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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Sub-components of the gross power equation are described below. | Sub-components of the gross power equation are described below. | ||
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=== Charge motion power === | === Charge motion power === | ||
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[[File:TumblingMillPower8.png|thumb|450px|Figure 1. Tumbling mill profile showing Hilden and Powell's assumed charge and slurry component shapes when <math>U<1</math>, along with key dimensions.]] | [[File:TumblingMillPower8.png|thumb|450px|Figure 1. Tumbling mill profile showing Hilden and Powell's assumed charge and slurry component shapes when <math>U<1</math>, along with key dimensions.]] | ||
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The remaining parameters of the charge motion power equations are described below. | The remaining parameters of the charge motion power equations are described below. | ||
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=== Charge position === | === Charge position === | ||
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The charge shoulder and toe positions and inner surface radius are calculated using Morrell's (1996) method.{{Morrell (1996a)}} | The charge shoulder and toe positions and inner surface radius are calculated using Morrell's (1996) method.{{Morrell (1996a)}} | ||
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:<math>\beta = \frac{\left ( \dfrac{2 \pi + \theta_{\rm S} - \theta_{\rm T}}{\pi N_{\rm m}} \right )}{\sqrt{\dfrac{R}{g} \left ( 1 + \sqrt{1 - \dfrac{2 \pi J_{\rm T}}{2 \pi + \theta_{\rm S} - \theta_{\rm T}} } \right ) (\sin \theta_{\rm S} - \sin \theta_{\rm T})} + \left( \dfrac{2 \pi + \theta_{\rm S} - \theta_{\rm T}}{\pi N_{\rm m}} \right )}</math> | :<math>\beta = \frac{\left ( \dfrac{2 \pi + \theta_{\rm S} - \theta_{\rm T}}{\pi N_{\rm m}} \right )}{\sqrt{\dfrac{R}{g} \left ( 1 + \sqrt{1 - \dfrac{2 \pi J_{\rm T}}{2 \pi + \theta_{\rm S} - \theta_{\rm T}} } \right ) (\sin \theta_{\rm S} - \sin \theta_{\rm T})} + \left( \dfrac{2 \pi + \theta_{\rm S} - \theta_{\rm T}}{\pi N_{\rm m}} \right )}</math> | ||
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=== Slurry position === | === Slurry position === | ||
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The position of the '''shoulder of the slurry component''' is the same as the charge shoulder position. | The position of the '''shoulder of the slurry component''' is the same as the charge shoulder position. | ||
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* if <math>U>1</math> then there is an excess of slurry beyond the available charge void space, which manifests as a slurry pool, and <math>\theta_{\rm L} < \theta_{\rm T}</math>. | * if <math>U>1</math> then there is an excess of slurry beyond the available charge void space, which manifests as a slurry pool, and <math>\theta_{\rm L} < \theta_{\rm T}</math>. | ||
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==== Slurry toe when U≤1 ==== | ==== Slurry toe when U≤1 ==== | ||
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When the charge void space is partially or exactly filled with slurry (<math>U\leq1</math>), the position of the '''toe of the slurry component''', <math>\theta_{\rm L}</math> (rad), is: | When the charge void space is partially or exactly filled with slurry (<math>U\leq1</math>), the position of the '''toe of the slurry component''', <math>\theta_{\rm L}</math> (rad), is: | ||
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:<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> | :<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> | ||
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==== 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.]] | ||
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:<math>A_{\rm Sect} = \frac{(\theta_{\rm T} - \theta_{\rm L}}{2} . (R^2 - r_{\rm i}^2)</math> | :<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>. | ||
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=== Charge density === | === Charge density === | ||
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The masses of balls, <math>m_{\rm B}</math> (t), and coarse rocks, <math>m_{\rm R}</math> (t), in the charge are: | The masses of balls, <math>m_{\rm B}</math> (t), and coarse rocks, <math>m_{\rm R}</math> (t), in the charge are: | ||
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where: | where: | ||
* <math>J_{\rm B}</math> is the ball filling (v/v) | * <math>J_{\rm B}</math> is the ball filling (v/v) | ||
* <math>V_{\rm C}</math> is the volume | * <math>V_{\rm C}</math> is the volume of the cylindrical section of the mill (m<sup>3</sup>) | ||
* <math>\varepsilon</math> is the charge void fraction, typically 0.4 (v/v) | * <math>\varepsilon</math> is the charge void fraction, typically 0.4 (v/v) | ||
* <math>\rho_{\rm B}</math> is the Specific Gravity or density of the balls (- or t/m<sup>3</sup>) | * <math>\rho_{\rm B}</math> is the Specific Gravity or density of the balls (- or t/m<sup>3</sup>) | ||
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:<math>S_{\rm V} = \frac{S}{\rho_{\rm R} \left ( \dfrac{S}{\rho_{\rm R}} + \dfrac{1 - S}{\rho_{\rm L}} \right )}</math> | :<math>S_{\rm V} = \frac{S}{\rho_{\rm R} \left ( \dfrac{S}{\rho_{\rm R}} + \dfrac{1 - S}{\rho_{\rm L}} \right )}</math> | ||
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=== No-load power === | === No-load power === | ||
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The no-load power (<math>P_{\rm N}</math>) is estimated as: | The no-load power (<math>P_{\rm N}</math>) is estimated as: | ||
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| 6.1 | | 6.1 | ||
|} | |} | ||
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== Excel == | == Excel == | ||
Latest revision as of 08:15, 1 May 2025
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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Charge motion power
Charge position
Slurry position
Slurry toe when U≤1
Slurry toe when U>1
Charge density
No-load power
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.