Bond Ball-Mill Work Index Grinding Energy Formula

Size concentrator SAG and ball mills accurately—and never feed millimeters into Bond’s micron formula!

Crushing and grinding rock consumes up to 50% of a hard-rock mine’s total electricity! Between Rittinger’s (1867) surface-area law (dE ~ dD / D^2, fine grinding) and Kick’s (1885) volume-compression law (dE ~ dD / D, coarse crushing), Fred C. Bond (1952, Mining Engineering) formulated the industry-standard Third Theory of Comminution: Specific Grinding Work (kWh/short or metric tonne) to reduce 80%-passing feed size F_80 (in MICROMETERS, um!) to 80%-passing product size P_80 (in um!) for ore of Bond Work Index W_i (kWh/t to grind from infinite size to 100 um) is W = 10 * W_i * (1 / SQRT(P_80) - 1 / SQRT(F_80))! Look out for the #1 metallurgical spreadsheet error: entering F_80 and P_80 in millimeters instead of micrometers inflates 1 / SQRT(P_80) by SQRT(1000) = 31.62x!

Verify on Porphyry_Work_Index W_i = 15.0 kWh/t (B2), Mill_Feed_Size F_80 = 2,500 um (2.50 mm, SQRT = 50, B3), Cyclone_Overflow_Product P_80 = 100 um (0.10 mm, SQRT = 10, B4), Mill_Throughput Q_tph = 450 t/h (B5), and Motor_Drivetrain_Efficiency eta_drv = 0.90 (90.0%, B6): Size Reduction Ratio is 2,500 / 100 = 25.00x (E2); Bond Specific Grinding Work is 10 * 15.0 * (1/10 - 1/50) = 150 * 0.08 = 12.00 kWh/t (E3); Required Mill Pinion Shaft Power is 450 * 12.00 = 5,400.00 kW (E4); and Drawn Electrical Motor Power is 5,400 / 0.90 = 6,000.00 kW (6.00 MW in E5)! Test ultrafine liberation regrinding to P_80 = 25 um (SQRT = 5): Specific Energy surges 2.25x to 27.00 kWh/t (13,500 kW motor draw)!

Bond Specific Grinding Work & Drawn Motor Power Formula

Enforce W_i > 0, F_80 > P_80 > 0 (feed must be coarser than ground product!), Q_tph > 0, and 0 < eta_drv <= 1!

=LET(w_ix, B2, f_80, B3, p_80, B4, q_tp, B5, et_d, B6, IF(OR(w_ix <= 0, p_80 <= 0, f_80 <= p_80, q_tp <= 0, et_d <= 0, et_d > 1), "Invalid Comminution Circuit", LET(w_sp, 10 * w_ix * (1 / SQRT(p_80) - 1 / SQRT(f_80)), ROUND((q_tp * w_sp) / et_d, 2))))

Specific Energy Check at Infinite-Feed Reference Size

Notice why the factor 10 sits out front: as F_80 -> infinity and P_80 = 100 um, 10 * W_i * (1 / SQRT(100) - 0) = 10 * W_i * 0.10 = W_i exact by definition of Bond’s 100-micron standard!

=IF(ISTEXT(E5), E5, ROUND(10 * B2 * (1 / SQRT(B4) - 1 / SQRT(B3)), 4))

Further reading

• Arte Sheets Lesson — Open-Pit Cutoff Grade & Stripping Ratio Formula
• Arte Sheets Lesson — Mining Net Smelter Return (NSR) & TC/RC Formula