Tumbling Mill (Power, Hilden and Powell)

Description

This article describes the Hilden and Powell (2018) approach for estimating the power draw of a tumbling mill.

Hilden and Powell extended Morrell's (1996, 2016) approaches to include the following features:

• Partial filling of the charge void space with slurry in two directions, from the charge shoulder towards the toe and from the outer shell towards the mill centre
• Estimation of the fraction of a slurry pool which settles across the centre of the mill profile and thus does not contribute to power draw
• No-load power based on mill drive type and size

Model theory

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Excel

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

=mdMillPower_HildenPowell(Parameters as Range)

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

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:

$Parameters={\begin{bmatrix}D{\text{ (m)}}\\L{\text{ (m)}}\\d_{\rm {T}}{\text{ (m)}}\\\alpha _{c}{\text{ (degrees)}}\\J_{\rm {T}}{\text{ (v/v)}}\\J_{\rm {B}}{\text{ (v/v)}}\\\rho _{\rm {R}}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\rho _{\rm {L}}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\rho _{\rm {B}}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\varepsilon {\text{ (v/v)}}\\U{\text{ (v/v)}}\\S{\text{ (}}\%{\text{ w/w)}}\\\phi {\text{ (frac)}}\\{\text{Motor loss factor (kW/kW)}}\\{\text{Installed motor power (kW)}}\\k{\text{ (kW/kW)}}\\y{\text{ (frac)}}\\\end{bmatrix}},\;\;\;\;\;\;mdMillPower\_HildenPowell={\begin{bmatrix}P_{\rm {N}}{\text{ (kW)}}\\P_{\rm {Net}}{\text{ (kW)}}\\P{\text{ (kW)}}\\\theta _{S}{\text{ (rad)}}\\\theta _{\rm {T}}{\text{ (rad)}}\\\theta _{L}{\text{ (rad)}}\\r_{\rm {i}}{\text{ (m)}}\\r'_{\rm {i}}{\text{ (m)}}\\\end{bmatrix}}\;\;\;\;\;\;$ where: $P_{\rm {Net}}=k.(P_{\rm {L}}+P'_{\rm {L}}+P_{\rm {C}}+P'_{\rm {C}})$

Figure 5. Example showing the selection of the Parameters (blue frame), and Results (light blue frame) arrays in Excel.

SysCAD

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

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
HildenPowell
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
MillDiameter	Display	Diameter of the mill (inside liners).
BellyLength	Display	Length of the cylindrical section (belly) of the mill (inside liners).
TrunnionDiameter	Display	Diameter of the trunnion (inside liners).
ConeAngle	Display	Angular displacement of the cone surface from the vertical direction.
Jt	Display	Volumetric fraction of the mill occupied by balls and coarse rock (including voids).
BallLoadVol	Display	Volumetric fraction of the mill occupied by balls (including voids).
SolidsSG	Display	Specific Gravity or density of solids.
BallSG	Display	Specific Gravity or density of balls.
LiquidsSG	Display	Specific Gravity or density of liquids.
Voidage	Input/Display	Volumetric fraction of interstitial void space in the charge. Usually 0.4. Note: The Dynamic Perfect Mixing ball mill model uses Shi's dynamic charge porosity estimation to align power draw predictions with the adopted slurry filling approach.
U	Display	Volumetric fraction of interstitial grinding media voidage occupied by slurry.
Cw	Display	Mass fraction of solids in discharge slurry.
FracCS	Display	Fraction critical speed of the mill.
MillDriveLosses	Input	Mill drive motor power loss factor.
MotorInstalledPower	Input	Maximum power draw of installed mill motor.
NetPowerAdjust	Input	Lumped calibration parameter accounting for power losses. Found to be a value of 1.26 for industrial mills.
y	Input	Profile shape factor.
ThetaShoulder	Display	Angle of the charge shoulder.
ThetaToe	Display	Angle of the charge toe.
ThetaSlurryToe	Display	Angle of the toes of the slurry component
ChargeSurfaceRadius	Display	Radius of the inner surface of the charge.
SlurrySurfaceRadius	Display	Radius of the inner surface of the slurry component.
NoLoadPower	Display	Power input to the motor when the mill is empty (no balls, rocks or slurry).
NetPower	Display	Charge motion power, including losses.
GrossPower	Display	Power input to the motor.

See also

• Morrell Continuum model