A tumbling grinding mill rotates a drum containing material and grinding media. Contact, rolling and falling media break the feed into smaller particles. This calculator estimates an ideal critical speed from the internal diameter and converts your selected percentages into rpm. It covers tumbling mills, rather than roller or stirred mills.
What the speed calculation means
Critical speed is a geometric reference for the motion of media at the top of the drum. It does not select a motor, predict tonnes per hour or determine a suitable operating speed for a particular ore. The two percentage inputs are a comparison range chosen by you.
Grinding Mill Critical Speed Interactive Calculator
Enter internal diameter and two percentages of critical speed to compare rpm. The animation uses a simplified two-dimensional disc-contact model to illustrate lifting and tumbling. It is not calibrated to a particular mill.
Equation Used
The calculator balances gravity with the centripetal acceleration needed to carry a point-sized particle through the top of a circular path. It uses the diameter-only approximation; the chosen percentages are not operating recommendations.
- D is the internal mill diameter in metres.
- The 42.3 constant gives critical speed in rpm for SI units.
- Operating range is expressed as percent of critical speed.
- Use the internal diameter available inside the liners. Finite media diameter, slurry, lifters and interactions between particles are not included.
How a tumbling mill grinds
The shell carries the charge upward through contact with the lining and lifter bars. Some material rolls down the charge surface; some is released and falls. Cascading describes rolling and sliding motion, while cataracting describes media falling across the interior. Both contact forces and impacts can contribute to breakage.
The motion depends on speed, fill, media size, liner profile and material properties. Diameter alone cannot determine the release points or product fineness. In the animation, identical discs move under gravity and simplified contact with the rotating liner, lifters and other discs. Media size, fill, friction and rebound are illustrative fixed choices; slurry, ore breakage and three-dimensional movement are omitted. The drawing cannot predict an actual charge distribution or process performance.
Main components
- Shell: the rotating drum that contains the charge.
- Linings and lifters: replaceable surfaces that protect the shell and influence charge movement.
- Grinding media: balls, rods, pebbles or suitably competent feed material, according to mill type.
- Supports and drive: bearings and the mechanical or direct drive that rotate the shell.
- Feed and discharge: passages that introduce material and remove ground product.
Where tumbling mills are used
Tumbling mills are used in mineral processing, cement production and other material-reduction processes. Ball mills use balls as grinding media; rod mills use rods. Autogenous mills use the feed itself as grinding media, and semi-autogenous mills add a ball charge.
These arrangements differ from vertical roller mills, which press material between rollers and a grinding surface, and stirred mills, which move media with an internal agitator. The drum critical-speed formula does not choose an operating speed for those machines.
Critical-speed approximation
Nc = (60/2π)√(2g/D) ≈ 42.3/√D
N = (p/100)Nc
D is the internal diameter in metres, g is gravitational acceleration, Nc is critical speed in rpm, and p is the chosen percentage. At the top of the ideal circular path, gravity alone supplies the required inward acceleration at this threshold.
A finite ball of diameter d has its center on a smaller circle. Under the same ideal assumptions, the corresponding expression is 42.3/√(D − d), with both diameters in metres. This calculator uses D only and therefore neglects ball size. It does not automatically subtract liner thickness: enter the diameter inside the liners.
Critical speed is not a structural speed limit or a guarantee that all media move together. Actual charge behavior, grinding performance and machine limits require more than this one-particle model.
Example: a four-metre internal diameter
Enter D = 4.0 m and percentage limits of 65% and 78%.
- Critical speed: 42.3/√4 = 21.15 rpm.
- Lower comparison speed: 0.65 × 21.15 = 13.7475 rpm.
- Upper comparison speed: 0.78 × 21.15 = 16.497 rpm.
- Mid-band speed: 0.715 × 21.15 = 15.12225 rpm.
The display rounds these values to two decimal places. The middle value is the arithmetic midpoint of the two entered percentages, not an optimized setting.
For comparison outside this calculator, including an 80 mm ball in the ideal expression gives 42.3/√(4 − 0.08) ≈ 21.36 rpm. The small difference comes from the smaller circle followed by the ball’s center. Neither result predicts throughput, power or liner life.
Using the result in a larger design
| Quantity | What is needed |
|---|---|
| Approximate critical rpm | Internal diameter, with the stated point-media approximation. |
| A chosen speed as a percentage | The entered percentage multiplied by critical rpm. |
| Product size and throughput | Feed characteristics, circuit arrangement, charge and operating data. |
| Drive power and mechanical limits | Equipment design and manufacturer information. |
| Liner selection and wear | Mill geometry, media, material, impact conditions and service data. |
Frequently asked questions
What diameter should I enter?
Enter the internal diameter inside the liners in metres. The tool does not subtract lining thickness from an outside shell measurement.
Does it include the grinding-ball diameter?
No. It uses 42.3/√D. The finite-ball expression uses D minus the ball diameter, in consistent units, as explained in the formula section.
What happens if I enter the percentage limits in reverse order?
The calculator sorts the two values before calculating the lower and upper speeds. The midpoint stays the same.
Is the midpoint the best operating speed?
No. It is simply the average of your selected limits. Ore properties, media charge, lifters and process requirements are not inputs.
Why does a larger mill have a lower critical rpm?
The ideal result varies inversely with the square root of diameter. Quadrupling diameter halves critical rpm.
Can this calculate power or grind fineness?
No. The outputs are speeds only. A process and equipment model is needed for power, throughput and particle-size predictions.
Does the animation run at the calculated rpm?
The shell follows the calculated midpoint rpm, slowed to one third for viewing. Gravity is scaled consistently with that slower playback. Diameter and the midpoint percentage affect the motion; reversing the two percentage inputs leaves the midpoint unchanged. The simplified two-dimensional contacts are illustrative and do not predict real media trajectories.
Does the formula apply to the roller mill in the video?
No. That related video shows rollers acting on a grinding surface. This calculation is for media carried around inside a tumbling drum.
References and further reading
Metso: mill liners describes the role of grinding type, speed, mill size and impact conditions in liner design.
Metso: optimization and laboratory services discusses how charge, speed, power and ore properties are considered together when evaluating performance.
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