Tumbling Mill (Power, Morrell Discrete Shell)

Description

This article describes the Morrell Discrete Shell (Morrell D) method for estimating the power draw of a tumbling mill. The Morrel D model adopts a more sophisticated treatment of charge dynamics than the Morrell Continuum method:

• The total charge is subdivided into discrete layers, or shells, whose size and position may individually vary due to grinding media size, mill speed, lift bar geometry and other conditions.
• The rate at which potential and kinetic energy is imparted to each shell during mill rotation is calculated separately and subsequently summed to estimate the total mill power draw.

Model theory

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Excel

The Morrell Discrete Shell mill power model may be invoked from the Excel formula bar with the following function call:

=mdMillPower_MorrellD(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:

[math]\displaystyle{ Parameters= \begin{bmatrix} D\text{ (m)}\\ L\text{ (m)}\\ D_{\rm t}\text{ (m)}\\ \alpha_{c}\text{ (degrees)}\\ h\text{ (m)}\\ \rho\text{ (rad)}\\ \text{Mill speed (rpm)}\\ J_{\rm t}\text{ (v/v)}\\ J_{\rm B}\text{ (v/v)}\\ U\text{ (v/v)}\\ \bar x\text{ (m)}\\ \rho_{\rm o}\text{ (t/m}^{\text{3}}\text{)}\\ \rho_{\rm L}\text{ (t/m}^{\text{3}}\text{)}\\ \rho_{\rm B}\text{ (t/m}^{\text{3}}\text{)}\\ \text{Discharge pulp density (}\% \text{ w/w)}\\ \text{Discharge mechanism}\\ \gamma\\ Q_{\rm V}\text{ (m}^{\text{3}}\text{/h)}\\ \end{bmatrix},\;\;\;\;\;\; mdMillPower\_MorrellD= \begin{bmatrix} \begin{bmatrix} \text{Gross power (kW)}\\ \text{No-loadpower (kW)}\\ \text{Net impact power (kW)}\\ \text{Net att/abr power (kW)}\\ \end{bmatrix} \begin{bmatrix} r_1 \text{ (m)}\\ \vdots\\ r_n \text{ (m)} \end{bmatrix} \begin{bmatrix} \theta_{\rm S1} \text{ (rad)}\\ \vdots\\ \theta_{{\rm S}n} \text{ (rad)} \end{bmatrix} \begin{bmatrix} \theta_{\rm T1} \text{ (rad)}\\ \vdots\\ \theta_{{\rm T}n} \text{ (rad)} \end{bmatrix} \begin{bmatrix} P_1 \text{ (kW)}\\ \vdots\\ P_n \text{ (kW)} \end{bmatrix} \end{bmatrix}\;\;\;\;\;\;\;\;\;\;\;\; }[/math] where: [math]\displaystyle{ D_{\rm t} }[/math] is the diameter of the discharge trunnion (m), i.e. [math]\displaystyle{ D_{\rm t} = 2 r_{\rm t} }[/math] [math]\displaystyle{ h }[/math] is the lifter height (m) [math]\displaystyle{ \rho }[/math] is the lifter face angle (deg.) [math]\displaystyle{ \text{Mill speed} }[/math] is the rotational speed of the mill (rpm) [math]\displaystyle{ \rho_{\rm L} }[/math] is the density of liquids (t/m3) [math]\displaystyle{ \text{Discharge pulp density} }[/math] is the mass fraction of solids in the discharge pulp (% w/w) [math]\displaystyle{ \text{Discharge mechanism} }[/math] is the discharge configuration type, 0 = Grate, 1 = Overflow [math]\displaystyle{ Q_{\rm V} }[/math] is the volumetric discharge flow rate of pulp from the mill (m3/h) [math]\displaystyle{ \text{Net impact power} }[/math] is the power draw attributable to the impact mechanism (kW) [math]\displaystyle{ \text{Net att/abr power} }[/math] is the power draw attributable to the attrition/abrasion mechanisms (kW) [math]\displaystyle{ n }[/math] is the number of discrete shells [math]\displaystyle{ r_i }[/math] is the radius of shell [math]\displaystyle{ i }[/math] (m) [math]\displaystyle{ \theta_{{\rm S}i} }[/math] is the angular position of the shoulder of shell [math]\displaystyle{ i }[/math] (rad) [math]\displaystyle{ \theta_{{\rm T}i} }[/math] is the angular position of the toe of shell [math]\displaystyle{ i }[/math] (rad) [math]\displaystyle{ P_i }[/math] is the power drawn by shell [math]\displaystyle{ i }[/math] (kW)

Figure 4. Example showing the selection of the Parameters (blue frame), and Results (light blue frame) arrays in Excel.

SysCAD

The Morrell Discrete Shell 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
MorrellC
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
MillDiameter	Input/Display	Diameter of the mill (inside liners).
BellyLength	Input/Display	Length of the cylindrical section (belly) of the mill (inside liners).
TrunnionDiameter	Input/Display	Diameter of the trunnion (inside liners).
ConeAngle	Input/Display	Angular displacement of the cone surface from the vertical direction.
LifterHeight	Input	Height of the lifters above the liner surface
LifterAngle	Input	Angle between the face and the base of the lifter
MillRPM	Input/Display	Mill rotational speed
Jt	Input/Display	Volumetric fraction of the mill occupied by balls and coarse rock (including voids).
BallLoadVol	Input/Display	Volumetric fraction of the mill occupied by balls (including voids).
VoidFillFraction	Input/Display	Volumetric fraction of interstitial grinding media voidage occupied by slurry.
MeanMediaSize	Input	Characteristic media size, [math]\displaystyle{ \bar x }[/math]
SolidsSG	Display	Specific Gravity or density of solids.
LiquidsSG	Display	Specific Gravity or density of liquids.
BallSG	Input/Display	Specific Gravity or density of balls.
Cw	Display	Mass fraction of solids in discharge slurry.
DischargeType	Grate/Overflow	Discharge configuration, grate or overflow.
Gamma	Input	Value of the slip coefficient, [math]\displaystyle{ \gamma }[/math]
ThetaShoulder	Display	Angle of the shoulder of the outermost charge shell.
ThetaToe	Display	Angle of the toe of the outermost charge toe
NoLoadPower	Display	Power input to the motor when the mill is empty (no balls, rocks or slurry).
NetImpactPower	Display	Net power attributable to impact
NetAttAbrPower	Display	Net power attributable to attrition/abrasion
NetPower	Display	Charge motion power.
GrossPower	Display	Power input to the motor.

See also

• Morrell Continuum model
• Morrell Empirical model