Determining the operating speed of a tube mill essentially involves finding a balance: on one hand, the grinding media (such as steel balls) must be lifted high enough to fall with maximum impact force for crushing; on the other hand, the speed must not be so high that the media centrifuge against the shell and lose their grinding action.

This process generally consists of three steps: first calculate the “critical speed”, then determine the “theoretical optimal speed”, and finally adjust based on actual operating conditions.

📖 Step 1: Understanding the “Critical Speed” (n₀)

This is the foundation for determining the operating speed. The critical speed is defined as the limiting speed at which the outermost layer of grinding media just rotates with the mill shell without falling. If the operating speed reaches or exceeds this value, the grinding action ceases.

The theoretical formula is:
n₀ = 42.4 / √D₀

Where:

• n₀ = critical speed (r/min)
• D₀ = effective inner diameter of the mill (m), i.e., the shell inner diameter minus twice the liner thickness

Note: This formula is derived from idealised assumptions and serves as a theoretical reference; it is not absolutely precise.

⚙️ Step 2: Calculating the “Theoretical Optimal Speed” (n)

This is the speed at which the grinding media achieve the best crushing effect. It is usually expressed as a percentage of the critical speed, and this percentage is called the “rotation rate” (ψ).

n = ψ × n₀

Regarding the optimal value of the rotation rate ψ, several mainstream views exist:

• Based on maximum impact force: To achieve the maximum falling height (i.e., maximum impact crushing work) for the outermost grinding media, ψ ≈ 76%.
• Based on overall efficiency (the layer‑contraction theory): Considering the mass distribution of the entire charge, it is believed that ψ ≈ 88% gives higher efficiency.
• Considering both energy and steel consumption: To save energy and reduce wear, a range of ψ = 65% ~ 76% is generally considered more economical.

In summary, ψ = 65% ~ 85% is a widely accepted range. For practical selection, refer to the following table:

🔧 Step 3: Determining the Final Speed Based on Actual Operating Conditions

Theoretical calculations provide a starting point, but the final speed must be adjusted according to actual conditions:

• Refer to empirical formulas: For certain specific mills, industry empirical formulas exist. For example, for wet intermittent ball mills:
  n = 21.2 / √D (r/min)
• Conduct experimental optimisation: This is the most reliable method. Adjust the speed slightly around the theoretical value and observe changes in mill output, product quality, and energy consumption to identify the “optimal operating speed” for your specific conditions. For example, one case study successfully increased output by slightly raising the speed of a Φ1.8m×6.4m mill.
• Consider other mechanical factors: The final speed setting is also limited by mechanical conditions such as the transmission ratio.

In short, the procedure for determining the operating speed of a tube mill is: calculate the critical speed → choose a rotation rate of 65%–85% as a starting point based on conditions → fine‑tune during actual production.

Table 2.12 Critical speed, theoretical optimal speed, and operating speed for common tube mills