View source for HPGR (Torres)

== Description ==

This article describes the '''Torres and Casali''' (2009) model for comminution by High Pressure Grinding Rolls (HPGR).{{Torres and Casali (2009)}}

== Model theory ==

{{Under construction|section}}
<!-- SME handbook has a reference to cake density factor being 80-85% -->

== Excel ==

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

<syntaxhighlight lang="vb">=mdUnit_HPGR_Torres(Parameters as Range, Size as Range, HPGRFeed as Range, OreSG as Range, Selection as Range, Breakage as Range)</syntaxhighlight>

{{Excel (Text, Help, No Arguments)}}

=== Inputs ===

{{Excel (Text, Inputs)}}

:<math>Parameters=
\begin{bmatrix}
D\text{ (m)}\\
L\text{ (m)}\\
R_{\rm p}\text{  (bar)}\\
\rho_{\rm a}\text{ (t/m}^3\text{)}\\
\text{Cake density factor (frac)}\\
a\text{ (frac)}\\
\text{FracIP (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}
</math>


:<math>
Selection = 
\begin{bmatrix}
\begin{bmatrix}
S^{\rm E}_{1}\\
\zeta_1\\
\zeta_2\\
d_{{\rm p}1}\text{ (mm)}\\
\end{bmatrix}_1
\dots
\begin{bmatrix}
S^{\rm E}_{1}\\
\zeta_1\\
\zeta_2\\
d_{{\rm p}1}\text{ (mm)}\\
\end{bmatrix}_m
\end{bmatrix},\;\;\;\;\;\;

Breakage = 
\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}


</math>


where:
* <math>D</math> is the diameter of the rolls (m)
* <math>L</math> is the length of the rolls (m)
* <math>R_{\rm p}</math> is the operating pressure of the rolls (bar)
* <math>\rho_{\rm a}</math> is the bulk density of the feed solids (t/m<sup>3</sup>)
* <math>\text{Cake density factor}</math> is the ratio of cake bulk density to solids phase density (frac)
* <math>a</math> is the edge fraction (frac)
* <math>FracIP</math> is the fraction of the interparticle compression angle applied to compute power draw (deg/deg)
* <math>N_{\rm B}</math> is the number of model blocks
* <math>n</math> is the number of size intervals
* <math>m</math> is the number of ore types
* <math>d_i</math> is the size of the square mesh interval that mass is retained on (mm)
* <math>d_{i+1}<d_i<d_{i-1}</math>, i.e. descending size order from top size (<math>d_{1}</math>) to sub mesh (<math>d_{n}=0</math> mm)
* <math>Q_{\rm M,F}</math> is feed solids mass flow rate by size and ore type (t/h)
* <math>\rho_{\rm S}</math> is the solid density (t/m<sup>3</sup>)
* <math>S^{\rm E}_1</math> is the value of the energy-specific selection function (<math>S^{\rm E}_i</math>) at a nominal reference particle size <math>\bar d_{{\rm p}1}</math> (mm)
* <math>\zeta_1</math> and <math>\zeta_2</math> are coefficients that characterise the shape of the energy-specific selection function curve
* <math>\beta_0</math>, <math>\beta_1</math>, <math>\beta_2</math>, and <math>\beta_{02}</math> are coefficients that determine the shape of the cumulative breakage function curve

=== Results ===

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

:<math>
\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^{\rm E}_1\\
\vdots\\
S^{\rm E}_n\\
\end{bmatrix}_1
\dots
\begin{bmatrix}
S^{\rm E}_1\\
\vdots\\
S^{\rm E}_n\\
\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}
</math>

where:
* <math>\delta</math> is the bulk density of discharge cake solids (t/m<sup>3</sup>)
* <math>x_{\rm c}</math> is the critical gap (m)
* <math>\alpha_{\rm IP}</math> is the interparticle compression angle (deg.)
* <math>G_{\rm s}</math> is the throughput (t/h)
* <math>U</math> is the circumferential rolls speed (m/s)
* <math>\text{Rolls rotational speed}</math> is the rotational speed of the rolls (rpm)
* <math>\dot m</math> is the specific throughput (t.s/m<sup>3</sup>.h)
* <math>F</math> is the compression force (kN)
* <math>W</math> is the specific comminution energy (kWh/t)
* <math>P</math> is power draw (kW)
* <math>Q_{\rm M,P}</math> is the mass flow rate of particles in the overall, combined product (t/h)
* <math>Q_{\rm M,E}</math> is the mass flow rate of particles in the ''edge'' product (t/h)
* <math>Q_{\rm M,C}</math> is the mass flow rate of particles in the ''centre'' product (t/h)
* <math>\bar d_{i}</math> is the [[Conversions|geometric mean size]] of particles in size interval <math>i</math> (mm)
* <math>S^{\rm E}</math> is the energy-specific selection function
* <math>B</math> is the cumulative breakage function (frac)
* <math>y</math> is block position (m)

=== Example ===

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

{|
|- style="vertical-align:top;"
| [[File:HPGRTorres1.png|left|frame|Figure 1. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]]|| [[File:HPGRTorres2.png|left|frame|Figure 2. Example showing the selection of the '''Size''' (red frame), '''HPGRFeed''' (purple frame) and '''OreSG''' (green frame) arrays in Excel.]] || [[File:HPGRTorres3.png|left|frame|Figure 3. Example showing the selection of the '''Selection''' (pink frame) and '''Breakage''' (brown frame) arrays in Excel.]] 
|- 
|colspan="3"| [[File:HPGRTorres4.png|left|frame|Figure 4. 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.

{{SysCAD (Page, HPGR, DLL*HPGR)}}

{{SysCAD (Page, HPGR, Torres, HPGR)}}

{{SysCAD (Page, Mill, Herbst-Fuerstenau, Selection)}}

{{SysCAD (Page, Mill, Herbst-Fuerstenau, Breakage)}}

{{SysCAD (Page, HPGR, Torres, Power)}}

{{SysCAD (Page, About)}}

== References ==

[[Category:Excel]]
[[Category:SysCAD]]