# Tumbling Mill (Speed)

## Description

This article describes mathematical methods for converting between the **rotational** and **fraction critical** speeds of a tumbling mill.

The *critical speed* of a horizontal tumbling mill is defined as the minimum speed at which a single element (ball, rod, rock etc.) will remain in contact with the mill wall for a full rotation. The critical speed is therefore a function of mill diameter, rotational speed and the force of gravity.

Tumbling mill speed is frequently referred to as the *fraction* of critical speed the mill is operating at.

The fraction critical speed of a mill has particular relevance to the mathematical modelling of grinding product size, power draw, and other properties.

## Model theory

The theoretical critical speed of a tumbling mill, (rev/min), is defined as:^{[1]}

where is the mill diameter *inside liners* (m) and is acceleration due to gravity (m/s^{2}).

Therefore, the fraction critical speed, (frac), given a rotational mill speed, (rev/min), is computed as:

and the rotational mill speed given a fraction critical speed is:

## Excel

The **fraction critical speed** of a tumbling mill may be computed from the Excel formula bar with the following function call:

```
=mdMillSpeed_FracCS(millDiameter As Double, millRPM as Double)
```

The *millDiameter*, *millRPM* and function results are defined for the fraction critical speed function below in matrix notation, along with example images showing the selection of the same cells in the Excel interface:

The **rotational speed** of a tumbling mill may be computed from the Excel formula bar with the following function call:

```
=mdMillSpeed_RPM(millDiameter As Double, fracCriticalSpeed as Double)
```

The *millDiameter*, *fracCriticalSpeed * and function results are defined for the fraction critical speed function below in matrix notation, along with example images showing the selection of the same cells in the Excel interface:

Invoking the function with no arguments will print Help text associated with the model, including a link to this page.

## See also

- Ball Mill (Perfect Mixing)
- Ball Mill (Perfect Mixing, Dynamic)
- Ball Mill (Overfilling)
- Tumbling Mill (Media Trajectory)
- Tumbling Mill (Power, Hogg and Fuerstenau)
- Tumbling Mill (Power, Morrell Continuum)
- Tumbling Mill (Power, Morrell Empirical)
- Tumbling Mill (Power, Morrell Discrete Shell)
- Tumbling Mill (Power, Hilden and Powell)
- Tumbling Mill (Slurry Flow)

## References

- ↑ King, R.P., 2012.
*Modeling and Simulation of Mineral Processing Systems*. Elsevier.