Ball Mill (Overfilling): Difference between revisions

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== Description ==
== Description ==


This article describes several methods for estimating the maximum volumetric flow capacity of an overflow ball mill, including the '''Shi''' (2016) and '''Arbiter''' (1990) approaches.{{Shi (2016)}}{{Arbiter (1990)}}
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.{{Shi (2016)}}{{Arbiter (1991)}}


== Model theory ==
== Model theory ==


=== Shi (2016) ===
{{Restricted content}}
 
<hide>
=== Shi method ===


[[File:BallMillOverfilling1.png|thumb|450px|Figure 1. Overflow discharge tumbling mill profile showing Shi's assumed charge and slurry pool areas.]]
[[File:BallMillOverfilling1.png|thumb|450px|Figure 1. Overflow discharge tumbling mill profile showing Shi's assumed charge and slurry pool areas.]]


Shi estimated the residence time of an overflow ball mill by considering the volume of slurry present in the grinding media interstices and slurry pool.
Shi (2016) estimated the residence time of an overflow ball mill by considering the volume of slurry present in the grinding media interstices and slurry pool.


This was accomplished by adopting Morrell's simplified charge geometry and calculating the volume of slurry resident in each of the areas A-C in Figure 1:{{Morrell (1996a)}}
This was accomplished by adopting Morrell's (1996) simplified charge geometry and calculating the volume of slurry resident in each of the areas A-C in Figure 1:{{Morrell (1996a)}}


:A. In the grinding media interstices above the slurry pool level
:A. In the grinding media interstices above the slurry pool level
Line 24: Line 27:


The simplified charge geometry is defined by three parameters:
The simplified charge geometry is defined by three parameters:
* the angular position of the ''toe'' of the charge, <math>\theta_t</math> (rad)
* the angular position of the ''toe'' of the charge, <math>\theta_{\rm t}</math> (rad),
* the angular position of the ''shoulder'' charge, <math>\theta_s</math> (rad)
* the angular position of the ''shoulder'' charge, <math>\theta_{\rm s}</math> (rad), and
* the radius of the ''inner charge surface'', <math>R_i</math> (m)
* the radius of the ''inner charge surface'', <math>R_i</math> (m).


Morrell's (1996) equations are used to define the charge position, i.e.{{Morrell (1996a)}}
Morrell's equations are used to define the charge position, i.e.{{Morrell (1996a)}}


The position of the '''toe''' of the charge is defined as:  
The position of the '''toe''' of the charge is defined as:  


:<math>\theta_t = 2.5307 \left ( 1.2796 - J_t \right ) \left (1 - {\rm e}^{-19.42(\phi_c - \phi)} \right ) + \frac{\pi}{2}</math>
:<math>\theta_{\rm t} = 2.5307 \left ( 1.2796 - J_{\rm t} \right ) \left (1 - {\rm e}^{-19.42(\phi_{\rm c} - \phi)} \right ) + \frac{\pi}{2}</math>


where:
where:
* <math>J_{t}</math> is the volumetric fraction of the mill occupied by balls and coarse rock (including void space and interstitial slurry) (v/v)
* <math>J_{t}</math> is the volumetric fraction of the mill occupied by balls and coarse rock (including void space and interstitial slurry) (v/v)
* <math>\phi</math> (frac) is the theoretical [[Tumbling Mill (Speed)|fraction critical speed]]
* <math>\phi</math> (frac) is the theoretical [[Tumbling Mill (Speed)|fraction critical speed]].


The fraction of critical speed at which centrifuging actually occurs, <math>\phi_c</math> (frac), is:
The fraction of critical speed at which centrifuging actually occurs, <math>\phi_{\rm c}</math> (frac), is:


:<math> \phi_c =
:<math> \phi_{\rm c} =
     \begin{cases}
     \begin{cases}
       \phi & \phi>0.35(3.364-J_t)\\
       \phi & \phi>0.35(3.364-J_{\rm t})\\
       0.35(3.364-J_t)& \phi \leq 0.35(3.364-J_t)\\
       0.35(3.364-J_{\rm t})& \phi \leq 0.35(3.364-J_{\rm t})\\
     \end{cases}
     \end{cases}
</math>
</math>
Line 49: Line 52:
The position of the '''shoulder''' of the charge is:  
The position of the '''shoulder''' of the charge is:  


:<math>\theta_s = \frac{\pi}{2} - \left ( \theta_t - \frac{\pi}{2} \right ) \big[ \left ( 0.336 + 0.1041 \phi \right ) + \left (1.54 - 2.5673 \phi \right ) J_t \big]</math>
:<math>\theta_{\rm s} = \frac{\pi}{2} - \left ( \theta_{\rm t} - \frac{\pi}{2} \right ) \big[ \left ( 0.336 + 0.1041 \phi \right ) + \left (1.54 - 2.5673 \phi \right ) J_{\rm t} \big]</math>


The inner '''charge surface radius''' is:
The inner '''charge surface radius''' is:


:<math>R_i = r_m \left ( 1 - \frac{2 \pi \beta J_t}{2 \pi + \theta_s - \theta_t} \right )^{0.5}</math>
:<math>R_i = r_{\rm m} \left ( 1 - \frac{2 \pi \beta J_{\rm t}}{2 \pi + \theta_{\rm s} - \theta_{\rm t}} \right )^{0.5}</math>


where the fraction of total charge in the active region, <math>\beta</math> (frac), is:
where <math>r_{\rm m}</math> is the radius of the mill (m), and the fraction of total charge in the active region, <math>\beta</math> (frac), is:


:<math>\beta = \frac{t_c}{t_f + t_c}</math>
:<math>\beta = \frac{t_{\rm c}}{t_{\rm f} + t_{\rm c}}</math>


The time taken to travel between the toe and shoulder of the charge during one revolution, <math>t_c</math> (s), is:
The time taken to travel between the toe and shoulder of the charge during one revolution, <math>t_{\rm c}</math> (s), is:


:<math>t_c = \frac{2 \pi - \theta_t + \theta_s}{2 \pi \bar N}</math>
:<math>t_{\rm c} = \frac{2 \pi - \theta_{\rm t} + \theta_{\rm s}}{2 \pi \bar N}</math>


where the mean rotational rate, <math>\bar N</math> (rev/s), is:
where the mean rotational rate, <math>\bar N</math> (rev/s), is:


:<math>\bar N \approx \frac{N_m}{2}</math>
:<math>\bar N \approx \frac{N_{\rm m}}{2}</math>  


The time taken to travel between the shoulder and toe of the charge in free flight during one revolution, <math>t_f</math> (s), is:
and <math>N_{\rm m}</math> is the rotational rate of the mill shell (rev/s).


:<math>t_f \approx \left ( \frac{2 \bar r \left (\sin \theta_s - \sin \theta_t \right )}{g} \right )^{0.5}</math>
The time taken to travel between the shoulder and toe of the charge in free flight during one revolution, <math>t_{\rm f}</math> (s), is:
 
:<math>t_{\rm f} \approx \left ( \frac{2 \bar r \left (\sin \theta_{\rm s} - \sin \theta_{\rm t} \right )}{g} \right )^{0.5}</math>


where the mean radial position <math>\bar r</math> (m), is:
where the mean radial position <math>\bar r</math> (m), is:
:<math>\bar r = \frac{r_m}{2} \left [ 1 + \left ( 1 - \frac{2 \pi J_t}{2 \pi + \theta_s - \theta_t} \right )^{0.5} \right ]</math>
 
:<math>\bar r = \frac{r_{\rm m}}{2} \left [ 1 + \left ( 1 - \frac{2 \pi J_{\rm t}}{2 \pi + \theta_{\rm s} - \theta_{\rm t}} \right )^{0.5} \right ]</math>


==== Slurry hold-up below pool level ====
==== Slurry hold-up below pool level ====
Line 78: Line 84:
The volume of slurry held up below the pool level in areas B and C, <math>V_1</math> (m<sup>3</sup>), is:
The volume of slurry held up below the pool level in areas B and C, <math>V_1</math> (m<sup>3</sup>), is:


:<math>V_1 = L_m (S_{pool} - S_{net\,ball})</math>
:<math>V_1 = L_{\rm m} (S_{\rm pool} - S_{\rm net\,ball})</math>


where:
where:
* <math>L_m</math> is effective mill length (m)
* <math>L_{\rm m}</math> is effective mill length (m)
* <math>S_{pool}</math> is cross-sectional area of the slurry pool (m<sup>2</sup>)
* <math>S_{\rm pool}</math> is cross-sectional area of the slurry pool (m<sup>2</sup>)
* <math>S_{net\,ball}</math> is cross-sectional area occupied by the grinding media, excluding balls, below the slurry pool level(m<sup>2</sup>)
* <math>S_{\rm net\,ball}</math> is cross-sectional area occupied by the grinding media, excluding balls, below the slurry pool level(m<sup>2</sup>)


This implementation of the Shi model computes <math>L_m</math> as an ''effective'' mill length from:
This implementation of the Shi model computes <math>L_{\rm m}</math> as an ''effective'' mill length from:


:<math>L_m = \dfrac{V_m}{\pi {R_m}^2}</math>
:<math>L_{\rm m} = \dfrac{V_{\rm m}}{\pi {R_{\rm m}}^2}</math>


where <math>V_m</math> is the volume of the mill (m<sup>3</sup>) and <math>R_m</math> is the radius of the mill (m).
where <math>V_{\rm m}</math> is the volume of the mill (m<sup>3</sup>) and <math>R_{\rm m}</math> is the radius of the mill (m).


The volume of the mill is calculated as the sum of a cylinder and two right circular frustums:{{Gupta and Yan (2016)}}
The volume of the mill is calculated as the sum of a cylinder and two right circular frustums:{{Gupta and Yan (2016)}}


:<math>V_m = \pi {R_m}^2L + 2 \cdot \bigg[ \dfrac{\pi}{3} (R_m - R_t) \cdot \tan \alpha_c \cdot \left ( {R_{m}}^{2} + R_{m} R_{t} + {R_{t}}^{2} \right) \bigg]</math>
:<math>V_{\rm m} = \pi {R_{\rm m}}^2L + 2 \cdot \bigg[ \dfrac{\pi}{3} (R_{\rm m} - R_{\rm t}) \cdot \tan \alpha_{\rm c} \cdot \left ( {R_{\rm m}}^{2} + R_{\rm m} R_{\rm t} + {R_{\rm t}}^{2} \right) \bigg]</math>


wher:
where:
* <math>L</math> is the length of the cylindrical (belly) section of the mill (m)
* <math>L</math> is the length of the cylindrical (belly) section of the mill (m)
* <math>R_t</math> is the radius of the discharge trunnion (m)
* <math>R_{\rm t}</math> is the radius of the discharge trunnion (m)
* <math>\alpha_c</math> is the cone angle, measured as the angular displacement of the cone surface from the vertical direction (rad)
* <math>\alpha_{\rm c}</math> is the cone angle, measured as the angular displacement of the cone surface from the vertical direction (rad)


<math>S_{pool}</math> is computed from:
<math>S_{\rm pool}</math> is computed from:


:<math>S_{pool} = 0.5 {R_m}^2 (\theta_p - \sin \theta_p)</math>
:<math>S_{\rm pool} = 0.5 {R_{\rm m}}^2 (\theta_{\rm p} - \sin \theta_{\rm p})</math>


:<math>\theta_p = \pi - 2 \alpha</math>
:<math>\theta_{\rm p} = \pi - 2 \alpha</math>


:<math>\alpha = \arcsin \left (\dfrac{R_t - \Delta h}{R_m} \right )</math>
:<math>\alpha = \arcsin \left (\dfrac{R_{\rm t} - \Delta h}{R_{\rm m}} \right )</math>


where <math>R_t</math> is the radius of the discharge trunnion (m), and <math>\Delta h</math> is the height of the slurry pool level over the trunnion lip (m).
where <math>R_{\rm t}</math> is the radius of the discharge trunnion (m), and <math>\Delta h</math> is the height of the slurry pool level over the trunnion lip (m).


The value of <math>\Delta h</math> may be estimated  as <math>0.67 h</math>, where <math>h</math> is the slurry height above the trunnion level inside the mill (m):{{Morrell (2016)}}
The value of <math>\Delta h</math> may be estimated  as <math>0.67 h</math>, where <math>h</math> is the slurry height above the trunnion level inside the mill (m):{{Morrell (2016)}}


:<math>h = k_{\mu} \cdot \frac{1}{0.67} \left ( \frac{2Q}{\pi R_t g^{0.5}} \right )^{\frac{2}{3}}</math>
:<math>h = k_{\mu} \cdot \frac{1}{0.67} \left ( \frac{2Q}{\pi R_{\rm t} g^{0.5}} \right )^{\frac{2}{3}}</math>


and:
and:
Line 119: Line 125:
* <math>g</math> is acceleration due to gravity (m/s<sup>2</sup>)
* <math>g</math> is acceleration due to gravity (m/s<sup>2</sup>)


<math>S_{net\,ball}</math> is computed as:
<math>S_{\rm net\,ball}</math> is computed as:


:<math>S_{net\,ball} = 0.5 \theta_b (1 - \epsilon_d)({R_m}^2 - {R_t}^2)</math>
:<math>S_{\rm net\,ball} = 0.5 \theta_{\rm b} (1 - \varepsilon_{\rm d})({R_{\rm m}}^2 - {R_{\rm t}}^2)</math>


:<math>\theta_b = 2 \pi - \alpha - \theta_t</math>
:<math>\theta_{\rm b} = 2 \pi - \alpha - \theta_{\rm t}</math>


where the ''dynamic charge porosity'', <math>\epsilon_d</math> (v/v), is:
where the ''dynamic charge porosity'', <math>\varepsilon_{\rm d}</math> (v/v), is:


:<math>\epsilon_d = 0.4 + 0.228 \left [ 1 - \exp \left (-0.315\dfrac{\phi}{J_t} \right) \right ]</math>
:<math>\varepsilon_{\rm d} = 0.4 + 0.228 \left [ 1 - \exp \left (-0.315\dfrac{\phi}{J_{\rm t}} \right) \right ]</math>


==== Slurry hold up above pool level ====
==== Slurry hold up above pool level ====
Line 133: Line 139:
The volume of slurry held up in the grinding media interstices above the slurry pool level in area A, <math>V_2</math> (m<sup>3</sup>), is:
The volume of slurry held up in the grinding media interstices above the slurry pool level in area A, <math>V_2</math> (m<sup>3</sup>), is:


:<math>V_2 = L_m . \epsilon_d . S_{ball\,above\,pool}</math>
:<math>V_2 = L_{\rm m} . \varepsilon_{\rm d} . S_{\rm ball\,above\,pool}</math>


where:
where:


:<math>S_{ball\,above\,pool} = 0.5 (\theta_s + \alpha)({R_m}^2 - {R_t}^2)</math>
:<math>S_{\rm ball\,above\,pool} = 0.5 (\theta_{\rm s} + \alpha)({R_{\rm m}}^2 - {R_{\rm t}}^2)</math>


==== Residence time and overfilling ====
==== Residence time and overfilling ====


The total volume of slurry hold up in the mill is <math>V_{total}=V_1+V_2</math> (m<sup>3</sup>).
The total volume of slurry hold up in the mill is <math>V_{\rm total}=V_1+V_2</math> (m<sup>3</sup>).


The residence time of slurry in the mill, <math>t_{Res}</math> (s), is then:
The residence time of slurry in the mill, <math>t_{\rm Res}</math> (s), is then:


:<math>t_{Res} = \dfrac{V_{total}}{Q}</math>
:<math>t_{\rm Res} = \dfrac{V_{\rm total}}{Q}</math>


The ''axial velocity'' of slurry through the mill, <math>v_{ax}</math> (m/s), is:
The ''axial velocity'' of slurry through the mill, <math>v_{\rm ax}</math> (m/s), is:


:<math>v_{ax} = \dfrac{L_m}{t_{Res}}</math>
:<math>v_{\rm ax} = \dfrac{L_{\rm m}}{t_{\rm Res}}</math>


Shi suggests the following residence time limits for overflow discharge ball mills (s):
Shi suggests the following residence time limits for overflow discharge ball mills (s):


:<math>
:<math>
t_{Limit} =  
t_{\rm Limit} =  
\begin{cases}
\begin{cases}
120\text{ s} & D_m<5.9\text{ m}\\
120\text{ s} & D_{\rm m}<5.9\text{ m}\\
60\text{ s} & D_m\geq5.9\text{ m}\\
60\text{ s} & D_{\rm m}\geq5.9\text{ m}\\
\end{cases}
\end{cases}
</math>
</math>


where <math>D_m</math> is the diameter of the mill (m), i.e. <math>D_m = 2 R_m</math>.
where <math>D_{\rm m}</math> is the diameter of the mill (m), i.e. <math>D_{\rm m} = 2 R_{\rm m}</math>.


The maximum volumetric flow rate of the mill, <math>Q_{Max}</math> (m<sup>3</sup>/s), at the residence time limit <math>t_{Limit}</math> may be back-calculated using the above relations. However, as the height of slurry above the trunnion lip (<math>h</math>), and hence residence time (<math>t_{Res}</math>), is a function of flow rate, no analytical solution is available and a numerical method is required for computation.
The maximum volumetric flow rate of the mill, <math>Q_{\rm Max}</math> (m<sup>3</sup>/s), at the residence time limit <math>t_{\rm Limit}</math> may be back-calculated using the above relations. However, as the height of slurry above the trunnion lip (<math>h</math>), and hence residence time (<math>t_{\rm Res}</math>), is a function of flow rate, no analytical solution is available and a numerical method is required for computation.


=== Arbiter (1990) ===
=== Arbiter method ===


Arbiter describes two approaches for identifying the overfilling of an overflow discharge ball mill:
Based on an analysis of industrial ball mills at five operations, Arbiter (1991) postulated that the axial velocity of pulp through a mill should be less than 2.2% of the mill's tangential velocity. The ''Arbiter Flow Number'', <math>N_{\rm q}</math>, is defined as the ratio of axial to tangential velocity and is determined from:
#. the ''critical flow number'' method
#. the ''axial flow velocity'' method


Arbiter's limits are rearranged to yield the following relations for the maximum volumetric discharge rate of slurry from a mill:
:<math>N_{\rm q} = \dfrac{\left (\dfrac{Q}{\frac{\pi {D_{\rm m}}^2}{4} (0.5 - 0.66 J_{\rm t})}\right )}{\pi N_{\rm m} D_{\rm m}}, \quad N_{\rm q} < 0.0217</math>


:<math>Q_{Max}^{N_q} = 0.0217 \dfrac{\pi^2}{4} {D_m}^3 N_{eq} (0.5 - 0.66 J_t) \cdot 60</math>
where:
* <math>Q</math> is the volumetric discharge rate of pulp from the mill (m3/s)
* <math>D_{\rm m}</math> is the mill diameter (m)
* <math>J_{\rm t}</math> is the volumetric charge fraction in the mill (v/v)
* <math>N_{\rm m}</math> is mill rotational speed (rev/min)


:<math>Q_{Max}^{v_{ax}} = 0.0769 \dfrac{\pi}{4} {D_m}^{2} (0.5 - 0.66 J_t) \cdot 3600</math>
The Arbiter Flow Number equation may be rearranged to yield the following relation for the maximum volumetric discharge rate of slurry from a mill prior to overloading, <math>Q_{N_{\rm q}}</math> (m<sup>3</sup>/h):


where:
:<math>Q_{N_{\rm q}} = 0.0217 \dfrac{\pi^2}{4} {D_{\rm m}}^3 N_{\rm m} (0.5 - 0.66 J_{\rm t}) \cdot 3600</math>
* <math>Q_{Max}^{N_q}</math> is the critical flow number volumetric slurry discharge rate limit (m<sup>3</sup>/h)
* <math>Q_{Max}^{v_{ax}}</math> is the axial velocity volumetric slurry discharge rate limit (m/s)
* <math>D_m</math> is mill diameter (m)
* <math>N_{eq}</math> is mill rotational speed (rev/min)
* <math>J_t</math> is the volumetric charge fraction in the mill (v/v)


Overfilling may be a risk if the actual volumetric flow rate from the mill approaches or exceeds either of <math>Q_{Max}^{N_q}</math> or <math>Q_{Max}^{v_{ax}}</math>.
Overfilling may be a risk if the actual volumetric flow rate to/from the mill approaches or exceeds <math>Q_{N_{\rm q}}</math>.


== Additional notes ==
== Additional notes ==
Line 196: Line 199:


Arbiter's relations were developed from a limited database of five mills, the largest of which was 5.3 m diameter x 6.4 m length.
Arbiter's relations were developed from a limited database of five mills, the largest of which was 5.3 m diameter x 6.4 m length.
</hide>


== Excel ==
== Excel ==


=== Shi ===
=== Shi method ===


The Shi overflow discharge ball mill overfilling model may be invoked from the Excel formula bar with the following function call:
The Shi overflow discharge ball mill overfilling model may be invoked from the Excel formula bar with the following function call:
Line 214: Line 218:
:<math>Parameters=
:<math>Parameters=
\begin{bmatrix}
\begin{bmatrix}
D_m\text{ (m)}\\
D_{\rm m}\text{ (m)}\\
L\text{ (m)}\\
L\text{ (m)}\\
R_t\text{ (m)}\\
R_{\rm t}\text{ (m)}\\
\alpha_{c}\text{ (degrees)}\\
\alpha_{c}\text{ (degrees)}\\
\phi\text{ (frac)}\\
\phi\text{ (frac)}\\
J_t\text{ (v/v)}\\
J_{\rm t}\text{ (v/v)}\\
Q\text{ (m}^{3}\text{/h)}\\
Q\text{ (m}^{3}\text{/h)}\\
k_{\mu}\\
k_{\mu}\\
Line 226: Line 230:
mdMillOverfilling\_Shi=
mdMillOverfilling\_Shi=
\begin{bmatrix}
\begin{bmatrix}
v_{ax}\text{ (m/s)}\\
v_{\rm ax}\text{ (m/s)}\\
t_{Res}\text{ (s)}\\
t_{\rm Res}\text{ (s)}\\
t_{Limit}\text{ (s)}\\
t_{\rm Limit}\text{ (s)}\\
Q_{Max}\text{ (m}^{3}\text{/h)}\\
Q_{\rm Max}\text{ (m}^{3}\text{/h)}\\
V_{1}\text{ (m}^{3}\text{)}\\
V_{1}\text{ (m}^{3}\text{)}\\
V_{2}\text{ (m}^{3}\text{)}\\
V_{2}\text{ (m}^{3}\text{)}\\
V_{total}\text{ (m}^{3}\text{)}\\
V_{\rm total}\text{ (m}^{3}\text{)}\\
\end{bmatrix}\;\;\;\;\;\;
\end{bmatrix}\;\;\;\;\;\;
</math>
</math>
Line 239: Line 243:
|}
|}


=== Arbiter ===
=== Arbiter method ===


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


<syntaxhighlight lang="vb">=mdMillOverfilling_Arbiter(Parameters as Range)(Parameters as Range)</syntaxhighlight>
<syntaxhighlight lang="vb">=mdMillOverfilling_Arbiter(Parameters as Range)</syntaxhighlight>


{{Excel (Text, Help, No Arguments)}}
{{Excel (Text, Help, No Arguments)}}
Line 254: Line 258:
:<math>Parameters=
:<math>Parameters=
\begin{bmatrix}
\begin{bmatrix}
D_m\text{ (m)}\\
D_{\rm m}\text{ (m)}\\
\phi\text{ (frac)}\\
\phi\text{ (frac)}\\
J_t\text{ (v/v)}\\
J_{\rm t}\text{ (v/v)}\\
\end{bmatrix},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;


mdMillOverfilling\_Arbiter=
mdMillOverfilling\_Arbiter=
\begin{bmatrix}
\begin{bmatrix}
Q_{Max}^{N_q}\text{ (m}^{3}\text{/h)}\\
Q_{N_{\rm q}}\text{ (m}^{3}\text{/h)}\\
Q_{Max}^{v_{ax}}\text{ (m}^{3}\text{/h)}\\
\end{bmatrix}\;\;\;\;\;\;
\end{bmatrix}\;\;\;\;\;\;
</math>
</math>
Line 273: Line 276:
The Ball Mill Overfilling Indicator model is an optional calculation for tumbling mill units. If selected, the input and display parameters below are shown.
The Ball Mill Overfilling Indicator model is an optional calculation for tumbling mill units. If selected, the input and display parameters below are shown.


{{SysCAD_Table_Header}}
{{SysCAD (Text, Table Header)}}
 
|-
! colspan="3" style="text-align:left;" |''Overfilling''
 
{{SysCAD (Text, Help Link)}}
 
|-
|-
! colspan="3" style="text-align:left;" |''Shi''
! colspan="3" style="text-align:left;" |''Shi''
Line 362: Line 371:
|style="background: #eaecf0" | Display
|style="background: #eaecf0" | Display
|Estimated volumetric flow rate of slurry (solids + liquids) at Arbiters critical flow number limit.
|Estimated volumetric flow rate of slurry (solids + liquids) at Arbiters critical flow number limit.
|-
|Feed.SLQv.VmaxLimit
|style="background: #eaecf0" | Display
|Estimated volumetric flow rate of slurry (solids + liquids) at Arbiters axial velocity limit.
|}
|}


Latest revision as of 11:21, 4 December 2024

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.[1][2]

Model theory

Nuvola apps important blue.svg.png This content is available to registered users. Please log in to view.

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:

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:

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 ButtonModelHelp.png 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

References

  1. ↑ Shi, F., 2016. An overfilling indicator for wet overflow ball mills. Minerals Engineering, 95, pp.146-154.
  2. ↑ Arbiter, N., 1991. Dimensionality in ball mill dynamics. Mining, Metallurgy & Exploration, 8(2), pp.77-81.
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