Tumbling Mill (Power, Morrell Continuum)

Description

This article describes the Morrell Continuum (Morrell C) method for estimating the power draw of a tumbling mill.

The Morrell C model is a theoretical approach based on the dynamics of the grinding charge. By making simplifying assumptions about the charge shape, composition and position, Morrell developed analytical solutions to equations describing the rate at which potential and kinetic energy is imparted to the charge during mill rotation.

Model theory

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Excel

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

=mdMillPower_MorrellC(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:

$Parameters={\begin{bmatrix}D{\text{ (m)}}\\L{\text{ (m)}}\\D_{\rm {t}}{\text{ (m)}}\\\alpha _{c}{\text{ (degrees)}}\\\phi {\text{ (frac)}}\\J_{\rm {t}}{\text{ (v/v)}}\\J_{\rm {B}}{\text{ (v/v)}}\\E{\text{ (v/v)}}\\U{\text{ (v/v)}}\\{\text{Discharge pulp density (}}\%{\text{ w/w)}}\\\rho _{\rm {o}}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\rho _{\rm {L}}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\rho _{\rm {B}}{\text{ (t/m}}^{\text{3}}{\text{)}}\\k{\text{ (kW/kW)}}\\{\text{Grate discharge}}\\\end{bmatrix}},\;\;\;\;\;\;mdMillPower\_MorrellC={\begin{bmatrix}{\text{No-load power (kW)}}\\{\text{Net power (kW)}}\\{\text{Gross power (kW)}}\\\theta _{\rm {S}}{\text{ (rad)}}\\\theta _{\rm {T}}{\text{ (rad)}}\\\theta _{\rm {TO}}{\text{ (rad)}}\\r_{\rm {i}}{\text{ (m)}}\\\end{bmatrix}}\;\;\;\;\;\;$ where: $D_{\rm {t}}$ is the diameter of the discharge trunnion (m) ${\text{Discharge pulp density}}$ is the mass fraction of solids in the discharge pulp (% w/w) $\rho _{\rm {L}}$ is the density of liquids (t/m3) ${\text{Discharge mechanism}}$ is true if the mill is configured with a grate discharge, false if an overflow discharge ${\text{Net power (kW) = Gross power (kW) - No-load power (kW)}}$

Figure 3. Example showing the selection of the Parameters (blue frame), and Results (light blue frame) arrays in Excel.

SysCAD

The Morrell Continuum 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).
FracCS	Input/Display	Fraction critical speed of the mill.
Jt	Input/Display	Volumetric fraction of the mill occupied by balls and coarse rock (including voids).
Jb	Input/Display	Volumetric fraction of the mill occupied by balls (including voids).
Voidage	Input/Display	Volumetric fraction of interstitial void space in the charge. Usually 0.4.
VoidFillFraction	Input/Display	Volumetric fraction of interstitial grinding media voidage occupied by slurry.
ConeAngle	Input/Display	Angular displacement of the cone surface from the vertical direction.
DischargePulpDensity	Display	Mass fraction of solids in discharge slurry.
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.
DischargeType	Grate/Overflow	Discharge configuration, grate or overflow.
NetPowerAdjust	Input	Lumped calibration parameter accounting for power losses. Found to be a value of 1.26 for industrial mills.
ThetaShoulder	Display	Angle of the charge shoulder.
ThetaToe	Display	Angle of the charge toe.
ThetaSlurryToe	Display	Angle of the slurry pool surface at the toe.
ChargeSurfaceRadius	Display	Radius of the inner surface of the charge.
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 Empirical method