**A DROPPED CHARGE PROTECTION SYSTEM AND A MONITORING SYSTEM**

VAN ZYL, Paul Hendrik Stephanus (14 Partridge Avenue, Park Exttension 2, 1619 Kempton, ZA)

*;*

**B02C17/24***;*

**B02C13/31***;*

**B02C17/18***;*

**B02C23/04**

**B02C25/00**2004-08-05 |

DE102004015057A1 | 2005-10-20 | |||

US20030052205A1 | 2003-03-20 | |||

US20020175232A1 | 2002-11-28 | |||

DE60224561T2 | 2008-04-30 |

Claims 1. A dropped charge protection system, wherein the system includes calculating an angle of repose of a charge of a grinding mill during start-up and tripping the mill motor when the angle of repose of the charge exceeds a maximum allowable angle. 2. A dropped charge protection system as claimed in claim 1 , wherein the system includes plotting the calculated angle of repose relative an angle of rotation of the mill shell. 3. A dropped charge protection system as claimed in any one of the preceding claims, wherein the angle of repose of the charge is determined by solving the non-linear differential equation of T= Ja + mgr sinθ, wherein T is the air-gap torque applied to the motor rotor by the electric field; α is the angular acceleration of the mill around the centre of rotation of the mill shell and is determined from d/dt(ω). ω is the angular speed of the mill shell around the centre of rotation of the mill shell and is determined from d/dt(φ). 4. A dropped charge protection system as claimed in claim 3, wherein J is the moment of inertia [kgm 5. A dropped charge protection system as claimed in any one of claims 3 to 4, wherein θ = φ before the charge has tumbled and wherein it rotates with the mill shell and, and wherein φ is the angular position of the mill shell around the centre of rotation of the mill shell. 6. A dropped charge protection system as claimed in any one of claims 3 to 5, wherein the torque T causes the acceleration of all rotating masses (Ja), and, the pendulum-like raising of the charge (mgrsinθ). 7. A dropped charge protection system as claimed in any one of the preceding claims, wherein the tripping criterion in the equation T= Ja + mgrsinθ is the angle of repose (θ). 8. A dropped charge protection system as claimed in claim 7, wherein solving θ, includes determining the system parameters J and mgr and the system variables T and α measured in real time and/or calculated from measurable quantities in real time. 9. A dropped charge protection system as claimed in any one of claims 3 to 8, wherein the torque (T) is calculated using the formula T= P/ω wherein P is the power of the motor and ω is the angular speed of the motor. 10. A dropped charge protection system as claimed in any one of claims 3 to 9, wherein any one or more of θ and/or α and/or ω is measured through the use of rotary encoders, magnetic pick-ups and the like on the motor shaft or elsewhere in the drive train. 11. A dropped charge protection system as claimed in any one of claims 3 to 10, wherein T and any one or more of φ and/or α and/or ω are calculated from the rotor current of the mill motor in real time, making both the instantaneous measurement of P and the use of rotary encoders, magnetic pick-ups and the like on the motor shaft or elsewhere in the drive train unnecessary, in the case of a wound-rotor motor and if the rotor current is accessible. 12. A dropped charge protection system as claimed in any one of claims 3 to 11 , wherein the torque (T) produced by the wound-rotor motor is directly proportional to the rotor current. 13. A dropped charge protection system as claimed in any one of the preceding claims, wherein the mill motor includes a liquid resistance starter (LRS) in series with the motor rotor windings. 14. A dropped charge protection system as claimed in claim 13, wherein the LRS controls the rotor current and thereby controls the amount of torque produced by the motor as the torque is proportional to the rotor current. 15. A dropped charge protection system as claimed in any one of the preceding claims, wherein a power factor (the ratio of the real power to the apparent power,) in the rotor circuit is close to unity (where unity = 1 ) and the torque is determined by the formula T=(l/l 16. A dropped charge protection system as claimed in any one of claims 3 to 15, wherein α is determined from ω by means of differentiation (d/dt(ω)). 17. A dropped charge protection system as claimed in any one of claims 3 to 16, wherein the mill rotation speed (ω) is determined from a motor speed (n) and the gear ratio. 18. A dropped charge protection system as claimed in claim 17, wherein the motor speed (n) is calculated from the rotor current using the formula f - f n = 19. A dropped charge protection system as claimed in claim 18, wherein the frequency of the rotor current of the motor (f 20. A dropped charge protection system as claimed in any one of claims 3 to 19, wherein the moment of inertia of all rotating mass (J), the mass of the charge (m) and the radius from the centre of the mill's axis of rotation to the centre of gravity of the charge (r) is unknown. 21. A dropped charge protection system as claimed in any one of claims 3 to 20, wherein J and mgr are dependent on r but r is not readily determinable due to the non-homogenous state of the charge. 22. A dropped charge protection system as claimed in any one of claims 3 to 21 , wherein J and mgr are determined dynamically within the first few degrees of mill rotation, before the possibility of a dropped charge exists, so that the system can start calculating θ timeously. 23. A dropped charge protection system as claimed in any one of claims 3 to 22, wherein φ is determined if it is to be used in the calculation of J and mgr. 24. A dropped charge protection system as claimed in any one of claims 3 to 23, wherein, in the period before tumbling it is known that θ = φ and θ is therefore known. 25. A dropped charge protection system as claimed in any one of claims 3 to 24, wherein the mill shell's rotation φ is determined by integration of ω where the integration of ω is the taking the integral of ω with respect to time. 26. A dropped charge protection system as claimed in any one of claims 3 to 25, wherein at a small mill shell rotation of 1 °, φ = θ = 1 ° and sin (1 °) =0.017 and the contribution of mgrsinθ to T = Ja + mgr sinθ is relatively small, resulting in T = Ja + mgr sinθ being simplified to T = Ja and J is therefore be calculated from T the formula J = - . a 27. A dropped charge protection system as claimed in any one of claims 3 to 26, wherein st a relatively bigger mill shell rotation, of φ=10°, the charge has not have yet rotated enough to tumble, but sin (10°) =0.173 and the contribution of mgrsinθ is 10 times bigger in the equation T = Ja + mgr sinθ and is no longer negligible. 28. A dropped charge protection system as claimed in any one of claims 3 to 27, wherein mgr is calculated from the eq πuation mg and the angle of repose are 10°. 29. A dropped charge protection system as claimed in any one of claims 3 to 28, wherein it is possible to calculate θ once J and mgr have been calculated, plot θ relative an angle of rotation of the mill shell (φ) and trip the mill motor when the angle of repose of the charge exceeds a maximum allowable angle. 30. A dropped charge protection system as claimed in any one of claims 3 to 29, wherein tumbling will have occurred when φ is no longer equal to θ, and this is used as a criterion to determine if start-up of the mill has been safe and successful. 31. A dropped charge protection system as claimed in any one of the preceding claims, wherein the dropped charge protection system continues to record the rotor current after tumbling and facilitates evaluation of the rotor current and resultant torque. 32. A control system for controlling the torque applied to starting a grinding mill, wherein the system includes using a pre-determined angle of repose, controlling a real angle of repose of a charge such that the real angle of repose coincides with the pre-determined angle of repose through the manipulation of the torque of the motor and wherein the angle of repose is controlled in such a way as to encourage tumbling of the charge. 33. A control system for controlling the torque applied to starting a grinding mill as claimed in claim 32, wherein the torque is the actuating signal and the angle of repose θ is the controlled signal. 34. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 32 to 33, wherein the angle of repose of the charge is determined by solving the non-linear differential equation of T= Ja + mgr sinθ, wherein T is the air-gap torque applied to the motor rotor by the electric field; α is the angular acceleration of the mill around the centre of rotation of the mill and is determined from d/dt(ω) and ω is the angular speed of the mill shell around the centre of rotation of the mill shell and is determined from d/dt(φ). 35. A control system for controlling the torque applied to starting a grinding mill as claimed in claim 34, wherein J is the moment of inertia [kgm 36. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 35, wherein prior to the tumbling of the charge, the charge rotates with the mill and θ = φ, and wherein φ is the angular position of the mill around the centre of rotation of the mill shell. 37. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 36, wherein the torque T affects the acceleration of all rotating masses (Ja), and, the pendulum-like raising of the charge (mgrsinθ). 38. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 37, wherein the controlled variable in the equation T= Ja + mgrsinθ is the angle of repose (θ). 39. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 38, wherein solving θ, requires the determining of the system parameters J and mgr and the system variables T and α, measured in real time and/or calculated from measurable quantities in real time. 40. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 39, wherein the torque (T) is calculated using the formula T= P/ω wherein P is the power of the motor and ω is the angular speed of the motor. 41. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 40, wherein any one or more of θ and/or α and/or ω are measured through the use of rotary encoders, magnetic pick-ups and the like on the motor shaft or elsewhere in the drive train. 42. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 41 , wherein T and any one or more of φ and/or α and/or ω are calculated from the rotor current of the mill motor in real time. 43. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 42, wherein the torque (T) produced by the wound-rotor motor is directly proportional to the rotor current. 44. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 43, wherein the mill motor includes a liquid resistance starter (LRS) in series with the motor rotor windings. 45. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 44, wherein the LRS controls the rotor current and thereby control the amount of torque produced by the motor as the torque is proportional to the rotor current. 46. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 45, wherein the power factor (the ratio of the real power to the apparent power,) in the rotor circuit may be close to unity (where unity = 1 ) and the torque is therefore determined by the formula T=(l/lrated)T 47. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 46, wherein α is determined from ω by differentiation (d/dt(ω)). 48. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 47, wherein the mill rotation speed (ω) is determined from the motor speed (n) and the gear ratio. 49. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 48, wherein the motor speed (n) is f - f calculated from the rotor current using the formula P wherein f 50. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 49, wherein the frequency of the rotor current of the motor (f 51. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 50, wherein the moment of inertia of all rotating mass (J), the mass of the charge (m) and the radius from the centre of the mill's axis of rotation to the centre of gravity of the charge (r) is unknown. 52. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 51 , wherein J and mgr are dependent on r but r is not be readily determinable due to the non-homogenous state of the charge. 53. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 52, wherein J and mgr are determined dynamically within the first few degrees of mill rotation, before the possibility of a dropped charge exists, so that the system can start calculating θ timeously. 54. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 53, wherein It is to be appreciated from this specification that φ must be determined if it is to be used in the calculation of J and mgr and in the period before tumbling it is known that θ = φ and θ is therefore known. 55. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 54, wherein the mill shell's rotation φ is also determinable by integration of ω where the integration of ω is the taking the integral of ω with respect to time. 56. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 55, wherein at a small mill shell rotation of 1 °, φ = θ = 1 ° and sin (1 °) =0.017 and the contribution of mgrsinθ to T = Ja + mgr sinθ is relatively small resulting in T = Ja + mgr sinθ being simplified to T = T Ja and J may therefore be calculated from the formula J = - . a 57. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 56, wherein at a relatively bigger mill shell rotation, of φ=10°, the charge may not have yet rotated enough to tumble, but sin (10°) =0.173 and the contribution of mgrsinθ is therefore 10 times bigger in the equation T = Ja + mgr sinθ and can no longer be neglected. 58. A control system for controlling the torque applied to starting a grinding mill as claimed in claim 57, wherein mgr is calculated from the equation mgr = as both the mill and the angle of repose are 10°. sin(10°) 59. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 58, wherein the calculation of mgr permits the calculation of the amount of torque (T) necessary to keep φ at an optimum angle for the charge to tumble. 60. A control system for controlling the torque applied to starting a grinding mill as claimed in any one of claims 34 to 59, wherein controlling the liquid resistance starter, permits the rotor current to be controlled, thereby to apply the correct amount of torque to bring φ to this optimum angle. |

Field of the invention

The invention is in the field of systems that are used to monitor and protect mills from damage caused by dropped charges.

Background to the invention

The inventor is aware of the potential damage that may be caused to a mill when a charge becomes solidified or semi-solidified and drops as a solid mass instead of tumbling through the rotation of the drum. The dropped charge (also known as a frozen / baked / locked or cemented charge) consists of the mined ore, water and grinding balls and may cause damage to the drum and/or the drive.

Damage to the drive and/or the drum leads to down time of the mill and production loss.

Electronic systems that protect gearless mill drives (GMD) from dropped charges are known. GMD are however significantly more expensive than geared mills. The potential damage to a geared drive by a dropped charge may be a contributing factor for mines opting for a GMD despite the high capital outlay.

Moreover, mechanical systems that prevent dropped charges in geared mills are known. These are however relatively costly and are generally thought to be ineffective.

The inventor believes that a need exists for a dropped charge protection system that can be used effectively in a geared mill arrangement.

Summary of the invention

Definitions for purpose of interpreting this specification: The angle of repose is defined for the purpose of this invention as the angle between the vector from the mill's axis of rotation to the centre of gravity of the charge and the gravitational vector

According to an aspect of the invention there is provided a dropped charge protection system, wherein the system includes calculating an angle of repose of a charge of a grinding mill during start-up and tripping the mill motor when the angle of repose of the charge exceeds a maximum allowable angle

The dropped charge protection system may include plotting the calculated angle of repose relative an angle of rotation of the mill shell

The angle of repose of the charge may be determined by solving the non-linear differential equation of T= Ja + mgr sinθ, wherein

T ι$ the air-gap torque applied to the motor rotor by the electric field, α is the angular acceleration of the mill around the centre of rotation of the mill shell and may be determined from d/dt{ω) o> is the angular speed of the mill shell around the centre of rotation of the mill shell and may be determined from d/dt{φ);

J is the moment of inertia [kgm ^{2 }] of all the rotating mass referenced to the mill shell side of the drive train, m is the mass of the charge, g is the gravitational constant; r is the radius from the milt shell's axis of rotation to the centre of gravity of the charge, and θ Is the rotation of the centre of gravity of the charge around the mill shell's axis of rotation which was defined above as the angle of repose. Before the charge has tumbled, it rotates with the mill shell and θ = φ, and wherein φ is the angular position of the mill shell around the centre of rotation of the mill shell;

The torque T may cause the acceleration of all rotating masses (Ja), and, the pendulum-like raising of the charge (mgrsinθ)

It is to be appreciated from this specification that the tripping criterion in the equation T= Ja + mgrsinθ is the angle of repose (θ). In order to solve θ, the system parameters J and mgr must be determined and the system variables T and α measured in real time or calculated from measurable quantities in real time.

The torque (T) may be calculated using the formula T= P/ω wherein P is the power of the motor and ω is the angular speed of the motor.

Any one or more of θ and/or α and/or ω may be measured through the use of rotary encoders, magnetic pick-ups and the like on the motor shaft or elsewhere in the drive train.

T and any one or more of φ and/or α and/or ω may be calculated from the rotor current of the mill motor in real time, making both the instantaneous measurement of P and the use of rotary encoders, magnetic pick-ups and the like on the motor shaft or elsewhere in the drive train unnecessary in the case of a wound-rotor motor if the rotor current is accessible.

The torque (T) produced by the wound-rotor motor may be directly proportional to the rotor current.

The mill motor may include a liquid resistance starter (LRS) in series with the motor rotor windings.

The LRS may control the rotor current and thereby control the amount of torque produced by the motor as the torque may be proportional to the rotor current.

The power factor (the ratio of the real power to the apparent power,) in the rotor circuit may be close to unity (where unity = 1 ) and the torque may therefore be determined by the formula wherein T is the air-gap Torque or T _{aιrgap }, I is the rotor current and l _{rated } is the rated rotor current at rated torque, produced at rated power.

α may be determined from ω by differentiation (d/dt(ω))

The mill rotation speed (ω) may be determined from the motor speed (n) and the gear ratio.

The motor speed (n) may be calculated from the rotor current using f s stem ^{~ } f rotor the formula n = ^{system } ^^ ^{χ } 60 [rpm], wherein f _{sys }te _{m } is the frequency of the system

P

(line frequency), f _{rotor } is the frequency of the rotor current of the motor, and p is the number of pole pairs of the motor.

The frequency of the rotor current of the motor (f _{ro }to _{r }) may be determined by inverting the period of a measured sine wave cycle of the rotor current.

The moment of inertia of all rotating mass (J), the mass of the charge (m) and the radius from the centre of the mill's axis of rotation to the centre of gravity of the charge (r) may be unknown.

J and mgr may be dependent on r but r may not be readily determinable due to the non-homogenous state of the charge.

J and mgr may be determined dynamically within the first few degrees of mill rotation, before the possibility of a dropped charge exists, so that the system can start calculating θ timeously.

It is to be appreciated from this specification that φ must be determined if it is to be used in the calculation of J and mgr. In the period before tumbling it is known that θ = φ and θ is therefore known.

The mill shell's rotation φ may also be determined by integration of ω where the integration of ω is the taking the integral of ω with respect to time. At a small mill shell rotation of 1°, ψ = θ = 1 ° and sin (1 °) =0.017 and the contribution of mgrsinθ to T = Ja + mgr sinθ may be relatively small resulting in T = Ja + mgr sinθ being simplified to T = Ja and J may therefore be calculated from the

T formula J = — . a

It is however to be appreciated from this specification that although φ = 1° was used, the result holds for any angle of φ = θ small enough that mgrsinθ can be neglected from T = Ja + mgr sinθ.

At a relatively bigger mill shell rotation, of φ=10°, the charge may not have yet rotated enough to tumble, but sin (10°) =0.173 and the contribution of mgrsinθ is therefore 10 times bigger in the equation T = Ja + mgr sinθ and can no longer be neglected.

mgr may therefore be calculated from the equation mgr = as sin(10°) both the mill and the angle of repose are 10°.

It is once again to be appreciated from the specification that the calculation is not limited to φ = 10°. The result will hold for any angle of φ = θ wherein said angle is large enough that mgrsinθ can not be neglected from T = Ja + mgrsinθ, but small enough that the charge has not yet tumbled.

As soon as J and mgr have been calculated, it is possible to calculate θ, plot θ relative an angle of rotation of the mill shell (φ) and trip the mill motor when the angle of repose of the charge exceeds a maximum allowable angle.

Tumbling may have occurred when φ is no longer equal to θ, and this may be used as a criterion to determine if start-up of the mill has been safe and successful. The dropped charge protection system may continue to record the rotor current after tumbling and facilitate evaluation of the rotor current and resultant torque

According to another aspect of the invention there is provided a control system controlling the torque applied to starting a grinding mill, wherein the system includes using a pre-determined angle of repose, controlling a real angle of repose of a charge such that the real angle of repose coincides with the pre-determined angle of repose through the manipulation of the torque of the motor and wherein the angle of repose is controlled in such a way as to encourage tumbling of the charge.

The torque may be the actuating signal and the angle of repose θ may be the controlled signal.

The angle of repose of the charge may be determined by solving the nonlinear differential equation of T= Ja + mgr sinθ, wherein

T is the air-gap torque applied to the motor rotor by the electric field; α is the angular acceleration of the mill around the centre of rotation of the mill and may be determined from d/dt(ω). ω is the angular speed of the mill shell around the centre of rotation of the mill shell and may be determined from d/dt(φ);

J is the moment of inertia [kgm ^{2 }] of all the rotating mass referenced to the mill side of the drive train; m is the mass of the charge; g is the gravitational constant; r is the radius from the mill's axis of rotation to the centre of gravity of the charge; and θ is the rotation of the centre of gravity of the charge around the mill's axis of rotation which was defined above as the angle of repose. Before the charge has tumbled, it rotates with the mill and θ = φ, and wherein φ is the angular position of the mill around the centre of rotation of the mill shell;

The torque T may effect the acceleration of all rotating masses (Ja), and, the pendulum-like raising of the charge (mgrsinθ) It is to be appreciated from this specification that the controlled variable in the equation T= Ja + mgrsinθ is the angle of repose (θ). In order to solve θ, the system parameters J and mgr must be determined and the system variables T and α measured in real time or calculated from measurable quantities in real time.

The torque (T) may be calculated using the formula T= P/ω wherein P is the power of the motor and ω is the angular speed of the motor.

Any one or more of θ and/or α and/or ω may be measured through the use of rotary encoders, magnetic pick-ups and the like on the motor shaft or elsewhere in the drive train.

T and any one or more of φ and/or α and/or ω may be calculated from the rotor current of the mill motor in real time, making both the instantaneous measurement of P and the use of rotary encoders, magnetic pick-ups and the like on the motor shaft or elsewhere in the drive train unnecessary in the case of a wound-rotor motor as the rotor current is accessible.

The torque (T) produced by the wound-rotor motor may be directly proportional to the rotor current.

The mill motor may include a liquid resistance starter (LRS) in series with the motor rotor windings.

The LRS may control the rotor current and thereby control the amount of torque produced by the motor as the torque is proportional to the rotor current.

The power factor (the ratio of the real power to the apparent power,) in the rotor circuit may be close to unity (where unity = 1 ) and the torque may therefore be determined by the formula T=(l/l _{ra }ted)T _{ra }ted wherein T is the air-gap Torque or T _{aιrgap }, I is the rotor current and l _{rate }d is the rated rotor current at rated torque, produced at rated power.

α may be determined from ω by differentiation (d/dt(ω)) The mill rotation speed (ω) may be determined from the motor speed (n) and the gear ratio.

The motor speed (n) may be calculated from the rotor current using f s stem ^{~ } f rotor the formula n = ^{system } ^^ ^{χ } 60 [rpm], wherein f _{system } is the frequency of the system

P

(line frequency), f _{rotor } is the frequency of the rotor current of the motor, and p is the number of pole pairs of the motor.

The frequency of the rotor current of the motor (f _{rot }o _{r }) may be determined by inverting the period of a measured sine wave cycle of the rotor current.

The moment of inertia of all rotating mass (J), the mass of the charge (m) and the radius from the centre of the mill's axis of rotation to the centre of gravity of the charge (r) may be unknown.

J and mgr may be dependent on r but r may not be readily determinable due to the non-homogenous state of the charge.

J and mgr may be determined dynamically within the first few degrees of mill rotation, before the possibility of a dropped charge exists, so that the system can start calculating θ timeously.

It is to be appreciated from this specification that φ must be determined if it is to be used in the calculation of J and mgr. In the period before tumbling it is known that θ = φ and θ is therefore known.

The mill shell's rotation φ may also be determined by integration of ω where the integration of ω is the taking the integral of ω with respect to time.

At a small mill shell rotation of 1°, φ = θ = 1 ° and sin (1 °) =0.017 and the contribution of mgrsinθ to T = Ja + mgr sinθ may be relatively small resulting in T = Ja + mgr sinθ being simplified to T = Ja and J may therefore be calculated from the formula

a

It is however to be appreciated from this specification that although φ = 1° was used, the result holds for any angle of φ = θ small enough that mgrsinθ can be neglected from T = Ja + mgr sinθ.

At a relatively bigger mill shell rotation, of φ=10°, the charge may not have yet rotated enough to tumble, but sin (10°) =0.173 and the contribution of mgrsinθ is therefore 10 times bigger in the equation T = Ja + mgr sinθ and can no longer be neglected.

mgr may therefore be calculated from the equation mgr = as sin(10°) both the mill and the angle or repose are 10°.

It is once again to be appreciated from the specification that the calculation is not limited to φ = 10°. The result will hold for any angle of φ = θ wherein said angle is large enough that mgrsinθ can not be neglected from T = Ja + mgrsinθ, but small enough that the charge has not yet tumbled.

As soon as mgr have been calculated, it is possible to calculate the amount of torque (T) necessary to keep φ at an optimum angle for the charge to tumble. By controlling the liquid resistance starter, the rotor current can be controlled to apply the correct amount of torque to bring φ to this optimum angle.

Unrelated to the issue of dropped charge, another advantage of this system is that with a small additional software algorithm and no additional hardware cost, the rotor current and therefore torque can be controlled such as to eliminate overtorque transients and arcing of the LRS electrodes, which is a common problem with present generation LRSs. The inventor believes that the invention has the advantage of providing a reliable and satisfactory dropped charge protection system for geared mills that are driven by wound rotor induction motors. Thereafter, the current is still recorded, and from this the engineer/operator is able to evaluate the rotor current and therefore the torque.

Furthermore, the inventor believes that the system provides an accurate evaluation of the liquid resistance starter performance and allows for control of the LRS and the resultant rotor current and therefore the torque of the motor. Over-torque transients will be caused if the LRS decreases its resistance too rapidly during start-up of the motor, causing the current of the motor to increase too rapidly, with a resultant undesirable high torque.

Example and detailed description of drawings.

The invention will be further explained by way of the following non-limiting working example and drawings of a dropped charge protection relay and monitoring system, wherein

Figure 1 shows the start-up graphs of a grinding mill in accordance with the invention; and

Figure 2 is a screen shot of a Human Machine Interface that depicts a graph of the charge's angle of repose (θ) relative the mill shell angle of rotation (φ). The screenshot also shows a graphic representation of the θ and φ in a simulated mill shell.

A dropped charge protection relay system, wherein the system calculates an angle of repose of a charge of a grinding mill during start-up, plots the angle of repose of the charge relative an angle of rotation of the mill and trips the mill motor when the angle of repose of the charge exceeds a maximum allowable angle.

Measurements and certain calculated values are recorded at a sampling rate of 1kHz for the duration of the mill start-up.

The angle of repose of the charge is determined by solving the non-linear differential equation of T= Ja + mgr sinθ, wherein T is the air-gap torque applied to the motor rotor by the electric field; α is the angular acceleration of the mill around the centre of rotation of the mill shell and may be determined from d/dt(ω)and wherein ω is the angular speed of the mill around the centre of rotation of the mill and may be determined from d/dt(φ);

J is the moment of inertia [kgm ^{2 }] of all the rotating mass referenced to the mill side of the drive train; m is the mass of the charge; g is the gravitational constant; r is the radius from the mill's axis of rotation to the centre of gravity of the charge; and θ is the rotation of the centre of gravity of the charge around the mill's axis of rotation which was defined above as the angle of repose. Before the charge has tumbled, it rotates with the mill and θ = φ and wherein φ is the angular position of the mill shell around the centre of rotation of the mill shell;

The torque T causes the acceleration of all rotating masses (Ja), and, the pendulum-like raising of the charge (mgrsinθ)

The tripping criterion in the equation T= Ja + mgrsinθ is the angle of repose (θ). In order to solve θ, the system parameters J and mgr must be determined and the system variables T and α measured in real time or calculated from measurable quantities in real time.

In this example the torque (T) is not calculated using the formula T= P/ω wherein P is the power of the motor and ω is the angular speed of the motor.

It is to be appreciated from this specification that any one or more of θ and/or α and/or ω can be measured through the use of rotary encoders, magnetic pickups and the like on the motor shaft or elsewhere in the drive train, but neither is this done in the example.

As a matter of fact, in this example, T, φ a, α and ω are calculated from the rotor current of the mill motor in real time, making both the instantaneous measurement of P and the use of rotary encoders, magnetic pick-ups and the like on the motor shaft or elsewhere in the drive train unnecessary in the case of a wound-rotor motor as the rotor current is accessible.

As the mill's motor rotor circuit includes a liquid resistance starter (LRS) in series with the motor rotor windings during start-up, the power factor of the rotor circuit is close to unity (=1), and therefore the torque (T) produced by the wound rotor motor is directly proportional to the rotor current.

The power factor (the ratio of the real power to the apparent power,) in the rotor circuit is close to unity (where unity = 1 ) and the torque is therefore determinable by the formula T=(l/l _{ra }ted)T _{ra }ted wherein T is the air-gap Torque or T _{aιrgap }, I is the rotor current and l _{rate }d is the rated rotor current at rated torque, produced at rated power.

In this working example of the invention, α is determined from ω by differentiation (d/dt(ω)) and the mill rotation speed (ω) is determined from the motor speed (n) and the gear ratio.

The motor speed (n) is calculated from the rotor current using f s stem ^{~ } f rotor the formula n = ^{system } ^^ ^{χ } 60 [rpm], wherein f _{system } is the frequency of the system

P

(line frequency), f _{rator } is the frequency of the rotor current of the motor, and p is the number of pole pairs of the motor. (In the case of a 6 pole motor, p = 3 and in the case of an 8 pole motor p = 4.)

The frequency of the rotor current of the motor (f _{rotor }) is determined by inverting the period of a measured sine wave cycle of the rotor current.

The moment of inertia of all rotating mass (J), the mass of the charge (m) and the radius from the centre of the mill's axis of rotation to the centre of gravity of the charge (r) are unknown at the moment of start-up. J and mgr are dependent on r but r may not be readily determinable due to the non-homogenous state of the charge.

J and mgr are therefore determined dynamically within the first few degrees of mill rotation, before the possibility of a dropped charge exists, so that the system can start calculating θ timeously.

Furthermore, φ must be determined if it is to be used in the calculation of J and mgr. In the period before tumbling it is known that θ = φ and θ is therefore known.

The mill rotation φ is determined through the integration of ω where the integration of ω is the taking the integral of ω with respect to time.

At a small mill shell rotation of 1°, φ = θ = 1 ° and sin (1 °) =0.017 and the contribution of mgrsinθ to T = Ja + mgr sinθ is relatively small resulting in T = Ja + mgr sinθ being simplified to T = Ja and J is therefore be calculated from the

T formula J = - . This example determines J at φ=θ=1°. a

It is however to be appreciated from this specification that although φ = 1° was used, the result holds for any angle of φ = θ small enough that mgrsinθ can be neglected from T = Ja + mgr sinθ.

At a relatively bigger mill shell rotation, of φ=10°, the charge has not yet rotated enough to tumble, but sin (10°) =0.173 and the contribution of mgrsinθ is therefore 10 times bigger in the equation T = Ja + mgr sinθ and can no longer be neglected.

mgr is therefore calculated from the equation mgr = as both the a sin(10°) angle of rotation of the mill shell φ and the angle of repose θ are 10°, in this example. It is once again to be appreciated from the specification that the calculation is not limited to φ = 10°. The result will hold for any angle of φ = θ wherein said angle is large enough that mgrsinθ can not be neglected from T = Ja + mgrsinθ, but small enough that the charge has not yet tumbled.

As soon as J and mgr have been calculated, it is possible to calculate θ, plot θ relative an angle of rotation of the mill shell (φ) and trip the mill motor when the angle of repose of the charge exceeds a maximum allowable θ value.

It is however to be appreciated from this specification that the invention also allows for the control the angle of rotation of the mill shell φ. to facilitate tumbling of the charge.

It is also to be appreciated from this specification that the invention also allows for the control of the torque of the motor until the motor is at full speed. Controlling the torque of the motor minimizes the risk of over-torque transients and mechanical failure.

Over-torque can occur at any time that the LRS is not presenting enough resistance to the rotor circuit to limit the rotor current (and therefore torque) to a safe value, even at the moment the motor is switched on. Typically, in order to evaluate the risk of torque transients, the engineer/operator would study the value of the rotor current during the entire start-up.

The following table shows the various measured and calculated values during the start-up of a grinding mill. Values at start-up:

[ms] IM Iw1 Ib1 Vpp1 Ir2 Iw2 Ib2 Vpp2 T n PI Jest Rot Tacc Toob mgrest Toob(Rot) OreAngle Tumble rdt

OOOOO -0739 -17582 -0192 5119 -16666 19619 31845 -1551 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

00001 -1057 -18530 0446 1958 -19531 18039 32164 -1235 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

00002 -0420 -18214 0127 2274 -19850 17723 32483 -1551 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

00003 -1057 -16950 0127 1010 -21442 17092 32483 -1867 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

00004 -1057 -16318 -0192 -2467 -24943 17092 33758 -1235 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

10 00005 -0420 -16002 -0192 -2783 -25262 15828 33439 -1235 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

00006 -0420 -16318 -0192 -2151 -24307 15512 33120 -1235 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

00007 -0420 -16634 0446 -1835 -23670 15828 31845 -1867 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

00008 -0420 -16634 0446 -1519 -23670 14565 31526 -2815 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

00009 0854 -16318 -0829 -1203 -25262 14880 31207 -1235 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

15 00010 -3924 -17266 0127 -0571 -23033 15196 31845 -2183 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

00011 0854 -17266 -0192 1010 -20805 15828 31526 -1551 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

00012 7226 -18214 -0510 1010 -17940 18039 32483 -1235 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

00013 -26544 -18846 -0510 2590 -16985 18039 31845 -1235 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

00014 409598 -18530 -0510 5751 -12528 19935 31526 -1235 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000000

20 0

00015 667651 -18846 0127 7332 -12209 20251 30889 -2183 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

00016 498482 -19478 -0510 8596 -12528 19935 30889 -1867 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000000 0

00017 360217 -18530 0446 6700 -12846 20566 32164 -2183 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

25 00018 141987 -18846 -0192 5751 -12846 20566 30889 -1235 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000000 0

00019 -78792 -19162 0127 6068 -11254 20251 31526 -1867 20000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000000

00020 -280137 -18530 -0510 5435 -12528 20566 31207 -1867 20000 0000 0000 NaN 0000 0000 0000 0000 0000 0000 000010 0

30 00021 -423818 -18530 -0192 2590 -14756 19303 32164 -0919 20000 0000 0000 NaN 0000 0000 0000 0000 0000 0000 000010 0

00022 -535960 -17582 -0192 1958 -15074 19935 31845 -0919 20000 0000 0000 NaN 0000 0000 0000 0000 0000 0000 000010 0

00023 -599995 -16950 -0829 1010 -16030 19619 33758 -0602 20000 0000 0000 NaN 0000 0000 0000 0000 0000 0000 000010

35 0

00024 -594579 -16950 -0192 -1519 -19213 18671 32164 -0602 20000 0000 0000 NaN 0000 0000 0000 0000 0000 0000 000010 0

00025 -558579 -15686 -0192 -1835 -20805 17723 33120 -1235 20000 0000 0000 NaN 0000 0000 0000 0000 0000 0000 000010 0

00026 -485 623 -16 634 -0 192 -2 783 -21 760 17 723 33439 -1 235 20 000 0 000 0 000 NaN 0 000 0 000 0 000 0 000 0 000 0 000 0 000 1 0 0

00027 -366 792 -17 266 -0 192 -1 835 -23 033 17408 32 483 -0 602 20 000 0 000 0 000 NaN 0 000 0 000 0 000 0 000 0 000 0 000 0 000 1 0 0

00028 -213 871 -16 318 0 127 -1 835 -23 033 16 144 31 207 -1 551 20 000 0 000 0 000 NaN 0 000 0 000 0 000 0 000 0 000 0 000 0 000 1 0 0

00029 -30 686 -16 318 -0 510 -1 519 -25 262 16460 34 077 -0 602 20 000 0 000 0 000 NaN 0 000 0 000 0 000 0 000 0 000 0 000 0 000 1 0 0

The recording includes some pre-trigger values. It can be seen that only at t - 12ms the motor is started, and there the current is only 7.2A. Al calculated 10 values are still zero.

Values around 1° of mill rotation. (Estimation of J is finalized at this point, and therefore calculation of Tacc and Toob may begin. At this stage the angle of repose θ cannot yet be calculated from mgrsinθ because mgr is not known yet, and is set equal to mill rotation Rot=φ by the DCPR)

15 [ms] IM Iw1 Ib1 Vpp1 Ir2 Iw2 Ib2 Vpp2 T n III Jest Rot Tacc Toob mgrest Toob(Rot) OreAngle Tumble r d t

00867 -542 650 -16 950 0446 -2 151 -23 033 15 196 32 164 -1 867 21 221 42 705 456 347 2013 124 0 988 0 000 0000 0 000 0 000 0 988 0000 1 00

00868 -613694 -16 950 0 127 -2 783 -23 988 15 196 32 164 -1 867 21 221 42 769 456438 2013 364 0 991 0 000 0000 0 000 0 000 0 991 0 000 1 00

00869 -650013 -16 950 0 127 -1 519 -22 397 15 828 32 164 -2 183 21 221 42 832 456 528 2013 608 0 994 0 000 0000 0 000 0 000 0 994 0 000 1 00

00870 -620385 -16 318 0 127 -2 151 -21 442 17 092 33439 -0 919 21 221 42 895 456 618 2013 855 0 998 0 000 0000 0 000 0 000 0 998 0 000 1 00

20 00871 -530544 -15 686 -0 192 -0 887 -19850 18 355 34 077 -1 235 21 221 42 958 456 707 2014 106 1 001 0000 0 000 0 000 0 000 1 001 0 000 1 00

00872 ^03 110 -17 582 1 083 0 694 -16985 19 619 32 483 -1 867 21 221 43 022 456 796 2014 106 1 004 0000 0 000 0 000 0 000 1 004 0 000 1 00

00873 -214 508 -17 266 -0 192 1 326 -15 393 20 882 32 801 -1 867 21 221 43085 456 884 2014 106 1 007 0 000 0000 0 000 0 000 1 007 0000 1 00

00874 -20491 -17 898 0 127 1 010 -14 438 20251 32 801 -0 602 21 221 43 148 456972 2014 106 1 010 13288 542 1172 806 66541 078 0000 1 010 0 000 1 0 0

00875 129 562 -19478 -0 192 4 487 -12 209 22 146 33 120 -0 919 21 221 43 210 457 060 2014 106 1 013 13274 087 1190028 67310 973 0 000 1 013 0 000 1 0 0

25 00876 317 208 -19478 -0 192 6 384 -11 573 21 198 31 845 -1 235 21 221 43 273 457 147 2014 106 1 016 13259818 1207 049 68064 526 0 000 1 016 0 000 1 0 0

00877 485421 -19 162 0 127 6700 -12 528 21 514 32 164 -1 551 21 221 43 336 457 233 2014 106 1 019 13245741 1223 861 68801 393 0 000 1 019 0 000 1 0 0

00878 576 217 -19 162 -0 192 6 384 -15074 20 566 31 207 -2 499 21 221 43 399 457 319 2014 106 1 022 13231 865 1240455 69521 268 0 000 1 022 0 000 1 0 0

00879 631 969 -18 530 -0510 5435 -17 303 19 935 31 845 -0 919 21 221 43461 457 404 2014 106 1 025 13218 196 1256824 70223 882 0 000 1 025 0 000 1 0 0

00880 632 606 -17 582 0 127 4 803 -17 303 19 303 31 526 -0 919 21 221 43 524 457 489 2014 106 1 029 13204 740 1272 962 70909 004 0 000 1 029 0 000 1 0 0

30 00881 563474 -18 846 -0510 4 803 -17 621 18 671 30 889 -1 551 21 221 43 586 457 573 2014 106 1 032 13191 505 1288861 71576439 0 000 1 032 0 000 1 0 0

00882 458 978 -17 898 0 764 2 274 -19213 15 828 30 570 -1 235 21 221 43 649 457 657 2014 106 1 035 13178494 1304 517 72226 026 0 000 1 035 0 000 1 0 0

35

Values around 10° of mill rotation. (Estimation of mgr is finilized at this point, and therefore calculation of θ may now start independently from φ.

[ms] IM Iw1 Ib1 Vpp1 Ir2 Iw2 Ib2 Vpp2 T π III Jest Rot Tacc Toob mgrest Toob(Rot) OreAngle Tumble r d t

02367 -665942 -16 950 -0 510 0378 -16 030 19 935 32 483 -1 867 22 731 116487 629222 2014 106 9970 5174 507 14737 837 85121 016 0000 9970 0 000 1 0 0

02368 -806438 -16 634 0 127 -1 519 -17 621 19 935 32483 -0 919 22 731 116 512 629255 2014 106 9979 5168315 14745 100 85092 049 0000 9979 0 000 1 0 0

02369 -899464 -15 686 -0 510 -2 783 -18576 19 935 34 077 -0 919 22 731 116 536 629 289 2014 106 9987 5162 071 14752 412 85063 399 0 000 9987 0000 1 0 0

02370 -907 747 -16 318 -0 192 -2 783 -19531 19 303 32 164 -1 551 22 731 116 561 629 323 2014 106 9996 5155771 14759 776 85035 081 0 000 9996 0000 1 0 0

02371 -857411 -17 266 0 127 -0887 -19 850 18 355 32 164 -1 867 22 731 116 585 629356 2014 106 10 004 5149414 14767 193 85007 107 0 000 10 004 0 000 1 00

02372 -742 721 -16 634 -0 510 -2 467 -23670 17 723 32 164 -1 235 22 731 116 609 629 390 2014 106 10 012 5149414 14768 250 85007 107 14779468 10 005 0008 1 00

10 02373 -545836 -16 318 -0 192 -1 519 -23988 17 723 32 483 -2 183 22 731 116 634 629423 2014 106 10 021 5149414 14769 303 85007 107 14791 746 10 005 0015 1 00

02374 -353093 -16 002 -0 192 -0 571 -23033 17 408 32 801 -2 499 22 731 116 658 629456 2014 106 10 029 5129 974 14789 793 85007 107 14804 026 10 019 0010 1 00

02375 -123712 -17 582 -0 829 0378 -20 805 17 408 31 207 -1 867 22 731 116 682 629489 2014 106 10 038 5123362 14797450 85007 107 14816 308 10 025 0013 1 00

02376 154 730 -17 266 -0 192 0 694 -21 442 16460 30 570 -2 499 22 731 116 707 629 522 2014 106 10 046 5116 681 14805 174 85007 107 14828592 10 030 0016 1 00

02377 386 660 -16 950 -0829 1 642 -19850 17 092 31 526 -2 499 22 731 116 731 629 555 2014 106 10 054 5109 928 14812 965 85007 107 14840878 10 035 0019 1 00

15 02378 584 182 -18 214 -0510 4 803 -18258 18 039 30 570 -1 867 22 731 116 755 629 588 2014 106 10 063 5103 101 14820827 85007 107 14853 167 10 041 0022 1 00

02379 754 305 -18 530 0446 7 016 -14 119 18 039 29 295 -2 183 22 731 116 779 629620 2014 106 10 071 5096 197 14828761 85007 107 14865458 10 046 0025 1 00

02380 855 615 -18 530 -0510 8 280 -12 528 20 882 31 207 -1 867 22 731 116 803 629 653 2014 106 10 080 5089 213 14836771 85007 107 14877 751 10 052 0028 1 00

02381 891 615 -17 582 -0510 5435 -12 846 20 251 31 207 -2 499 22 731 116 827 629 685 2014 106 10 088 5082 149 14844 858 85007 107 14890046 10 057 0031 1 00

02382 870 270 -17 898 -0510 5435 -11 254 20 566 30 251 -1 867 22 731 116 851 629 717 2014 106 10 097 5075 001 14853024 85007 107 14902 344 10 063 0034 1 00

20 02383 765 137 -18 530 -0510 6 700 -11 891 21 830 31 207 -1 551 22 731 116 875 629 749 2014 106 10 105 5067 767 14861 272 85007 107 14914 644 10 068 0037 1 00

02384 598 836 -17 898 -0829 5 119 -12 846 22 462 31 207 -1 867 22 731 116 899 629 781 2014 106 10 113 5060445 14869603 85007 107 14926946 10 074 0039 1 00

02385 400 677 -17 898 -0 192 1 958 -14 119 20 566 30 889 -0 602 22 731 116 923 629 813 2014 106 10 122 5053 033 14878020 85007 107 14939250 10 080 0042 1 00

02386 167 474 -17 266 -0829 1 642 -15711 19 935 32 164 -1 551 22 731 116 947 629 844 2014 106 10 130 5045 529 14886526 85007 107 14951 556 10 086 0045 1 00

02387 -76 562 -16634 -0829 1 010 -17 940 18987 31 845 -0919 22 716 116971 629 876 2014 106 10 139 5037 930 14895 121 85007 107 14963 865 10 092 0 047 1 0 0

25 02388 -318048 -16 634 0 127 -1 519 -20 805 18 671 31 845 -1 235 22 716 116 995 629907 2014 106 10 147 5030234 14903808 85007 107 14976 175 10 098 0050 1 00

30

Values around 12s By this time the values of θ and φ have diverged dramatically, θ still being safely below 30° while the mill shell has already rotated almost 60°

[ms] IM Iw1 Ib1 Vpp1 Ir2 Iw2 Ib2 Vpp2 T n III Jest Rot Tacc Toob mgrest Toob(Rot) OreAngle Tumble r i t

11990 -1311 075 -16318 0 127 -1 835 -17 621 18 987 34 077 -1 867 26 203 213763 1831 718 2014 106 58434 20944 536 37021 999 85007 107 72429453 25818 32 616 1 1 1

11991 -728703 -17 582 -0 192 -0 571 -16030 19 619 34 395 -1 867 26203 213 863 1831 856 2014 106 58450 20948 866 37022 042 85007 107 72441 414 25818 32 631 1 1 1

11992 -89 305 -16634 0446 0 378 -15 711 19303 33 758 -1 867 26 203 213 962 1831 994 2014 106 58465 20953 245 37022 036 85007 107 72453376 25818 32 647 1 1

11993 542 129 -17 266 1 083 1 642 -16030 19 619 33439 -1 867 26203 214 062 1832 132 2014 106 58480 20957 680 37021 972 85007 107 72465 338 25 818 32 662 1 1 1

11994 1146 164 -17 898 0 127 2 906 -15 074 20 882 34 077 -2 499 26203 214 161 1832 270 2014 106 58496 20962 177 37021 844 85007 107 72477 300 25 818 32 678 1 1 1

11995 1704 004 -18 846 0446 3 855 -16 030 18 355 31 845 -2 815 26203 214 260 1832408 2014 106 58 511 20966744 37021 646 85007 107 72489 263 25 818 32 693 1 1 1

10 11996 2128 358 -18 846 0 127 6 068 -15 711 18 671 32 164 -2 499 26203 214 360 1832 546 2014 106 58 527 20971 386 37021 372 85007 107 72501 226 25 818 32 709 1 1 1

11997 2440 571 -19478 0 127 7 016 -16 348 18 987 32 164 -2 499 26203 214459 1832 684 2014 106 58 542 20976 106 37021 017 85007 107 72513 189 25 817 32 725 1 1 1

11998 2621 845 -17 266 0 127 3 855 -15 074 19 303 33439 -1 867 26203 214 559 1832 822 2014 106 58 558 20980910 37020 577 85007 107 72525 153 25 817 32 741 1 1 1

11999 2640 960 -17 266 0 127 4 803 -16 348 18 987 33439 -1 551 26203 214 658 1832 960 2014 106 58 573 20985799 37020 049 85007 107 72537 116 25 817 32 756 1 1 1

12000 2510 022 -19 162 0 127 4 487 -16 348 18 039 32483 -2 815 26203 214 758 1833098 2014 106 58 589 20990778 37019430 85007 107 72549 080 25 816 32 772 1 1 1

15 12001 2225 526 -17 898 0 127 2 274 -18 576 17 723 32 801 -3448 26203 214 857 1833235 2014 106 58 604 20995847 37018 719 85007 107 72561 045 25 816 32 788 1 1 1

12002 1833 987 -16 950 -0 192 1 326 -19213 17 092 33439 -2 815 26 203 214 957 1833373 2014 106 58 620 21001 007 37017 914 85007 107 72573 009 25815 32 804 1 1 1

12003 1347 190 -17 266 0 127 0 694 -20 168 17 092 32483 -2 499 26203 215057 1833510 2014 106 58 635 21006260 37017 014 85007 107 72584 974 25 814 32 821 1 1 1

12004 768 323 -16 318 0 764 -2 151 -22 397 15 828 33439 -2 499 26 203 215 156 1833 648 2014 106 58 651 21011 604 37016 018 85007 107 72596 939 25814 32 837 1 1 1

20

25

Tumbling has occurred when φ is no longer equal to θ, and this may be used as a criterion to determine if start-up of the mill has been safe and successful.

In Figure 1 , line 12 on graph 10 is representative of the result of the calculation mgrsinφ and line 14 is representative of the result of the calculation mgrsinθ. Tumbling of the charge has occurred at the point in time marked 16. Line 18 represents the result of the formula Ja.

Graph 20 in Figure 1 shows the plotted angle φ 22 and the plotted angle θ 24.

In the screenshot shown in Figure 2 the graph 28 depicts the graphical representation of the charge's angle of repose (θ) relative the the mill shell's angle of rotation (φ). It can be seen that the angle θ relative the angle φ is a 45° line before tumbling occurs. The drop in the graph 32 denotes the angle of φ at which at which tumbling has occurred.

The graphic representation 34 shows θ 36 and φ 38 in a simulated mill shell.

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