Reference ID: MET-46BA | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The critical speed of a ball mill is the rotational speed at which the centrifugal force exactly balances the gravitational force on the grinding media. At this speed the media are held against the mill wall and no further grinding occurs. Operating a mill below the critical speed ensures that the media cascade and tumble, providing the impact and attrition forces required for efficient size reduction. This calculation is a fundamental step in the design and operation of grinding circuits in mineral processing, cement manufacturing, and other bulk-material industries.
Methodology & Formulas
The calculation follows a straightforward sequence of physics-based relationships, all expressed in SI units.
Define the mill geometry:
\[ R = \frac{D}{2} \]
where D is the internal diameter of the mill and R is the radius.
Compute the critical speed in revolutions per second:
\[ N_{c,\;{\rm rev/s}} = \frac{1}{2\pi}\,\sqrt{\frac{g}{R}} \]
with g representing the acceleration due to gravity.
Convert the critical speed to revolutions per minute (rpm):
\[ N_c = 60\,N_{c,\;{\rm rev/s}} \]
Apply an operating-speed factor k (typically between kmin and kmax) to obtain the recommended operating speed:
\[ N_{\rm opt} = k\,N_c \]
Validity Checks
Before using the results, the input parameters should be verified against typical engineering limits. The table below summarizes the required ranges and the associated warning messages.
Parameter
Valid Range (symbolic)
Warning Message
Radius \(R\)
\(R_{\min} \le R \le R_{\max}\)
Radius out of valid range.
Diameter \(D\)
\(D_{\min} \le D \le D_{\max}\)
Diameter out of valid range.
Operating factor \(k\)
\(k_{\min} \le k \le k_{\max}\)
Operating factor k out of recommended range.
Critical speed \(N_c\)
\(N_{c,\min} \le N_c \le N_{c,\max}\)
Critical speed outside typical operating window.
Result Summary
After performing the calculations, the key outputs are:
The critical speed is the rotational speed at which the centrifugal force acting on the grinding media equals the gravitational force. At this speed the media will cling to the mill wall and will not fall, causing the grinding action to cease.
The most common empirical formula is:
Critical Speed (rpm) = 76.6 / √D
where D is the internal diameter of the mill (in meters). Some engineers prefer the metric version:
Critical Speed (rpm) = 42.3 / √D
with D expressed in feet. Both equations assume a smooth, cylindrical mill and standard steel media.
Operating at 70-80 % of the critical speed provides the optimal balance between:
Impact – media are lifted and then fall, creating high-energy collisions.
Attrition – the tumbling action creates shear forces that further reduce particle size.
Power consumption – speeds above this range increase motor load without proportional grinding benefit.
Running too low results in excessive sliding and low impact; running too high causes the media to stick to the wall, reducing grinding action.
Several real-world conditions modify the theoretical critical speed:
Mill liner geometry – raised liners or lifters change the effective diameter.
Media size and density – larger or denser balls alter the centrifugal balance.
Load level – a partially filled mill has a different mass distribution than a full one.
Operating temperature – thermal expansion of the mill shell slightly increases the diameter.
Engineers often apply a correction factor (typically 0.9-0.95) to the calculated speed to account for these influences.
Worked Example – Sizing a Ball Mill for a Copper Concentrator
A metallurgical team is selecting a ball mill for a new 500 t/d copper concentrator. To avoid over-loading the mill, they must verify that the chosen mill will operate below its critical speed. The mill shell has an inside diameter of 1.5 m and the liners will be 75 mm thick, giving a grinding-charge path diameter of 1.35 m. The team decides to run the mill at 70 % of the theoretical critical speed.
Knowns
Gravitational acceleration, g = 9.81 m s-2
Mill inside diameter, D = 1.5 m
Fraction of critical speed, k = 0.7 (–)
Effective (mean) radius of the grinding charge, R = 0.75 m
Step-by-step calculation
Critical speed in rad s-1 is obtained from the classical formula
\[ \omega_c = \sqrt{ \frac{g}{R} } = \sqrt{ \frac{9.81}{0.75} } = 3.621 \; \text{rad s}^{-1} \]
Convert to revolutions per second
\[ N_c = \frac{\omega_c}{2\pi} = \frac{3.621}{2\pi} = 0.576 \; \text{rev s}^{-1} \]
Convert to revolutions per minute
\[ N_c = 0.576 \times 60 = 34.5 \; \text{rpm} \]
Apply the 70 % operating factor to obtain the recommended mill speed
\[ N_{\text{opt}} = k \times N_c = 0.7 \times 34.5 = 24.2 \; \text{rpm} \]
Final Answer
The ball mill must not exceed 34.5 rpm; the selected operating speed is 24.2 rpm.
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise."— Charles de Gaulle
