View source for Tumbling Mill (Power, Hogg and Fuerstenau)

== Description ==

This article describes the '''Hogg and Fuerstenau''' (1972) method for estimating the power draw of a tumbling mill.{{Hogg_and_Fuerstenau_(1972)}}

== Model theory ==

[[File:TumblingMillDimensions1.png|thumb|450px|Figure 1. Tumbling mill profile showing the charge shape and lift angle assumptions of the Hogg and Fuerstenau power draw model.]]

The Hogg and Fuerstenau method adopts the simplified charge geometry shown in Figure 1, and computes mill power as:{{Sepulveda_(2001)}}

:<math>P_{\rm net}=\eta P_{\rm gross} = 0.238D^{3.5} \left (\frac{L}{D} \right ) N_{\rm c} \rho_{\rm ap} (J - 1.065J^2)sin \alpha</math>

where:
* <math>P_{\rm net}</math> is the net power draw of the mill, i.e. excluding mill drive losses (kW)
* <math>P_{\rm gross}</math> is the gross power draw of the mill, i.e. including mill drive losses (kW)
* <math>\eta</math> is the mill drive efficiency (kW/kW)
* <math>D</math> is mill diameter (ft)
* <math>L</math> is mill length (ft)
* <math>N_{\rm c}</math> is mill [[Tumbling Mill (Speed)|fraction critical speed]] (frac)
* <math>\rho_{\rm ap}</math> is the apparent density of the mill charge (t/m<sup>3</sup>)
* <math>J</math> is the volumetric fraction of the mill filled with charge, including balls and interstitial voids between balls (v/v)
* <math>\alpha</math> is the charge lift angle (degrees)

The apparent charge density, <math>\rho_{\rm ap}</math>, may be computed as:

:<math>
\rho_{\rm ap} = 
\begin{cases}
\dfrac{(1-f_{\rm v}) \rho_{\rm b}J_{\rm b} + (1-f_{\rm v}) \rho_{\rm m} (J - J_{\rm b}) + \rho_{\rm p} J_{\rm p} f_{\rm v} J}{J} & \text{for autogenous or semi-autogenous mills}\\
\dfrac{(1-f_{\rm v}) \rho_{\rm b}J_{\rm b} + \rho_{\rm p}J_{\rm p}f_{\rm v}J_{\rm b} + \rho_{\rm p}(J-J_{\rm b})}{J} & \text{for ball mills}
\end{cases}
</math>

where:
* <math>f_{\rm v}</math> is the volumetric fraction of interstitial void space in the charge (usually 0.4) (v/v)
* <math>\rho_{\rm b}</math> is the density of balls (t/m<sup>3</sup>)
* <math>\rho_{\rm m}</math> is the density of solid ore particles (t/m<sup>3</sup>)
* <math>\rho_{\rm p}</math> is the density of slurry (t/m<sup>3</sup>)
* <math>J_{\rm b}</math> is the volumetric fraction of the mill filled with balls, including slurry and interstitial voids between balls (v/v)
* <math>J_{\rm p}</math> is the volumetric fraction of the interstitial charge void space occupied by slurry (v/v)

Slurry density, <math>\rho_{\rm p}</math>, may be computed as:

:<math>\rho_{\rm p} = \dfrac{1}{\dfrac{f_{\rm s}}{\rho_{\rm m}} + (1-f_{\rm s})}</math>

where <math>f_{\rm s}</math> is the mass fraction of solids in the slurry (w/w).

The net power draw, <math>P_{\rm net}</math>, may be separated into its contributing constituents:

:<math>P_{\rm b}=\left ( \frac{(1-f_{\rm v}) \rho_{\rm b} J_{\rm b} }{\rho_{\rm ap}J} \right ) \cdot P_{\rm net}</math>

:<math>P_{\rm r}=\left ( \frac{(1-f_{\rm v}) \rho_{\rm m} (J - J_{\rm b})}{\rho_{\rm ap}J} \right ) \cdot P_{\rm net}</math>

:<math>P_{\rm s}=\left ( \frac{\rho_{\rm p} J_{\rm p}f_{\rm v} J_{\rm b} }{\rho_{\rm ap}J} \cdot  \right ) \cdot P_{\rm net}</math>

:<math>P_{\rm o}=\left ( \frac{\rho_{\rm p} (J - J_{\rm b})}{\rho_{\rm ap}J} \cdot  \right ) \cdot P_{\rm net}</math>

where:
* <math>P_{\rm b}</math> is the power drawn by the ball component of the mill load (kW)
* <math>P_{\rm r}</math> is the power drawn by the rock (coarse ore) component of the mill load (kW). Autogenous and semi-autogenous mills only.
* <math>P_{\rm s}</math> is the power drawn by the interstitial slurry component of the mill load (kW)
* <math>P_{\rm o}</math> is the power drawn by the overfilling, or excess, slurry component of the mill load (when the interstitial voids are completely filled with slurry (kW). Ball mills only.

<!-- 
== Additional notes==

The Hogg and Fuerstenau model requires an estimation of the ''lift angle'' parameter, <math>\alpha</math>, which may not be readily measurable in practice.

As a guide, Doll (2016) suggests the following lift angles based on a large database of industrial mill dimensions and observed power draws:{{Doll_(2016)}}

:<math> \alpha=
    \begin{cases}
      43^{\circ} & \mbox{for } \frac{L}{D} \leq 1\\
      47^{\circ}  & \mbox{for } 1 < \frac{L}{D} \leq 2\\
      47^{\circ}  \mbox{ to } 50^{\circ}  & \mbox{for }\frac{L}{D} > 2\\
    \end{cases}
</math>

Lift angle may alternatively be employed as a 'fitting parameter' for existing power draw measurements and subsequently used to predict power draw under differing conditions, at the user's discretion.
-->

== Excel ==

The Hogg and Fuerstenau mill power model may be invoked from the Excel formula bar with the following function call:

<syntaxhighlight lang="vb">=mdMillPower_HoggFuerstenau(Parameters as Range)</syntaxhighlight>

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

The ''Parameters'' array and model results are defined below in matrix notation, along with an example image showing the selection of the same arrays in the Excel interface:

{| 
|- style="vertical-align:top;"
|
:<math>Parameters=
\begin{bmatrix}
\text{Mill type}\\
D\text{ (m)}\\
L\text{ (m)}\\
N_{\rm c}\text{ (frac)}\\
J\text{ (v/v)}\\
J_{\rm b}\text{ (v/v)}\\
J_{\rm p}\text{ (v/v)}\\
\alpha\text{ (deg.)}\\
(1-\eta)\text{ (kW/kW)}\\
f_{\rm s}\text{ (w/w)}\\
\rho_{\rm m}\text{ (t/m}^{\text{3}}\text{)}\\
\rho_{\rm b}\text{ (t/m}^{\text{3}}\text{)}\\
\end{bmatrix},\;\;\;\;\;\;

mdMillPower\_HoggFuerstenau=
\begin{bmatrix}
P_{\rm gross}\text{ (kw)}\\
P_{\rm net}\text{ (kw)}\\
P_{\rm b}\text{ (kw)}\\
P_{\rm r}\text{ (kw)}\\
P_{\rm s}\text{ (kw)}\\
P_{\rm o}\text{ (kw)}\\
\text{Charge volume (m}^{\text{3}}\text{)}\\
\text{Ball mass (t)}\\
\text{Rock mass (t)}\\
\text{Interstital slurry mass (t)}\\
\text{Above balls slurry mass (t)}\\
\rho_{\rm p}\text{ (t/m}^{\text{3}}\text{)}\\
\rho_{\rm ap}\text{ (t/m}^{\text{3}}\text{)}\\
\end{bmatrix}\;\;\;\;\;\;\;\;\;\;\;\;
</math>


where:
* <math>\text{Mill type}</math> is the type of mill, ''0 = autogenous or semi-autogenous'', ''1 = ball''
* <math>\text{Charge volume}</math> is the volume of the charge (m<sup>3</sup>)
* <math>\text{Ball mass}</math> is the mass of balls in the mill (t)
* <math>\text{Rock mass}</math> is the mass of rocks in the mill (t)
* <math>\text{Interstitial slurry mass}</math> is the mass of slurry in the interstitial charge void space (t)
* <math>\text{Above slurry mass}</math> is the mass of excess slurry above the charge (t)

| [[File:TumblingMillPower1.png|left|frame|Figure 2. Example showing the selection of the '''Parameters''' (blue frame), and '''Results''' (light blue frame) arrays in Excel.]]
|}

Please note the Excel function expects the mill diameter (<math>D</math>) and length (<math>L</math>) values in units of '''meters''', which are internally converted to feet for use in the power equation presented above.

== SysCAD ==

The Hogg and Fuerstenau power model is an optional calculation for tumbling mill units. If selected, the input and display parameters below are shown.

{{SysCAD (Text, Table Header)}}
|-
! colspan="3" style="text-align:left;" |''HoggFuerstenau''

{{SysCAD (Text, Help Link)}}

|-
|MillType
|style="background: #eaecf0" | AG/SAG or Ball
|Type of mill, AG/SAG (''RockMass'', no ''AboveBallsSLMass'') or Ball (''AboveBallsSLMass'', no ''RockMass'').
|-
|MillDiameter
|style="background: #eaecf0" | Input/Display
|Diameter of the mill (inside liners).
|-
|MillLength
|style="background: #eaecf0" | Input/Display
|Length of the mill (inside liners).
|-
|FracCS
|style="background: #eaecf0" | Input/Display
|Fraction critical speed of the mill.
|-
|ChargeFilling
|style="background: #eaecf0" | Input/Display
|Volumetric fraction of the mill filled with charge, including balls and interstitial voids between balls.
|-
|BallFilling
|style="background: #eaecf0" | Input/Display
|Volumetric fraction of the mill filled with balls, including slurry and interstitial voids between balls.
|-
|InterstialSLFill
|style="background: #eaecf0" | Input/Display
|Volumetric fraction of the interstitial charge void space occupied by slurry.
|-
|LiftAngle
|Input
|Lift angle of the charge.
|-
|PowerLosses
|Input
|Fraction of power lost due to mill drive efficiency (<math>=1-\eta</math>).
|-
|SolidsFraction
|style="background: #eaecf0" | Display
|Mass fraction of solids in the slurry phase.
|-
|OreDensity
|style="background: #eaecf0" | Display
|Density of solids.
|-
|BallDensity
|style="background: #eaecf0" | Input/Display
|Density of balls.
|-
|ChargeVolume
|style="background: #eaecf0" | Display
|Volume of charge in the mill.
|-
|BallMass
|style="background: #eaecf0" | Display
|Mass of balls in the mill.
|-
|RockMass
|style="background: #eaecf0" | Display
|Mass of coarse rocks in the mill (AG/SAG only).
|-
|InterstitialSLMass
|style="background: #eaecf0" | Display
|Mass of slurry occupying charge void space.
|-
|AboveBallsSLMass
|style="background: #eaecf0" | Display
|Mass of excess slurry resident outside/above charge (Ball mill only).
|-
|RhoSlurry
|style="background: #eaecf0" | Display
|Density of slurry phase.
|-
|ApparentDensity
|style="background: #eaecf0" | Display
|Density of total charge (including balls, rock, slurry and voids).
|-
|BallsPower
|style="background: #eaecf0" | Display
|Power drawn by the ball component of the mill load.
|-
|RocksPower
|style="background: #eaecf0" | Display
|Power drawn by the rock component of the mill load.
|-
|SlurryPower
|style="background: #eaecf0" | Display
|Power drawn by the interstitial slurry component of the mill load.
|-
|OverfillingPower
|style="background: #eaecf0" | Display
|Power drawn by the overfilling, or excess, slurry component of the mill load (when the interstitial voids are completely filled with slurry).
|-
|NetPower
|style="background: #eaecf0" | Display
|Net power drawn by the mill (excluding drive inefficiencies).
|-
|GrossPower
|style="background: #eaecf0" | Display
|Gross power drawn by the mill (including drive inefficiencies).
|}

== See also ==
* [[Mill (Herbst-Fuerstenau)]]
* [[Ball Mill (Perfect Mixing)]]
* [[Ball Mill (Perfect Mixing, Dynamic)]]

== References ==

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