Tumbling Mill (Power, Morrell Empirical)

Description

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

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.

Morrell's Empirical relationship for tumbling mill power draw is:

${\text{Gross power (kW)}}={\text{No-load power}}+\left(KD^{2.5}L_{e}\rho _{c}\alpha \delta \right)$

where:

${\text{No-load power}}=1.68D^{2.05}\left[\phi (0.667L_{d}+L\right]^{0.82}$

$\alpha ={\frac {J_{t}(\omega -J_{t})}{\omega ^{2}}}$

$\omega =2\left(2.9863\phi -2.2129\phi ^{2}-0.49267\right)$

$\delta =\phi \left(1-\left[1-\phi _{max}^{*}\right]{\rm {e}}^{-19.42(\phi _{max}^{*}-\phi )}\right)$

$\phi _{max}^{*}=0.954-0.135J_{t}$

$L_{e}=L\left(1+2.28J_{t}\left[1-J_{t}\right]{\frac {L_{d}}{L}}\right)$

$\rho _{c}={\frac {J_{t}\rho _{o}\left(1=E+EUS\right)+J_{B}\left(\rho _{B}-\rho _{o}\right)\left(1-E\right)}{J_{t}}}+{\frac {J_{t}EU\left(1-S\right)}{J_{t}}}$

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)
• $L_{e}$ is the effective length of the mill (m)
• $\rho _{c}$ is the density of the total charge (t/m³)
• $\phi$ is fraction critical speed (frac)
• $L_{d}$ 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
• $J_{t}$ is the volumetric fraction of the mill occupied by balls and coarse rock (v/v)
• $\phi _{max}^{*}$ is the fraction of critical speed at which power draw is maximum (frac)
• $\rho _{o}$ is the density of ore (t/m³)
• $\rho _{B}$ is the density of balls (t/m³)
• $E$ 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, $L_{d}$ (m), is:

$L_{d}=(r_{m}-r_{t})\tan \alpha _{c}$

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:

$Parameters={\begin{bmatrix}D{\text{ (m)}}\\L{\text{ (m)}}\\D_{t}{\text{ (m)}}\\\alpha _{c}{\text{ (degrees)}}\\\phi {\text{ (frac)}}\\J_{t}{\text{ (v/v)}}\\J_{B}{\text{ (v/v)}}\\E{\text{ (v/v)}}\\U{\text{ (v/v)}}\\{\text{Discharge pulp density (}}\%{\text{ w/w)}}\\\rho _{o}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\rho _{L}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\rho _{B}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\end{bmatrix}},\;\;\;\;\;\;mdMillPower\_MorrellE={\begin{bmatrix}{\text{No-load power (kW)}}\\{\text{Gross power (grate) (kW)}}\\{\text{Gross power (overflow) (kW)}}\\\end{bmatrix}}\;\;\;\;\;\;$ where: $D_{t}$ is the diameter of the discharge trunnion (m) ${\text{Discharge pulp density}}$ is the mass fraction of solids in the discharge pulp (% w/w) $\rho _{L}$ is the density of liquids (t/m3) ${\text{Gross power (grate)}}$ is the power the mill would draw if configured with a grate discharge (kW) ${\text{Gross power (overflow)}}$ is the power the mill would draw if configured with an overflow discharge (kW)

Figure 2. Example showing the selection of the Parameters (blue frame), and Results (light blue frame) arrays in Excel.

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
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	Display	Charge motion power, power associated with the movement of the charge.
GrossPower	Display	Gross power input to the motor.

See also

• Morrell Continuum method