Tumbling Mill (Power, Hilden and Powell): Difference between revisions
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== Model theory == | == Model theory == | ||
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Hilden and Powell's approach adopts Morrell's (1996) expression for tumbling mill power draw:{{Morrell (1996a)}} | Hilden and Powell's approach adopts Morrell's (1996) expression for tumbling mill power draw:{{Morrell (1996a)}} | ||
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Using Morrell's (1996) equations, the power drawn by the '''charge component''' in the cylindrical and cone end sections of the mill, <math>P_{\rm L}</math> and <math>P_{\rm C}</math>, are:{{Morrell (1996a)}} | Using Morrell's (1996) equations, the power drawn by the '''charge component''' in the cylindrical and cone end sections of the mill, <math>P_{\rm L}</math> and <math>P_{\rm C}</math>, are:{{Morrell (1996a)}} | ||
:<math>P_{\rm L} = \frac{\pi g L \rho_{\rm c} N_{\rm m} R}{3(R - | :<math>P_{\rm L} = \frac{\pi g L \rho_{\rm c} N_{\rm m} R}{3(R - zr_{\rm i})} \left [ 2R^3 -3zR^2r_{\rm i} + r_{\rm i}^3(3z-2) \right ].(\sin \theta_{\rm S} - \sin \theta_{\rm T}) + L \rho_{\rm c} \left ( \frac{\pi N_{\rm m} R}{R - zr_{\rm i}} \right )^3 \left [ (R-zr_{\rm i})^4 - r_{\rm i}^4(z-1)^4 \right ] </math> | ||
:<math>P_{\rm C} = \frac{\pi g L_{\rm d} \rho_{\rm c} N_{\rm m}}{3(R-r_{\rm T})}(R^4 - | :<math>P_{\rm C} = \frac{\pi g L_{\rm d} \rho_{\rm c} N_{\rm m}}{3(R-r_{\rm T})}(R^4 - 4Rr_{\rm i}^3 + 3r_{\rm i}^4)(\sin \theta_{\rm S} - \sin \theta_{\rm T}) + \frac{2 \pi^3 N_{\rm m}^3 L_{\rm d} \rho_{\rm c}}{5(R-r_{\rm T})}(R^5 - 5Rr_{\rm i}^4 + 4r_{\rm i}^5)</math> | ||
where: | where: | ||
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* <math>R</math> is the mill radius inside liners (m), which is half the mill diameter <math>D</math>, i.e. <math>R=0.5D</math> | * <math>R</math> is the mill radius inside liners (m), which is half the mill diameter <math>D</math>, i.e. <math>R=0.5D</math> | ||
* <math>r_{\rm T}</math> is the trunnion radius inside liners (m), which is half the trunnion diameter <math>d_{\rm T}</math>, i.e. <math>r_{\rm T}=0.5d_{\rm T}</math> | * <math>r_{\rm T}</math> is the trunnion radius inside liners (m), which is half the trunnion diameter <math>d_{\rm T}</math>, i.e. <math>r_{\rm T}=0.5d_{\rm T}</math> | ||
* <math> | * <math>r_{\rm i}</math> is the inner surface radius of the charge (m) | ||
* <math>\theta_{\rm S}</math> is the angle of the charge shoulder (radians) | * <math>\theta_{\rm S}</math> is the angle of the charge shoulder (radians) | ||
* <math>\theta_{\rm T}</math> is the angle of the charge toe (radians) | * <math>\theta_{\rm T}</math> is the angle of the charge toe (radians) | ||
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where <math>J_{\rm T}</math> is the total mill filling of balls plus rocks (v/v). | where <math>J_{\rm T}</math> is the total mill filling of balls plus rocks (v/v). | ||
The power drawn by the '''slurry component''' in the cylindrical and cone end sections, <math>P'_{\rm L}</math> and <math>P'_{\rm C}</math>, is computed by substituting the slurry component properties <math>\rho'_c</math>, <math>r' | The power drawn by the '''slurry component''' in the cylindrical and cone end sections, <math>P'_{\rm L}</math> and <math>P'_{\rm C}</math>, is computed by substituting the slurry component properties <math>\rho'_c</math>, <math>r'_{\rm i}</math>, <math>z'</math>, and <math>\theta_{\rm L}</math> for the equivalent charge properties <math>\rho_{\rm c}</math>, <math>r_{\rm i}</math>, <math>z</math>, and <math>\theta_{\rm T}</math> in the same equations for <math>P_{\rm L}</math> and <math>P_{\rm C}</math>, where: | ||
* <math>\rho'_c</math> is the density of the slurry component in the mill (t/m<sup>3</sup>) | * <math>\rho'_c</math> is the density of the slurry component in the mill (t/m<sup>3</sup>) | ||
* <math>r' | * <math>r'_{\rm i}</math> is the inner surface radius of the slurry component (m) | ||
* <math>z'</math> is an empirical term that relates the rotational rate of mass at radial positions within the slurry component to the rotational rate of the mill | * <math>z'</math> is an empirical term that relates the rotational rate of mass at radial positions within the slurry component to the rotational rate of the mill | ||
* <math>\theta_{\rm L}</math> is the angle of the toe of the slurry component (radians) | * <math>\theta_{\rm L}</math> is the angle of the toe of the slurry component (radians) | ||
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:<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> | :<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 surface '''radius of the charge''', <math> | The inner surface '''radius of the charge''', <math>r_{\rm i}</math> (m) is: | ||
:<math> | :<math>r_{\rm i} = R \sqrt{1 - \frac{2 \pi \beta J_{\rm T}}{2 \pi + \theta_{\rm S} - \theta_{\rm T}}}</math> | ||
where the active load fraction, <math>\beta</math> (frac), is: | where the active load fraction, <math>\beta</math> (frac), is: | ||
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where <math>y</math> is the profile shape factor (frac). | where <math>y</math> is the profile shape factor (frac). | ||
The inner surface '''radius of the slurry component''', <math>r' | The inner surface '''radius of the slurry component''', <math>r'_{\rm i}</math> (m), is: | ||
:<math>r' | :<math>r'_{\rm i} = \sqrt{R^2 - \frac{U(R^2-r_{\rm i}^2).(2\pi+\theta_{\rm S}-\theta_{\rm T})}{2\pi+\theta_{\rm S}-\theta_{\rm L}}}</math> | ||
==== Slurry toe when U>1 ==== | ==== Slurry toe when U>1 ==== | ||
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[[File:TumblingMillPower9.png|thumb|450px|Figure 3. Tumbling mill profile showing Hilden and Powell's assumed charge and slurry component shapes when <math>U>1</math>, along with key dimensions and areas.]] | [[File:TumblingMillPower9.png|thumb|450px|Figure 3. Tumbling mill profile showing Hilden and Powell's assumed charge and slurry component shapes when <math>U>1</math>, along with key dimensions and areas.]] | ||
[[File:TumblingMillPower10.png|thumb|450px|Figure 4. Tumbling mill profile showing Hilden and Powell's assumed charge and slurry component shapes when <math>U>1</math>, <math>h< | [[File:TumblingMillPower10.png|thumb|450px|Figure 4. Tumbling mill profile showing Hilden and Powell's assumed charge and slurry component shapes when <math>U>1</math>, <math>h<r_{\rm i}</math> and <math>h_2<r_{\rm i}</math>, along with key dimensions and areas.]] | ||
When the charge void space is overfilled (<math>U>1</math>), a slurry pool forms. | When the charge void space is overfilled (<math>U>1</math>), a slurry pool forms. | ||
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Only the slurry in the annular sector (<math>A_{\rm Sect}</math>) contributes to power draw. | Only the slurry in the annular sector (<math>A_{\rm Sect}</math>) contributes to power draw. | ||
The inner surface '''radius of the slurry component''' filling the charge void space and extending into the annular sector is equal to the charge radius, i.e. <math>r' | The inner surface '''radius of the slurry component''' filling the charge void space and extending into the annular sector is equal to the charge radius, i.e. <math>r'_{\rm i} = r_{\rm i}</math>. | ||
Locating the position of the '''toe of the slurry component''', <math>\theta_{\rm L}</math>, then becomes a problem of determining the distribution of slurry between the two areas, <math>A_{\rm Seg}</math> and <math>A_{\rm Sect}</math>. | Locating the position of the '''toe of the slurry component''', <math>\theta_{\rm L}</math>, then becomes a problem of determining the distribution of slurry between the two areas, <math>A_{\rm Seg}</math> and <math>A_{\rm Sect}</math>. | ||
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The total area of excess slurry, <math>A_x</math> (m<sup>2</sup>), in the pool is: | The total area of excess slurry, <math>A_x</math> (m<sup>2</sup>), in the pool is: | ||
:<math>A_x = A_{\rm Seg} + A_{\rm Sect} = (U - 1) . \frac{1}{2} (R^2 - | :<math>A_x = A_{\rm Seg} + A_{\rm Sect} = (U - 1) . \frac{1}{2} (R^2 - r_{\rm i}^2).(2 \pi + \theta_{\rm S} - \theta_{\rm T})</math> | ||
The area of the annular segment, <math>A_{\rm Seg}</math>, is: | The area of the annular segment, <math>A_{\rm Seg}</math>, is: | ||
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:<math>A_{\rm Seg} = | :<math>A_{\rm Seg} = | ||
\begin{cases} | \begin{cases} | ||
r_{\rm i}^2 . \arccos \left ( \frac{h}{r_{\rm i}} \right ) - h . \sqrt{r_{\rm i}^2 - h^2} & \text{if }h<r_{\rm i}\\ | |||
\left ( | \left ( r_{\rm i}^2.\arccos \left ( \frac{h}{r_{\rm i}} \right ) - h.\sqrt{r_{\rm i}^2 - h^2} \right ) - \left ( r_{\rm i}^2.\arccos \left ( \frac{h_2}{r_{\rm i}} \right ) - h_2.\sqrt{r_{\rm i}^2 - h_2^2} \right ) & \text{if }h<r_{\rm i} \text{ and }h_2<r_{\rm i} | ||
\end{cases} | \end{cases} | ||
</math> | </math> | ||
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where the parameters <math>h</math> and <math>h_2</math>, as shown in Figure 4, are: | where the parameters <math>h</math> and <math>h_2</math>, as shown in Figure 4, are: | ||
:<math>h = - \left ( \frac{R + | :<math>h = - \left ( \frac{R + r_{\rm i}}{2} \right ) . \sin \theta_{\rm L}</math> | ||
:<math>h_2 = - \left ( \frac{R + | :<math>h_2 = - \left ( \frac{R + r_{\rm i}}{2} \right ) . \sin \theta_{\rm T}</math> | ||
The area of the circular sector, <math>A_{\rm Sect}</math>, is: | The area of the circular sector, <math>A_{\rm Sect}</math>, is: | ||
:<math>A_{\rm Sect} = \frac{(\theta_{\rm T} - \theta_{\rm L}}{2} . (R^2 - | :<math>A_{\rm Sect} = \frac{(\theta_{\rm T} - \theta_{\rm L}}{2} . (R^2 - r_{\rm i}^2)</math> | ||
The slurry toe parameter <math>\theta_{\rm L}</math> cannot be algebraically isolated in the above equations. Therefore, an iterative numerical method is used to solve the value of <math>\theta_{\rm L}</math> that yields the required value of <math>A_x</math> | The slurry toe parameter <math>\theta_{\rm L}</math> cannot be algebraically isolated in the above equations. Therefore, an iterative numerical method is used to solve the value of <math>\theta_{\rm L}</math> that yields the required value of <math>A_x</math>. | ||
=== Charge density === | === Charge density === | ||
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| 6.1 | | 6.1 | ||
|} | |} | ||
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== Excel == | == Excel == | ||
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\theta_{\rm T} \text{ (rad)}\\ | \theta_{\rm T} \text{ (rad)}\\ | ||
\theta_{L} \text{ (rad)}\\ | \theta_{L} \text{ (rad)}\\ | ||
r_{\rm i}\text{ (m)}\\ | |||
r' | r'_{\rm i}\text{ (m)}\\ | ||
\end{bmatrix}\;\;\;\;\;\; | \end{bmatrix}\;\;\;\;\;\; | ||
</math> | </math> | ||
Latest revision as of 11:16, 4 December 2024
Description
This article describes the Hilden and Powell (2018) approach for estimating the power draw of a tumbling mill.[1]
Hilden and Powell extended Morrell's (1996, 2016) approaches to include the following features:[2][3]
- Partial filling of the charge void space with slurry in two directions, from the charge shoulder towards the toe and from the outer shell towards the mill centre
- Estimation of the fraction of a slurry pool which settles across the centre of the mill profile and thus does not contribute to power draw
- No-load power based on mill drive type and size
Model theory
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Excel
The Hilden and Powell mill power model may be invoked from the Excel formula bar with the following function call:
=mdMillPower_HildenPowell(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:
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 Figure 5. Example showing the selection of the Parameters (blue frame), and Results (light blue frame) arrays in Excel. |
SysCAD
The Hilden and Powell 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 |
|---|---|---|
| HildenPowell | ||
| HelpLink | 
|
Opens a link to this page using the system default web browser. Note: Internet access is required. |
| MillDiameter | Display | Diameter of the mill (inside liners). |
| BellyLength | Display | Length of the cylindrical section (belly) of the mill (inside liners). |
| TrunnionDiameter | Display | Diameter of the trunnion (inside liners). |
| ConeAngle | Display | Angular displacement of the cone surface from the vertical direction. |
| Jt | Display | Volumetric fraction of the mill occupied by balls and coarse rock (including voids). |
| BallLoadVol | Display | Volumetric fraction of the mill occupied by balls (including voids). |
| SolidsSG | Display | Specific Gravity or density of solids. |
| BallSG | Display | Specific Gravity or density of balls. |
| LiquidsSG | Display | Specific Gravity or density of liquids. |
| Voidage | Input/Display | Volumetric fraction of interstitial void space in the charge. Usually 0.4. Note: The Dynamic Perfect Mixing ball mill model uses Shi's dynamic charge porosity estimation to align power draw predictions with the adopted slurry filling approach. |
| U | Display | Volumetric fraction of interstitial grinding media voidage occupied by slurry. |
| Cw | Display | Mass fraction of solids in discharge slurry. |
| FracCS | Display | Fraction critical speed of the mill. |
| MillDriveLosses | Input | Mill drive motor power loss factor. |
| MotorInstalledPower | Input | Maximum power draw of installed mill motor. |
| NetPowerAdjust | Input | Lumped calibration parameter accounting for power losses. Found to be a value of 1.26 for industrial mills. |
| y | Input | Profile shape factor. |
| ThetaShoulder | Display | Angle of the charge shoulder. |
| ThetaToe | Display | Angle of the charge toe. |
| ThetaSlurryToe | Display | Angle of the toes of the slurry component |
| ChargeSurfaceRadius | Display | Radius of the inner surface of the charge. |
| SlurrySurfaceRadius | Display | Radius of the inner surface of the slurry component. |
| 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
References
- ↑ Hilden, M.M. and Powell, M.S., 2018. A model of SAG mill power incorporating slurry filling and its application to real-time mill performance analysis. Paper presented at Comminution '18, Cape Town, South Africa, April 16-19, 2018.
- ↑ Morrell, S., 1996. Power draw of wet tumbling mills and its relationship to charge dynamics. Pt. 1: a continuum approach to mathematical modelling of mill power draw. Transactions of the Institution of Mining and Metallurgy. Section C. Mineral Processing and Extractive Metallurgy, 105.
- ↑ Morrell, S., 2016. Modelling the influence on power draw of the slurry phase in Autogenous (AG), Semi-autogenous (SAG) and ball mills. Minerals Engineering, 89, pp.148-156.