Ball Mill (Overfilling)

Description

This article describes several methods for estimating the maximum volumetric flow capacity of an overflow ball mill, including the Shi (2016) and Arbiter (1991) approaches.

Model theory

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Excel

Shi method

The Shi overflow discharge ball mill overfilling model may be invoked from the Excel formula bar with the following function call:

=mdMillOverfilling_Shi(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:

$Parameters={\begin{bmatrix}D_{\rm {m}}{\text{ (m)}}\\L{\text{ (m)}}\\R_{\rm {t}}{\text{ (m)}}\\\alpha _{c}{\text{ (degrees)}}\\\phi {\text{ (frac)}}\\J_{\rm {t}}{\text{ (v/v)}}\\Q{\text{ (m}}^{3}{\text{/h)}}\\k_{\mu }\\\end{bmatrix}},\;\;\;\;\;\;mdMillOverfilling\_Shi={\begin{bmatrix}v_{\rm {ax}}{\text{ (m/s)}}\\t_{\rm {Res}}{\text{ (s)}}\\t_{\rm {Limit}}{\text{ (s)}}\\Q_{\rm {Max}}{\text{ (m}}^{3}{\text{/h)}}\\V_{1}{\text{ (m}}^{3}{\text{)}}\\V_{2}{\text{ (m}}^{3}{\text{)}}\\V_{\rm {total}}{\text{ (m}}^{3}{\text{)}}\\\end{bmatrix}}\;\;\;\;\;\;$

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

Arbiter method

The Arbiter overflow discharge ball mill overfilling model may be invoked from the Excel formula bar with the following function call:

=mdMillOverfilling_Arbiter(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:

$Parameters={\begin{bmatrix}D_{\rm {m}}{\text{ (m)}}\\\phi {\text{ (frac)}}\\J_{\rm {t}}{\text{ (v/v)}}\\\end{bmatrix}},\;\;\;\;\;\;mdMillOverfilling\_Arbiter={\begin{bmatrix}Q_{N_{\rm {q}}}{\text{ (m}}^{3}{\text{/h)}}\\\end{bmatrix}}\;\;\;\;\;\;$

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

SysCAD

The Ball Mill Overfilling Indicator 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
Overfilling
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
Shi
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.
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).
ViscidityCoeff	Input	Coefficient of viscidity.
V1	Display	Volume of slurry below slurry pool level.
V2	Display	Volume of slurry above slurry pool level.
V1	Display	Total volume of slurry in charge and pool.
AxialVelocity	Display	Axial velocity of slurry flow through the charge and pool.
tRes	Display	Residence time of slurry in mill.
tLimit	Display	Shi's suggested lower limit of residence time for mill. Based on mill diameter.
Feed.SLQv	Display	Volumetric flow rate of slurry (solids + liquids) in mill feed.
Feed.SLQv.Limit	Display	Estimated volumetric flow rate of slurry (solids + liquids) in mill feed at tLimit.
Overfilled	True/False	Indicates if tRes is less than tLimit, i.e. mill is overfilled.
Arbiter
MillDiameter	Input/Display	Diameter of the mill (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).
Feed.SLQv	Display	Volumetric flow rate of slurry (solids + liquids) in mill feed.
Feed.SLQv.NqLimit	Display	Estimated volumetric flow rate of slurry (solids + liquids) at Arbiters critical flow number limit.

See also

• Tumbling Mill (Power, Morrell Continuum)
• Perfect Mixing ball mill model
• Mill (Herbst-Fuerstenau)