Mineral Comminution, SAG/Ball Grinding Mills & Liner Mechanics
SAG & Ball Mill Critical Centrifuging Speed (%Nc) Formula
Calculate Effective Ball-Center Orbit Diameter (D - d), Critical Centrifuging Speed N_c (RPM), Operating Percent Critical Speed (%Nc), and Grinding Motion Regime.
Run a ball mill at 100% Critical Speed and grinding stops dead—steel balls centrifuge against the shell!
Horizontal rotating SAG (Semi-Autogenous Grinding) and Ball Mills crush ore by lifting forged steel grinding balls of diameter d_ball (m) inside shell liners of internal diameter D_inside (m) and throwing them into parabolic free-fall impact onto the toe of the charge (Cataracting at 65%–80% N_c)! At the crown of the mill (effective ball-center radius R_eff = (D_inside - d_ball) / 2), outward centrifugal acceleration omega^2 * R_eff exactly balances downward gravity g = 9.81 m/s² at Critical Centrifuging Speed N_c = (30 / PI()) * SQRT((2 * 9.81) / (D_inside - d_ball)) = 42.30 / SQRT(D_inside - d_ball) RPM! Above 100% N_c, the entire ball charge pins solidly to the shell wall with ZERO grinding impact!
Verify on an 8.00 m Inside-Liner SAG/Ball Mill—Mill_Inside_Diameter D = 8.00 m (B2), Top_Size_Ball_Diameter d = 0.12 m (120 mm forged balls, B3 -> Effective Orbit Diameter D - d = 7.88 m in E2), and Variable-Frequency Drive Operating_Speed = 11.00 RPM (B4): Critical Speed N_c equals (30 / PI()) * SQRT(19.62 / 7.88) = 15.07 RPM (E3), Operating Percent Critical Speed hits (11.00 / 15.07) * 100 = 73.00% N_c (E4), and Motion Regime confirms "OPTIMAL CATARACTING" (E5)!
| Tumbling Mill Speed KPI (D = 8.00 m, d_ball = 0.12 m, N_op = 11.0 RPM) | Formula | Verified Output |
|---|---|---|
| Effective Orbit Diameter (m) | Inside_Liner_D_m - Ball_Dia_d_m | 7.88 m |
| Critical Centrifuging Speed Nc | (30 / PI()) * SQRT(19.62 / (D - d)) | 15.07 RPM |
| Operating % Critical Speed (%Nc) | (Operating_RPM / Critical_RPM) * 100 | 73.00% (106.18% @ 16.0 RPM!) |
| Charge Motion Grinding Regime | 65%–80% Cataracting vs >=100% Lockup | OPTIMAL CATARACTING |
Exact Ball-Center Critical Speed & %Nc Cataracting Formula
Validate Inside_Diameter D > 0.50 m, 0 <= Ball_Diameter d < 0.25 * D, and Operating_RPM > 0!
=LET(dm_i, B2, db_l, B3, n_op, B4, IF(OR(dm_i <= 0.5, db_l < 0, db_l >= 0.25 * dm_i, n_op <= 0), "Invalid Mill Geometry", LET(nc_r, (30 / PI()) * SQRT((2 * 9.81) / (dm_i - db_l)), ROUND((n_op / nc_r) * 100, 2))))VFD Overspeed Centrifuging Lockup Warning Check (Set Mill Speed = 16.0 RPM)
Overspeed Operating_RPM B4 from 11.00 RPM to 16.00 RPM: Operating % Critical Speed E4 crosses 100% to 106.18% N_c, immediately triggering "CENTRIFUGING LOCKUP" in E5 as grinding media pins continuously to the lifters!
=IF(ISTEXT(E4), E4, LET(nc_r, (30 / PI()) * SQRT((2 * 9.81) / (B2 - B3)), pc_n, (B4 / nc_r) * 100, IF(pc_n >= 100, "CENTRIFUGING LOCKUP", IF(AND(pc_n >= 65, pc_n <= 80), "OPTIMAL CATARACTING", "CASCADING ATTRITION"))))Try the example
Use the original values or change them to test your understanding.