_{2})_{20}{\text{ (}}\%{\text{)}}\\(t_{75})_{30}{\text{ (}}\%{\text{)}}&(t_{50})_{30}{\text{ (}}\%{\text{)}}&(t_{25})_{30}{\text{ (}}\%{\text{)}}&(t_{4})_{30}{\text{ (}}\%{\text{)}}&(t_{2})_{30}{\text{ (}}\%{\text{)}}\\\end{bmatrix}}_{\rm {Bed,m}}\\\end{bmatrix}},\;\;\;\;\;\;Bed\_Ecst10Size={\begin{bmatrix}{\begin{bmatrix}(d_{\rm {p}})_{1}{\text{ (mm)}}&\dots &(d_{\rm {p}})_{k}{\text{ (mm)}}\\\end{bmatrix}}\\{\begin{bmatrix}(Ecs_{1})_{10}{\text{ (kWh/t)}}&\dots &(Ecs_{k})_{10}{\text{ (kWh/t)}}\\(Ecs_{1})_{20}{\text{ (kWh/t)}}&\dots &(Ecs_{k})_{20}{\text{ (kWh/t)}}\\(Ecs_{1})_{30}{\text{ (kWh/t)}}&\dots &(Ecs_{k})_{30}{\text{ (kWh/t)}}\\\end{bmatrix}}_{\rm {Bed,1}}\\\vdots \\{\begin{bmatrix}(Ecs_{1})_{10}{\text{ (kWh/t)}}&\dots &(Ecs_{k})_{10}{\text{ (kWh/t)}}\\(Ecs_{1})_{20}{\text{ (kWh/t)}}&\dots &(Ecs_{k})_{20}{\text{ (kWh/t)}}\\(Ecs_{1})_{30}{\text{ (kWh/t)}}&\dots &(Ecs_{k})_{30}{\text{ (kWh/t)}}\\\end{bmatrix}}_{\rm {Bed,m}}\\\end{bmatrix}}$

where:

• $D$ is the diameter of the rolls (m)
• $L$ is the length of the rolls (m)
• $x_{\rm {g}}$ is the working gap (m)
• $\rho _{\rm {c}}$ is the bulk density of the feed solids (t/m³)
• $\rho _{\rm {g}}$ is the bulk density of the discharge cake solids (t/m³)
• $\gamma$ is the edge split factor (-)
• $CpA$ is a throughput relationship parameter
• $CpB$ is a throughput relationship parameter
• $P_{\rm {NoLoad}}$ is the no-load power draw (kW)
• $k_{\rm {p}}$ is the power coefficient (kW/kW)
• $c_{\rm {K_{1}}}$ is the $K_{1}$ scale factor (mm/mm)
• $K_{3}$ is an exponent determining the shape of the Whiten crusher classification function
• $t_{10}$ is a parameter of the Whiten crusher breakage function, defined as the fraction of broken particles which are smaller than one-tenth of the original particle size (%)
• $n$ is the number of size intervals
• $m$ is the number of ore types
• $d_{i}$ is the size of the square mesh interval that mass is retained on (mm)
• $d_{i+1}<d_{i}<d_{i-1}$, i.e. descending size order from top size ($d_{1}$) to sub mesh ($d_{n}=0$ mm)
• $Q_{\rm {M,F}}$ is the mass flow rate of particles in the crusher feed (t/h)
• $(t_{y})_{x}$ is the fraction of mass passing one-$y$th the size of a parent particle (%) when $x\%$ passes one-tenth the size of a parent particle (%), $y\in \{75,50,25,2\}$, $x\in \{10,20,30\}$
• $d_{\rm {p}}$ is the size of a parent particle subject to breakage for Energy-Size Reduction relationship testing (mm)
• $k$ is the number of particle sizes subject to breakage for Energy-Size Reduction relationship testing
• $(Ecs_{k})_{x}$ is the specific comminution energy required to break a parent particle of size $(d_{p})_{k}$ into a distribution of progeny particles with $x\%$ passing one-tenth of the parent size (kWh/t)
• the ${\rm {SP}}$ subscript indicates properties associated with single particle breakage
• the ${\rm {Bed}}$ subscript indicates properties associated with bed compression breakage

Results

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

${\mathit {mdUnit\_HPGR\_MorrellTondoShi}}={\begin{bmatrix}{\begin{array}{c}{\begin{bmatrix}x_{\rm {c}}{\text{ (m)}}\\f{\text{ (frac)}}\\Cp{\text{ (frac)}}\\Q{\text{ (t/h)}}\\P_{\rm {Pre}}{\text{ (kW)}}\\P_{\rm {Edge}}{\text{ (kW)}}\\P_{\rm {Bed}}{\text{ (kW)}}\\P_{\rm {Gross}}{\text{ (kW)}}\\E_{\rm {cs}}{\text{ (kWh/t)}}\\(K_{1})_{\rm {Pre}}{\text{ (mm)}}\\(K_{2})_{\rm {Pre}}{\text{ (mm)}}\\(K_{1})_{\rm {Edge}}{\text{ (mm)}}\\(K_{2})_{\rm {Edge}}{\text{ (mm)}}\\\end{bmatrix}}\end{array}}&{\begin{array}{cccc}{\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,P}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,P}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,P}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,P}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,E}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,E}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,E}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,E}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,B}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,B}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,B}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,B}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}\\\\\\\\\\\\\\\\\\\\\\\end{array}}\end{bmatrix}}$

where:

• $x_{\rm {c}}$ is the critical gap size (m)
• $f$ is the fraction of feed reporting to the edge crusher (frac)
• $Cp$ is the throughput scaling factor (frac)
• $P$ is power draw (kW)
• $E_{\rm {cs}}$ is the specific comminution energy (kWh/t)
• $Q_{\rm {M,P}}$ is the mass flow rate of particles in the overall, combined product (t/h)
• $Q_{\rm {M,E}}$ is the mass flow rate of particles in the edge product (t/h)
• $Q_{\rm {M,B}}$ is the mass flow rate of particles in the bed (centre) product (t/h)

Example

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

Figure 2. Example showing the selection of the Parameters (blue frame) array in Excel.
Figure 2. Example showing the selection of the Size (red frame) and HPGRFeed (purple frame) arrays in Excel.	Figure 4. Example showing the selection of the Single Particle Appearance (green frame), Single Particle Energy-Size Reduction (pink frame), Compressed Bed Appearance (brown frame), and Compressed Bed Energy-Size Reduction (teal frame) arrays in Excel.	Figure 5. 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.
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
MorrellTondoShi
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: Feed is divided by the number of parallel units before being sent to the unit model. Unit model product is multiplied back by the same value and returned to the SysCAD product stream. All unit model result values are shown per parallel unit.
Parameters
RollDiameter / D	Input	Diameter of the rolls.
RollLength / L	Input	Length of the rolls.
RollSpeed / U	Input	Circumferential velocity of the rolls.
Working gap / xg	Input	Working gap between the rolls.
FeedBulkDensity / Rhoc	Input	Bulk density of the feed solids.
CakeDensity / Rhog	Input	Bulk density of the discharge cake solids.
SplitFactor / g	Input	Edge split factor.
CpA	Input	Throughput relationship parameter.
CpB	Input	Throughput relationship parameter.
NoLoadPower	Input	No-load power draw of machine.
PowerCoeffEdge / kPEdge	Input	Power coefficient of edge crusher.
PowerCoeffBed / kPBed	Input	Power coefficient of bed crusher.
Whiten
K1Factor	Input	K1 parameter scaling factor for Pre and Edge crushers.
K1	Display	K1 parameter for Pre, Bed and Edge crushers.
K2	Display	K2 parameter for Pre, Bed and Edge crushers.
K3	Input	K3 parameter for Pre, Bed and Edge crushers.
t10	Input	t10 parameter for Pre, Bed and Edge crushers.
Results
CriticalGap / xc	Display	Critical gap.
EdgeSplit / f	Display	Edge split fraction.
Cp	Display	Throughput scaling factor.
Throughput / Q	Display	Maximum throughput of HPGR unit.
GrossPower	Display	Gross power draw of HPGR unit.
Ecs	Display	Specific comminution energy of HPGR unit.

Ore page

This page is used to define the crusher comminution properties of SysCAD species with the size distribution quality in the project.

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.
View
ColumnView	CheckBox	Arranges the Appearance and Energy-Size Reduction data in table format (default) or column view.
Appearance
OreSpecific	CheckBox	Ore-specific parameters, allows the Appearance data 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.
Appearance	Input	Appearance function data
Ecs
NumEcsSizes	Input	The number of original particle sizes for which Energy-Size Reduction data was obtained in the JK Drop Weight test (usually three or five).
OreSpecific	CheckBox	Ore-specific parameters, allows the Energy-Size Reduction data to be separately input for all species. Default is all species have the same set of single input properties.
Ecs	Input	Energy-Size Reduction data

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.