Description
This article describes the Morrell Empirical (Morrell E) method for estimating the power draw of a tumbling mill.^{[1]}
The Morrell E model is a set of empirical equations based on the performance of the theoretical Morrell Continuum model. The model was originally intended to be simpler, and therefore easier to use in practice, than the theoretical Morrell C method.
Model theory
Figure 1. Schematic of a tumbling mill showing key dimensions.
Morrell developed a simpler, empirical model of power input to the motor of a tumbling mill based on the theoretical Continuum approach.^{[1]}
Morrell's Empirical relationship for tumbling mill power draw is:
 ${\text{Gross power (kW)}}={\text{Noload power}}+\left(KD^{2.5}L_{\rm {e}}\rho _{\rm {c}}\alpha \delta \right)$
where:
 ${\text{Noload power}}=1.68D^{2.05}\left[\phi (0.667L_{\rm {d}}+L\right]^{0.82}$
 $\alpha ={\frac {J_{\rm {t}}(\omega J_{\rm {t}})}{\omega ^{2}}}$
 $\omega =2\left(2.9863\phi 2.2129\phi ^{2}0.49267\right)$
 $\delta =\phi \left(1\left[1\phi _{\rm {max}}^{*}\right]{\rm {e}}^{19.42(\phi _{\rm {max}}^{*}\phi )}\right)$
 $\phi _{\rm {max}}^{*}=0.9540.135J_{\rm {t}}$
 $L_{\rm {e}}=L\left(1+2.28J_{\rm {t}}\left[1J_{\rm {t}}\right]{\frac {L_{\rm {d}}}{L}}\right)$
 $\rho _{\rm {c}}={\frac {J_{\rm {t}}\rho _{\rm {o}}\left(1=E+EUS\right)+J_{\rm {B}}\left(\rho _{\rm {B}}\rho _{\rm {o}}\right)\left(1E\right)}{J_{\rm {t}}}}+{\frac {J_{\rm {t}}EU\left(1S\right)}{J_{\rm {t}}}}$
and
 $K$ is a calibration constant, $K=7.98$ for overflow mills and $K=9.10$ for grate mills
 $D$ is mill diameter inside liners (m)
 $L_{\rm {e}}$ is the effective length of the mill (m)
 $\rho _{\rm {c}}$ is the density of the total charge (t/m^{3})
 $\phi$ is fraction critical speed (frac)
 $L_{\rm {d}}$ is length of the cone end (m)
 $L$ is length of the cylindrical section (belly) of the mill inside liners (m)
 $\alpha$, $\omega$ and $\delta$ are empirical parameters
 $J_{\rm {t}}$ is the volumetric fraction of the mill occupied by balls and coarse rock (v/v)
 $\phi _{\rm {max}}^{*}$ is the fraction of critical speed at which power draw is maximum (frac)
 $\rho _{\rm {o}}$ is the density of ore (t/m^{3})
 $\rho _{\rm {B}}$ is the density of balls (t/m^{3})
 $E$ is volumetric fraction of interstitial void space in the charge, typically 0.4 (v/v)
 $U$ is the volumetric fraction of interstitial grinding media voidage occupied by slurry (v/v)
 $S$ is the volume fraction of solids in the mill discharge (v/v)
The length of the cone end, $L_{\rm {d}}$ (m), is:
 $L_{\rm {d}}=(r_{\rm {m}}r_{\rm {t}})\tan \alpha _{c}$
where $\alpha _{c}$ is the cone angle, measured as the angular displacement of the cone surface from the vertical direction.
Excel
The Morrell Empirical mill power model may be invoked from the Excel formula bar with the following function call:
=mdMillPower_MorrellE(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:
SysCAD
The Morrell Empirical 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

MorrellE

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.

NoLoadPower

Display

Power input to the motor when the mill is empty (no balls, rocks or slurry).

NetPower.Grate

Display

Charge motion power, including losses, for a grate discharge mill.

NetPower.Overflow

Display

Charge motion power, including losses, for an overflow discharge mill.

GrossPower.Grate

Display

Gross power input to the motor, grate discharge mill.

GrossPower.Overflow

Display

Gross power input to the motor, overflow discharge mill.

See also
References
 ↑ ^{1.0} ^{1.1} Morrell, S., 1996. Power draw of wet tumbling mills and its relationship to charge dynamics. Pt. 2: an empirical approach to modelling of mill power draw. Transactions of the Institution of Mining and Metallurgy. Section C. Mineral Processing and Extractive Metallurgy, 105.